application of problems solving

• Bipolar Disorder
• Therapy Center
• When To See a Therapist
• Types of Therapy
• Best Online Therapy
• Best Couples Therapy
• Best Family Therapy
• Managing Stress
• Sleep and Dreaming
• Understanding Emotions
• Self-Improvement
• Healthy Relationships
• Student Resources
• Personality Types
• Guided Meditations
• Verywell Mind Insights
• 2023 Verywell Mind 25
• Mental Health in the Classroom
• Editorial Process
• Meet Our Review Board
• Crisis Support

Problem-Solving Strategies and Obstacles

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Sean is a fact-checker and researcher with experience in sociology, field research, and data analytics.

JGI / Jamie Grill / Getty Images

• Application
• Improvement

From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.

What Is Problem-Solving?

In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.

A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.

Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.

The problem-solving process involves:

• Discovery of the problem
• Deciding to tackle the issue
• Seeking to understand the problem more fully
• Researching available options or solutions
• Taking action to resolve the issue

Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.

Problem-Solving Mental Processes

Several mental processes are at work during problem-solving. Among them are:

• Perceptually recognizing the problem
• Representing the problem in memory
• Considering relevant information that applies to the problem
• Identifying different aspects of the problem
• Labeling and describing the problem

Problem-Solving Strategies

There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.

An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.

In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.

One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.

There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.

Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.

If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.

While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.

Trial and Error

A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.

This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.

In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.

Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .

Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.

How to Apply Problem-Solving Strategies in Real Life

If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:

• Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
• Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
• Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
• Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.

Obstacles to Problem-Solving

Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:

• Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
• Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
• Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
• Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.

How to Improve Your Problem-Solving Skills

In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:

• Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
• Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
• Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
• Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
• Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
• Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.

You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.

Sarathy V. Real world problem-solving .  Front Hum Neurosci . 2018;12:261. doi:10.3389/fnhum.2018.00261

Dunbar K. Problem solving . A Companion to Cognitive Science . 2017. doi:10.1002/9781405164535.ch20

Stewart SL, Celebre A, Hirdes JP, Poss JW. Risk of suicide and self-harm in kids: The development of an algorithm to identify high-risk individuals within the children's mental health system . Child Psychiat Human Develop . 2020;51:913-924. doi:10.1007/s10578-020-00968-9

Rosenbusch H, Soldner F, Evans AM, Zeelenberg M. Supervised machine learning methods in psychology: A practical introduction with annotated R code . Soc Personal Psychol Compass . 2021;15(2):e12579. doi:10.1111/spc3.12579

Mishra S. Decision-making under risk: Integrating perspectives from biology, economics, and psychology . Personal Soc Psychol Rev . 2014;18(3):280-307. doi:10.1177/1088868314530517

Csikszentmihalyi M, Sawyer K. Creative insight: The social dimension of a solitary moment . In: The Systems Model of Creativity . 2015:73-98. doi:10.1007/978-94-017-9085-7_7

Chrysikou EG, Motyka K, Nigro C, Yang SI, Thompson-Schill SL. Functional fixedness in creative thinking tasks depends on stimulus modality .  Psychol Aesthet Creat Arts . 2016;10(4):425‐435. doi:10.1037/aca0000050

Huang F, Tang S, Hu Z. Unconditional perseveration of the short-term mental set in chunk decomposition .  Front Psychol . 2018;9:2568. doi:10.3389/fpsyg.2018.02568

National Alliance on Mental Illness. Warning signs and symptoms .

Mayer RE. Thinking, problem solving, cognition, 2nd ed .

Schooler JW, Ohlsson S, Brooks K. Thoughts beyond words: When language overshadows insight. J Experiment Psychol: General . 1993;122:166-183. doi:10.1037/0096-3445.2.166

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

What is Problem Solving? (Steps, Techniques, Examples)

By Status.net Editorial Team on May 7, 2023 — 5 minutes to read

What Is Problem Solving?

Definition and importance.

Problem solving is the process of finding solutions to obstacles or challenges you encounter in your life or work. It is a crucial skill that allows you to tackle complex situations, adapt to changes, and overcome difficulties with ease. Mastering this ability will contribute to both your personal and professional growth, leading to more successful outcomes and better decision-making.

Problem-Solving Steps

The problem-solving process typically includes the following steps:

• Identify the issue : Recognize the problem that needs to be solved.
• Analyze the situation : Examine the issue in depth, gather all relevant information, and consider any limitations or constraints that may be present.
• Generate potential solutions : Brainstorm a list of possible solutions to the issue, without immediately judging or evaluating them.
• Evaluate options : Weigh the pros and cons of each potential solution, considering factors such as feasibility, effectiveness, and potential risks.
• Select the best solution : Choose the option that best addresses the problem and aligns with your objectives.
• Implement the solution : Put the selected solution into action and monitor the results to ensure it resolves the issue.
• Review and learn : Reflect on the problem-solving process, identify any improvements or adjustments that can be made, and apply these learnings to future situations.

Defining the Problem

To start tackling a problem, first, identify and understand it. Analyzing the issue thoroughly helps to clarify its scope and nature. Ask questions to gather information and consider the problem from various angles. Some strategies to define the problem include:

• Brainstorming with others
• Asking the 5 Ws and 1 H (Who, What, When, Where, Why, and How)
• Analyzing cause and effect
• Creating a problem statement

Generating Solutions

Once the problem is clearly understood, brainstorm possible solutions. Think creatively and keep an open mind, as well as considering lessons from past experiences. Consider:

• Creating a list of potential ideas to solve the problem
• Grouping and categorizing similar solutions
• Prioritizing potential solutions based on feasibility, cost, and resources required
• Involving others to share diverse opinions and inputs

Evaluating and Selecting Solutions

Evaluate each potential solution, weighing its pros and cons. To facilitate decision-making, use techniques such as:

• SWOT analysis (Strengths, Weaknesses, Opportunities, Threats)
• Decision-making matrices
• Pros and cons lists
• Risk assessments

After evaluating, choose the most suitable solution based on effectiveness, cost, and time constraints.

Implementing and Monitoring the Solution

Implement the chosen solution and monitor its progress. Key actions include:

• Communicating the solution to relevant parties
• Setting timelines and milestones
• Assigning tasks and responsibilities
• Monitoring the solution and making adjustments as necessary
• Evaluating the effectiveness of the solution after implementation

Utilize feedback from stakeholders and consider potential improvements. Remember that problem-solving is an ongoing process that can always be refined and enhanced.

Problem-Solving Techniques

During each step, you may find it helpful to utilize various problem-solving techniques, such as:

• Brainstorming : A free-flowing, open-minded session where ideas are generated and listed without judgment, to encourage creativity and innovative thinking.
• Root cause analysis : A method that explores the underlying causes of a problem to find the most effective solution rather than addressing superficial symptoms.
• SWOT analysis : A tool used to evaluate the strengths, weaknesses, opportunities, and threats related to a problem or decision, providing a comprehensive view of the situation.
• Mind mapping : A visual technique that uses diagrams to organize and connect ideas, helping to identify patterns, relationships, and possible solutions.

Brainstorming

When facing a problem, start by conducting a brainstorming session. Gather your team and encourage an open discussion where everyone contributes ideas, no matter how outlandish they may seem. This helps you:

• Generate a diverse range of solutions
• Encourage all team members to participate
• Foster creative thinking

When brainstorming, remember to:

• Reserve judgment until the session is over
• Encourage wild ideas
• Combine and improve upon ideas

Root Cause Analysis

For effective problem-solving, identifying the root cause of the issue at hand is crucial. Try these methods:

• 5 Whys : Ask “why” five times to get to the underlying cause.
• Fishbone Diagram : Create a diagram representing the problem and break it down into categories of potential causes.
• Pareto Analysis : Determine the few most significant causes underlying the majority of problems.

SWOT Analysis

SWOT analysis helps you examine the Strengths, Weaknesses, Opportunities, and Threats related to your problem. To perform a SWOT analysis:

• List your problem’s strengths, such as relevant resources or strong partnerships.
• Identify its weaknesses, such as knowledge gaps or limited resources.
• Explore opportunities, like trends or new technologies, that could help solve the problem.
• Recognize potential threats, like competition or regulatory barriers.

SWOT analysis aids in understanding the internal and external factors affecting the problem, which can help guide your solution.

Mind Mapping

A mind map is a visual representation of your problem and potential solutions. It enables you to organize information in a structured and intuitive manner. To create a mind map:

• Write the problem in the center of a blank page.
• Draw branches from the central problem to related sub-problems or contributing factors.
• Add more branches to represent potential solutions or further ideas.

Mind mapping allows you to visually see connections between ideas and promotes creativity in problem-solving.

Examples of Problem Solving in Various Contexts

In the business world, you might encounter problems related to finances, operations, or communication. Applying problem-solving skills in these situations could look like:

• Identifying areas of improvement in your company’s financial performance and implementing cost-saving measures
• Resolving internal conflicts among team members by listening and understanding different perspectives, then proposing and negotiating solutions
• Streamlining a process for better productivity by removing redundancies, automating tasks, or re-allocating resources

In educational contexts, problem-solving can be seen in various aspects, such as:

• Addressing a gap in students’ understanding by employing diverse teaching methods to cater to different learning styles
• Developing a strategy for successful time management to balance academic responsibilities and extracurricular activities
• Seeking resources and support to provide equal opportunities for learners with special needs or disabilities

Everyday life is full of challenges that require problem-solving skills. Some examples include:

• Overcoming a personal obstacle, such as improving your fitness level, by establishing achievable goals, measuring progress, and adjusting your approach accordingly
• Navigating a new environment or city by researching your surroundings, asking for directions, or using technology like GPS to guide you
• Dealing with a sudden change, like a change in your work schedule, by assessing the situation, identifying potential impacts, and adapting your plans to accommodate the change.
• How to Resolve Employee Conflict at Work [Steps, Tips, Examples]
• How to Write Inspiring Core Values? 5 Steps with Examples
• 30 Employee Feedback Examples (Positive & Negative)

10.4 Solve Applications Modeled by Quadratic Equations

Learning objectives.

By the end of this section, you will be able to:

• Solve applications modeled by Quadratic Equations

Be Prepared 10.4

Before you get started, take this readiness quiz.

• The sum of two consecutive odd numbers is −100 −100 . Find the numbers. If you missed this problem, review Example 3.10 .
• The area of triangular mural is 64 square feet. The base is 16 feet. Find the height. If you missed this problem, review Example 3.36 .
• Find the length of the hypotenuse of a right triangle with legs 5 inches and 12 inches. If you missed this problem, review Example 3.39 .

Solve Applications of the Quadratic Formula

We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications. To get us started, we will copy our usual Problem Solving Strategy here so we can follow the steps.

Use the problem solving strategy.

• Step 1. Read the problem. Make sure all the words and ideas are understood.
• Step 2. Identify what we are looking for.
• Step 3. Name what we are looking for. Choose a variable to represent that quantity.
• Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
• Step 5. Solve the equation using good algebra techniques.
• Step 6. Check the answer in the problem and make sure it makes sense.
• Step 7. Answer the question with a complete sentence.

We have solved number applications that involved consecutive even integers and consecutive odd integers by modeling the situation with linear equations. Remember, we noticed each even integer is 2 more than the number preceding it. If we call the first one n, then the next one is n + 2 n + 2 . The next one would be n + 2 + 2 n + 2 + 2 or n + 4 n + 4 . This is also true when we use odd integers. One set of even integers and one set of odd integers are shown below.

Some applications of consecutive odd integers or consecutive even integers are modeled by quadratic equations. The notation above will be helpful as you name the variables.

Example 10.38

The product of two consecutive odd integers is 195. Find the integers.

Try It 10.75

The product of two consecutive odd integers is 99. Find the integers.

Try It 10.76

The product of two consecutive odd integers is 168. Find the integers.

We will use the formula for the area of a triangle to solve the next example.

Area of a Triangle

For a triangle with base b b and height h h , the area, A A , is given by the formula A = 1 2 b h A = 1 2 b h .

Recall that, when we solve geometry applications, it is helpful to draw the figure.

Example 10.39

An architect is designing the entryway of a restaurant. She wants to put a triangular window above the doorway. Due to energy restrictions, the window can have an area of 120 square feet and the architect wants the width to be 4 feet more than twice the height. Find the height and width of the window.

Notice that the solutions were integers. That tells us that we could have solved the equation by factoring.

When we wrote the equation in standard form, h 2 + 2 h − 120 = 0 h 2 + 2 h − 120 = 0 , we could have factored it. If we did, we would have solved the equation ( h + 12 ) ( h − 10 ) = 0 ( h + 12 ) ( h − 10 ) = 0 .

Try It 10.77

Find the dimensions of a triangle whose width is four more than six times its height and has an area of 208 square inches.

Try It 10.78

If a triangle that has an area of 110 square feet has a height that is two feet less than twice the width, what are its dimensions?

In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. If we get an irrational number as a solution to an application problem, we will use a calculator to get an approximate value.

The Pythagorean Theorem gives the relation between the legs and hypotenuse of a right triangle. We will use the Pythagorean Theorem to solve the next example.

Pythagorean Theorem

In any right triangle, where a a and b b are the lengths of the legs and c c is the length of the hypotenuse, a 2 + b 2 = c 2 a 2 + b 2 = c 2

Example 10.40

Rene is setting up a holiday light display. He wants to make a ‘tree’ in the shape of two right triangles, as shown below, and has two 10-foot strings of lights to use for the sides. He will attach the lights to the top of a pole and to two stakes on the ground. He wants the height of the pole to be the same as the distance from the base of the pole to each stake. How tall should the pole be?

Try It 10.79

The sun casts a shadow from a flag pole. The height of the flag pole is three times the length of its shadow. The distance between the end of the shadow and the top of the flag pole is 20 feet. Find the length of the shadow and the length of the flag pole. Round to the nearest tenth of a foot.

Try It 10.80

The distance between opposite corners of a rectangular field is four more than the width of the field. The length of the field is twice its width. Find the distance between the opposite corners. Round to the nearest tenth.

Example 10.41

Mike wants to put 150 square feet of artificial turf in his front yard. This is the maximum area of artificial turf allowed by his homeowners association. He wants to have a rectangular area of turf with length one foot less than three times the width. Find the length and width. Round to the nearest tenth of a foot.

Try It 10.81

The length of a 200 square foot rectangular vegetable garden is four feet less than twice the width. Find the length and width of the garden. Round to the nearest tenth of a foot.

Try It 10.82

A rectangular tablecloth has an area of 80 square feet. The width is 5 feet shorter than the length. What are the length and width of the tablecloth? Round to the nearest tenth of a foot.

The height of a projectile shot upwards is modeled by a quadratic equation. The initial velocity, v 0 v 0 , propels the object up until gravity causes the object to fall back down.

Projectile Motion

The height in feet, h h , of an object shot upwards into the air with initial velocity, v 0 v 0 , after t t seconds is given by the formula:

We can use the formula for projectile motion to find how many seconds it will take for a firework to reach a specific height.

Example 10.42

A firework is shot upwards with initial velocity 130 feet per second. How many seconds will it take to reach a height of 260 feet? Round to the nearest tenth of a second.

Try It 10.83

An arrow is shot from the ground into the air at an initial speed of 108 ft/sec. Use the formula h = −16 t 2 + v 0 t h = −16 t 2 + v 0 t to determine when the arrow will be 180 feet from the ground. Round the nearest tenth of a second.

Try It 10.84

A man throws a ball into the air with a velocity of 96 ft/sec. Use the formula h = −16 t 2 + v 0 t h = −16 t 2 + v 0 t to determine when the height of the ball will be 48 feet. Round to the nearest tenth of a second.

Access these online resources for additional instruction and practice with solving word problems using the quadratic equation:

• General Quadratic Word Problems
• Word problem: Solve a projectile problem using a quadratic equation

Section 10.4 Exercises

Practice makes perfect.

In the following exercises, solve by using methods of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth.

The product of two consecutive odd numbers is 255. Find the numbers.

The product of two consecutive even numbers is 360. Find the numbers.

The product of two consecutive even numbers is 624. Find the numbers.

The product of two consecutive odd numbers is 1023. Find the numbers.

The product of two consecutive odd numbers is 483. Find the numbers.

The product of two consecutive even numbers is 528. Find the numbers.

A triangle with area 45 square inches has a height that is two less than four times the width. Find the height and width of the triangle.

The width of a triangle is six more than twice the height. The area of the triangle is 88 square yards. Find the height and width of the triangle.

The hypotenuse of a right triangle is twice the length of one of its legs. The length of the other leg is three feet. Find the lengths of the three sides of the triangle.

The hypotenuse of a right triangle is 10 cm long. One of the triangle’s legs is three times the length of the other leg. Round to the nearest tenth. Find the lengths of the three sides of the triangle.

A farmer plans to fence off sections of a rectangular corral. The diagonal distance from one corner of the corral to the opposite corner is five yards longer than the width of the corral. The length of the corral is three times the width. Find the length of the diagonal of the corral.

Nautical flags are used to represent letters of the alphabet. The flag for the letter O consists of a yellow right triangle and a red right triangle which are sewn together along their hypotenuse to form a square. The adjoining side of the two triangles is three inches longer than a side of the flag. Find the length of the side of the flag.

The length of a rectangular driveway is five feet more than three times the width. The area is 350 square feet. Find the length and width of the driveway.

A rectangular lawn has area 140 square yards. Its width that is six less than twice the length. What are the length and width of the lawn?

A firework rocket is shot upward at a rate of 640 ft/sec. Use the projectile formula h = −16 t 2 + v 0 t h = −16 t 2 + v 0 t to determine when the height of the firework rocket will be 1200 feet.

An arrow is shot vertically upward at a rate of 220 feet per second. Use the projectile formula h = −16 t 2 + v 0 t h = −16 t 2 + v 0 t to determine when height of the arrow will be 400 feet.

Everyday Math

A bullet is fired straight up from a BB gun with initial velocity 1120 feet per second at an initial height of 8 feet. Use the formula h = −16 t 2 + v 0 t + 8 h = −16 t 2 + v 0 t + 8 to determine how many seconds it will take for the bullet to hit the ground. (That is, when will h = 0 h = 0 ?)

A city planner wants to build a bridge across a lake in a park. To find the length of the bridge, he makes a right triangle with one leg and the hypotenuse on land and the bridge as the other leg. The length of the hypotenuse is 340 feet and the leg is 160 feet. Find the length of the bridge.

Writing Exercises

Make up a problem involving the product of two consecutive odd integers. Start by choosing two consecutive odd integers. ⓐ What are your integers? ⓑ What is the product of your integers? ⓒ Solve the equation n ( n + 2 ) = p n ( n + 2 ) = p , where p p is the product you found in part (b). ⓓ Did you get the numbers you started with?

Make up a problem involving the product of two consecutive even integers. Start by choosing two consecutive even integers. ⓐ What are your integers? ⓑ What is the product of your integers? ⓒ Solve the equation n ( n + 2 ) = p n ( n + 2 ) = p , where p p is the product you found in part (b). ⓓ Did you get the numbers you started with?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/elementary-algebra/pages/1-introduction

• Authors: Lynn Marecek, MaryAnne Anthony-Smith
• Publisher/website: OpenStax
• Book title: Elementary Algebra
• Publication date: Feb 22, 2017
• Location: Houston, Texas
• Book URL: https://openstax.org/books/elementary-algebra/pages/1-introduction
• Section URL: https://openstax.org/books/elementary-algebra/pages/10-4-solve-applications-modeled-by-quadratic-equations

© Feb 9, 2022 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

How it works

Transform your enterprise with the scalable mindsets, skills, & behavior change that drive performance.

Explore how BetterUp connects to your core business systems.

We pair AI with the latest in human-centered coaching to drive powerful, lasting learning and behavior change.

Build leaders that accelerate team performance and engagement.

Unlock performance potential at scale with AI-powered curated growth journeys.

Build resilience, well-being and agility to drive performance across your entire enterprise.

Transform your business, starting with your sales leaders.

Unlock business impact from the top with executive coaching.

Foster a culture of inclusion and belonging.

Accelerate the performance and potential of your agencies and employees.

See how innovative organizations use BetterUp to build a thriving workforce.

Discover how BetterUp measurably impacts key business outcomes for organizations like yours.

A demo is the first step to transforming your business. Meet with us to develop a plan for attaining your goals.

• What is coaching?

Learn how 1:1 coaching works, who its for, and if it's right for you.

Accelerate your personal and professional growth with the expert guidance of a BetterUp Coach.

Types of Coaching

Navigate career transitions, accelerate your professional growth, and achieve your career goals with expert coaching.

Enhance your communication skills for better personal and professional relationships, with tailored coaching that focuses on your needs.

Find balance, resilience, and well-being in all areas of your life with holistic coaching designed to empower you.

Discover your perfect match : Take our 5-minute assessment and let us pair you with one of our top Coaches tailored just for you.

Research, expert insights, and resources to develop courageous leaders within your organization.

Best practices, research, and tools to fuel individual and business growth.

View on-demand BetterUp events and learn about upcoming live discussions.

The latest insights and ideas for building a high-performing workplace.

• BetterUp Briefing

The online magazine that helps you understand tomorrow's workforce trends, today.

Innovative research featured in peer-reviewed journals, press, and more.

Founded in 2022 to deepen the understanding of the intersection of well-being, purpose, and performance

We're on a mission to help everyone live with clarity, purpose, and passion.

Join us and create impactful change.

Read the buzz about BetterUp.

Meet the leadership that's passionate about empowering your workforce.

For Business

For Individuals

10 Problem-solving strategies to turn challenges on their head

Boost your productivity

Maximize your time and productivity with strategies from our expert coaches.

Jump to section

What is an example of problem-solving?

What are the 5 steps to problem-solving, 10 effective problem-solving strategies, what skills do efficient problem solvers have, how to improve your problem-solving skills.

Problems come in all shapes and sizes — from workplace conflict to budget cuts.

Creative problem-solving is one of the most in-demand skills in all roles and industries. It can boost an organization’s human capital and give it a competitive edge. 

Problem-solving strategies are ways of approaching problems that can help you look beyond the obvious answers and find the best solution to your problem . 

Let’s take a look at a five-step problem-solving process and how to combine it with proven problem-solving strategies. This will give you the tools and skills to solve even your most complex problems.

Good problem-solving is an essential part of the decision-making process . To see what a problem-solving process might look like in real life, let’s take a common problem for SaaS brands — decreasing customer churn rates.

To solve this problem, the company must first identify it. In this case, the problem is that the churn rate is too high. 

Next, they need to identify the root causes of the problem. This could be anything from their customer service experience to their email marketing campaigns. If there are several problems, they will need a separate problem-solving process for each one. 

Let’s say the problem is with email marketing — they’re not nurturing existing customers. Now that they’ve identified the problem, they can start using problem-solving strategies to look for solutions. 

This might look like coming up with special offers, discounts, or bonuses for existing customers. They need to find ways to remind them to use their products and services while providing added value. This will encourage customers to keep paying their monthly subscriptions.

They might also want to add incentives, such as access to a premium service at no extra cost after 12 months of membership. They could publish blog posts that help their customers solve common problems and share them as an email newsletter.

The company should set targets and a time frame in which to achieve them. This will allow leaders to measure progress and identify which actions yield the best results.

Perhaps you’ve got a problem you need to tackle. Or maybe you want to be prepared the next time one arises. Either way, it’s a good idea to get familiar with the five steps of problem-solving. 

Use this step-by-step problem-solving method with the strategies in the following section to find possible solutions to your problem.

1. Identify the problem

The first step is to know which problem you need to solve. Then, you need to find the root cause of the problem. 

The best course of action is to gather as much data as possible, speak to the people involved, and separate facts from opinions. 

Once this is done, formulate a statement that describes the problem. Use rational persuasion to make sure your team agrees .

2. Break the problem down 

Identifying the problem allows you to see which steps need to be taken to solve it. 

First, break the problem down into achievable blocks. Then, use strategic planning to set a time frame in which to solve the problem and establish a timeline for the completion of each stage.

3. Generate potential solutions

At this stage, the aim isn’t to evaluate possible solutions but to generate as many ideas as possible. 

Encourage your team to use creative thinking and be patient — the best solution may not be the first or most obvious one.

Use one or more of the different strategies in the following section to help come up with solutions — the more creative, the better.

4. Evaluate the possible solutions

Once you’ve generated potential solutions, narrow them down to a shortlist. Then, evaluate the options on your shortlist. 

There are usually many factors to consider. So when evaluating a solution, ask yourself the following questions:

• Will my team be on board with the proposition?
• Does the solution align with organizational goals ?
• Is the solution likely to achieve the desired outcomes?
• Is the solution realistic and possible with current resources and constraints?
• Will the solution solve the problem without causing additional unintended problems?

5. Implement and monitor the solutions

Once you’ve identified your solution and got buy-in from your team, it’s time to implement it. 

But the work doesn’t stop there. You need to monitor your solution to see whether it actually solves your problem. 

Request regular feedback from the team members involved and have a monitoring and evaluation plan in place to measure progress.

If the solution doesn’t achieve your desired results, start this step-by-step process again.

There are many different ways to approach problem-solving. Each is suitable for different types of problems. 

The most appropriate problem-solving techniques will depend on your specific problem. You may need to experiment with several strategies before you find a workable solution.

Here are 10 effective problem-solving strategies for you to try:

• Use a solution that worked before
• Brainstorming
• Work backward
• Use the Kipling method
• Draw the problem
• Use trial and error
• Sleep on it
• Get advice from your peers
• Use the Pareto principle
• Add successful solutions to your toolkit

Let’s break each of these down.

1. Use a solution that worked before

It might seem obvious, but if you’ve faced similar problems in the past, look back to what worked then. See if any of the solutions could apply to your current situation and, if so, replicate them.

2. Brainstorming

The more people you enlist to help solve the problem, the more potential solutions you can come up with.

Use different brainstorming techniques to workshop potential solutions with your team. They’ll likely bring something you haven’t thought of to the table.

3. Work backward

Working backward is a way to reverse engineer your problem. Imagine your problem has been solved, and make that the starting point.

Then, retrace your steps back to where you are now. This can help you see which course of action may be most effective.

4. Use the Kipling method

This is a method that poses six questions based on Rudyard Kipling’s poem, “ I Keep Six Honest Serving Men .” 

• What is the problem?
• Why is the problem important?
• When did the problem arise, and when does it need to be solved?
• How did the problem happen?
• Where is the problem occurring?
• Who does the problem affect?

Answering these questions can help you identify possible solutions.

5. Draw the problem

Sometimes it can be difficult to visualize all the components and moving parts of a problem and its solution. Drawing a diagram can help.

This technique is particularly helpful for solving process-related problems. For example, a product development team might want to decrease the time they take to fix bugs and create new iterations. Drawing the processes involved can help you see where improvements can be made.

6. Use trial-and-error

A trial-and-error approach can be useful when you have several possible solutions and want to test them to see which one works best.

7. Sleep on it

Finding the best solution to a problem is a process. Remember to take breaks and get enough rest . Sometimes, a walk around the block can bring inspiration, but you should sleep on it if possible.

A good night’s sleep helps us find creative solutions to problems. This is because when you sleep, your brain sorts through the day’s events and stores them as memories. This enables you to process your ideas at a subconscious level. 

If possible, give yourself a few days to develop and analyze possible solutions. You may find you have greater clarity after sleeping on it. Your mind will also be fresh, so you’ll be able to make better decisions.

8. Get advice from your peers

Getting input from a group of people can help you find solutions you may not have thought of on your own. 

For solo entrepreneurs or freelancers, this might look like hiring a coach or mentor or joining a mastermind group. 

For leaders , it might be consulting other members of the leadership team or working with a business coach .

It’s important to recognize you might not have all the skills, experience, or knowledge necessary to find a solution alone. 

9. Use the Pareto principle

The Pareto principle — also known as the 80/20 rule — can help you identify possible root causes and potential solutions for your problems.

Although it’s not a mathematical law, it’s a principle found throughout many aspects of business and life. For example, 20% of the sales reps in a company might close 80% of the sales. 

You may be able to narrow down the causes of your problem by applying the Pareto principle. This can also help you identify the most appropriate solutions.

10. Add successful solutions to your toolkit

Every situation is different, and the same solutions might not always work. But by keeping a record of successful problem-solving strategies, you can build up a solutions toolkit. 

These solutions may be applicable to future problems. Even if not, they may save you some of the time and work needed to come up with a new solution.

Improving problem-solving skills is essential for professional development — both yours and your team’s. Here are some of the key skills of effective problem solvers:

• Critical thinking and analytical skills
• Communication skills , including active listening
• Decision-making
• Planning and prioritization
• Emotional intelligence , including empathy and emotional regulation
• Time management
• Data analysis
• Research skills
• Project management

And they see problems as opportunities. Everyone is born with problem-solving skills. But accessing these abilities depends on how we view problems. Effective problem-solvers see problems as opportunities to learn and improve.

Ready to work on your problem-solving abilities? Get started with these seven tips.

1. Build your problem-solving skills

One of the best ways to improve your problem-solving skills is to learn from experts. Consider enrolling in organizational training , shadowing a mentor , or working with a coach .

2. Practice

Practice using your new problem-solving skills by applying them to smaller problems you might encounter in your daily life. 

Alternatively, imagine problematic scenarios that might arise at work and use problem-solving strategies to find hypothetical solutions.

3. Don’t try to find a solution right away

Often, the first solution you think of to solve a problem isn’t the most appropriate or effective.

Instead of thinking on the spot, give yourself time and use one or more of the problem-solving strategies above to activate your creative thinking. 

4. Ask for feedback

Receiving feedback is always important for learning and growth. Your perception of your problem-solving skills may be different from that of your colleagues. They can provide insights that help you improve. 

5. Learn new approaches and methodologies

There are entire books written about problem-solving methodologies if you want to take a deep dive into the subject. 

We recommend starting with “ Fixed — How to Perfect the Fine Art of Problem Solving ” by Amy E. Herman. 

6. Experiment

Tried-and-tested problem-solving techniques can be useful. However, they don’t teach you how to innovate and develop your own problem-solving approaches. 

Sometimes, an unconventional approach can lead to the development of a brilliant new idea or strategy. So don’t be afraid to suggest your most “out there” ideas.

7. Analyze the success of your competitors

Do you have competitors who have already solved the problem you’re facing? Look at what they did, and work backward to solve your own problem. 

For example, Netflix started in the 1990s as a DVD mail-rental company. Its main competitor at the time was Blockbuster. 

But when streaming became the norm in the early 2000s, both companies faced a crisis. Netflix innovated, unveiling its streaming service in 2007. 

If Blockbuster had followed Netflix’s example, it might have survived. Instead, it declared bankruptcy in 2010.

Use problem-solving strategies to uplevel your business

When facing a problem, it’s worth taking the time to find the right solution. 

Otherwise, we risk either running away from our problems or headlong into solutions. When we do this, we might miss out on other, better options.

Use the problem-solving strategies outlined above to find innovative solutions to your business’ most perplexing problems.

If you’re ready to take problem-solving to the next level, request a demo with BetterUp . Our expert coaches specialize in helping teams develop and implement strategies that work.

Elizabeth Perry, ACC

Elizabeth Perry is a Coach Community Manager at BetterUp. She uses strategic engagement strategies to cultivate a learning community across a global network of Coaches through in-person and virtual experiences, technology-enabled platforms, and strategic coaching industry partnerships. With over 3 years of coaching experience and a certification in transformative leadership and life coaching from Sofia University, Elizabeth leverages transpersonal psychology expertise to help coaches and clients gain awareness of their behavioral and thought patterns, discover their purpose and passions, and elevate their potential. She is a lifelong student of psychology, personal growth, and human potential as well as an ICF-certified ACC transpersonal life and leadership Coach.

8 creative solutions to your most challenging problems

5 problem-solving questions to prepare you for your next interview, what are metacognitive skills examples in everyday life, 31 examples of problem solving performance review phrases, what is lateral thinking 7 techniques to encourage creative ideas, learn what process mapping is and how to create one (+ examples), leadership activities that encourage employee engagement, how much do distractions cost 8 effects of lack of focus, can dreams help you solve problems 6 ways to try, similar articles, the pareto principle: how the 80/20 rule can help you do more with less, thinking outside the box: 8 ways to become a creative problem solver, experimentation brings innovation: create an experimental workplace, effective problem statements have these 5 components, contingency planning: 4 steps to prepare for the unexpected, stay connected with betterup, get our newsletter, event invites, plus product insights and research..

3100 E 5th Street, Suite 350 Austin, TX 78702

• Platform Overview
• Integrations
• Powered by AI
• BetterUp Lead
• BetterUp Manage™
• BetterUp Care™
• Sales Performance
• Diversity & Inclusion
• Case Studies
• Why BetterUp?
• About Coaching
• Find your Coach
• Career Coaching
• Communication Coaching
• Life Coaching
• News and Press
• Leadership Team
• Become a BetterUp Coach
• BetterUp Labs
• Center for Purpose & Performance
• Leadership Training
• Business Coaching
• Contact Support
• Contact Sales
• Privacy Policy
• Acceptable Use Policy
• Trust & Security
• Cookie Preferences

35 problem-solving techniques and methods for solving complex problems

Design your next session with SessionLab

Join the 150,000+ facilitators 
using SessionLab.

Recommended Articles

A step-by-step guide to planning a workshop, how to create an unforgettable training session in 8 simple steps, 47 useful online tools for workshop planning and meeting facilitation.

All teams and organizations encounter challenges as they grow. There are problems that might occur for teams when it comes to miscommunication or resolving business-critical issues . You may face challenges around growth , design , user engagement, and even team culture and happiness. In short, problem-solving techniques should be part of every team’s skillset.

Problem-solving methods are primarily designed to help a group or team through a process of first identifying problems and challenges , ideating possible solutions , and then evaluating the most suitable .

Finding effective solutions to complex problems isn’t easy, but by using the right process and techniques, you can help your team be more efficient in the process.

So how do you develop strategies that are engaging, and empower your team to solve problems effectively?

In this blog post, we share a series of problem-solving tools you can use in your next workshop or team meeting. You’ll also find some tips for facilitating the process and how to enable others to solve complex problems.

Let’s get started! 

How do you identify problems?

How do you identify the right solution.

• Tips for more effective problem-solving

Complete problem-solving methods

• Problem-solving techniques to identify and analyze problems
• Problem-solving techniques for developing solutions

Problem-solving warm-up activities

Closing activities for a problem-solving process.

Before you can move towards finding the right solution for a given problem, you first need to identify and define the problem you wish to solve. 

Here, you want to clearly articulate what the problem is and allow your group to do the same. Remember that everyone in a group is likely to have differing perspectives and alignment is necessary in order to help the group move forward. 

Identifying a problem accurately also requires that all members of a group are able to contribute their views in an open and safe manner. It can be scary for people to stand up and contribute, especially if the problems or challenges are emotive or personal in nature. Be sure to try and create a psychologically safe space for these kinds of discussions.

Remember that problem analysis and further discussion are also important. Not taking the time to fully analyze and discuss a challenge can result in the development of solutions that are not fit for purpose or do not address the underlying issue.

Successfully identifying and then analyzing a problem means facilitating a group through activities designed to help them clearly and honestly articulate their thoughts and produce usable insight.

With this data, you might then produce a problem statement that clearly describes the problem you wish to be addressed and also state the goal of any process you undertake to tackle this issue.  

Finding solutions is the end goal of any process. Complex organizational challenges can only be solved with an appropriate solution but discovering them requires using the right problem-solving tool.

After you’ve explored a problem and discussed ideas, you need to help a team discuss and choose the right solution. Consensus tools and methods such as those below help a group explore possible solutions before then voting for the best. They’re a great way to tap into the collective intelligence of the group for great results!

Remember that the process is often iterative. Great problem solvers often roadtest a viable solution in a measured way to see what works too. While you might not get the right solution on your first try, the methods below help teams land on the most likely to succeed solution while also holding space for improvement.

Every effective problem solving process begins with an agenda . A well-structured workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.

In SessionLab, it’s easy to go from an idea to a complete agenda . Start by dragging and dropping your core problem solving activities into place . Add timings, breaks and necessary materials before sharing your agenda with your colleagues.

The resulting agenda will be your guide to an effective and productive problem solving session that will also help you stay organized on the day!

Tips for more effective problem solving

Problem-solving activities are only one part of the puzzle. While a great method can help unlock your team’s ability to solve problems, without a thoughtful approach and strong facilitation the solutions may not be fit for purpose.

Let’s take a look at some problem-solving tips you can apply to any process to help it be a success!

Clearly define the problem

Jumping straight to solutions can be tempting, though without first clearly articulating a problem, the solution might not be the right one. Many of the problem-solving activities below include sections where the problem is explored and clearly defined before moving on.

This is a vital part of the problem-solving process and taking the time to fully define an issue can save time and effort later. A clear definition helps identify irrelevant information and it also ensures that your team sets off on the right track.

Don’t jump to conclusions

It’s easy for groups to exhibit cognitive bias or have preconceived ideas about both problems and potential solutions. Be sure to back up any problem statements or potential solutions with facts, research, and adequate forethought.

The best techniques ask participants to be methodical and challenge preconceived notions. Make sure you give the group enough time and space to collect relevant information and consider the problem in a new way. By approaching the process with a clear, rational mindset, you’ll often find that better solutions are more forthcoming.  

Try different approaches  

Problems come in all shapes and sizes and so too should the methods you use to solve them. If you find that one approach isn’t yielding results and your team isn’t finding different solutions, try mixing it up. You’ll be surprised at how using a new creative activity can unblock your team and generate great solutions.

Don’t take it personally 

Depending on the nature of your team or organizational problems, it’s easy for conversations to get heated. While it’s good for participants to be engaged in the discussions, ensure that emotions don’t run too high and that blame isn’t thrown around while finding solutions.

You’re all in it together, and even if your team or area is seeing problems, that isn’t necessarily a disparagement of you personally. Using facilitation skills to manage group dynamics is one effective method of helping conversations be more constructive.

Get the right people in the room

Your problem-solving method is often only as effective as the group using it. Getting the right people on the job and managing the number of people present is important too!

If the group is too small, you may not get enough different perspectives to effectively solve a problem. If the group is too large, you can go round and round during the ideation stages.

Creating the right group makeup is also important in ensuring you have the necessary expertise and skillset to both identify and follow up on potential solutions. Carefully consider who to include at each stage to help ensure your problem-solving method is followed and positioned for success.

Document everything

The best solutions can take refinement, iteration, and reflection to come out. Get into a habit of documenting your process in order to keep all the learnings from the session and to allow ideas to mature and develop. Many of the methods below involve the creation of documents or shared resources. Be sure to keep and share these so everyone can benefit from the work done!

Bring a facilitator 

Facilitation is all about making group processes easier. With a subject as potentially emotive and important as problem-solving, having an impartial third party in the form of a facilitator can make all the difference in finding great solutions and keeping the process moving. Consider bringing a facilitator to your problem-solving session to get better results and generate meaningful solutions!

Develop your problem-solving skills

It takes time and practice to be an effective problem solver. While some roles or participants might more naturally gravitate towards problem-solving, it can take development and planning to help everyone create better solutions.

You might develop a training program, run a problem-solving workshop or simply ask your team to practice using the techniques below. Check out our post on problem-solving skills to see how you and your group can develop the right mental process and be more resilient to issues too!

Design a great agenda

Workshops are a great format for solving problems. With the right approach, you can focus a group and help them find the solutions to their own problems. But designing a process can be time-consuming and finding the right activities can be difficult.

Check out our workshop planning guide to level-up your agenda design and start running more effective workshops. Need inspiration? Check out templates designed by expert facilitators to help you kickstart your process!

In this section, we’ll look at in-depth problem-solving methods that provide a complete end-to-end process for developing effective solutions. These will help guide your team from the discovery and definition of a problem through to delivering the right solution.

If you’re looking for an all-encompassing method or problem-solving model, these processes are a great place to start. They’ll ask your team to challenge preconceived ideas and adopt a mindset for solving problems more effectively.

• Six Thinking Hats
• Lightning Decision Jam
• Problem Definition Process
• Discovery & Action Dialogue

Design Sprint 2.0

• Open Space Technology

1. Six Thinking Hats

Individual approaches to solving a problem can be very different based on what team or role an individual holds. It can be easy for existing biases or perspectives to find their way into the mix, or for internal politics to direct a conversation.

Six Thinking Hats is a classic method for identifying the problems that need to be solved and enables your team to consider them from different angles, whether that is by focusing on facts and data, creative solutions, or by considering why a particular solution might not work.

Like all problem-solving frameworks, Six Thinking Hats is effective at helping teams remove roadblocks from a conversation or discussion and come to terms with all the aspects necessary to solve complex problems.

2. Lightning Decision Jam

Featured courtesy of Jonathan Courtney of AJ&Smart Berlin, Lightning Decision Jam is one of those strategies that should be in every facilitation toolbox. Exploring problems and finding solutions is often creative in nature, though as with any creative process, there is the potential to lose focus and get lost.

Unstructured discussions might get you there in the end, but it’s much more effective to use a method that creates a clear process and team focus.

In Lightning Decision Jam, participants are invited to begin by writing challenges, concerns, or mistakes on post-its without discussing them before then being invited by the moderator to present them to the group.

From there, the team vote on which problems to solve and are guided through steps that will allow them to reframe those problems, create solutions and then decide what to execute on. 

By deciding the problems that need to be solved as a team before moving on, this group process is great for ensuring the whole team is aligned and can take ownership over the next stages. 

Lightning Decision Jam (LDJ)   #action   #decision making   #problem solving   #issue analysis   #innovation   #design   #remote-friendly   The problem with anything that requires creative thinking is that it’s easy to get lost—lose focus and fall into the trap of having useless, open-ended, unstructured discussions. Here’s the most effective solution I’ve found: Replace all open, unstructured discussion with a clear process. What to use this exercise for: Anything which requires a group of people to make decisions, solve problems or discuss challenges. It’s always good to frame an LDJ session with a broad topic, here are some examples: The conversion flow of our checkout Our internal design process How we organise events Keeping up with our competition Improving sales flow

3. Problem Definition Process

While problems can be complex, the problem-solving methods you use to identify and solve those problems can often be simple in design. 

By taking the time to truly identify and define a problem before asking the group to reframe the challenge as an opportunity, this method is a great way to enable change.

Begin by identifying a focus question and exploring the ways in which it manifests before splitting into five teams who will each consider the problem using a different method: escape, reversal, exaggeration, distortion or wishful. Teams develop a problem objective and create ideas in line with their method before then feeding them back to the group.

This method is great for enabling in-depth discussions while also creating space for finding creative solutions too!

Problem Definition   #problem solving   #idea generation   #creativity   #online   #remote-friendly   A problem solving technique to define a problem, challenge or opportunity and to generate ideas.

4. The 5 Whys 

Sometimes, a group needs to go further with their strategies and analyze the root cause at the heart of organizational issues. An RCA or root cause analysis is the process of identifying what is at the heart of business problems or recurring challenges. 

The 5 Whys is a simple and effective method of helping a group go find the root cause of any problem or challenge and conduct analysis that will deliver results. 

By beginning with the creation of a problem statement and going through five stages to refine it, The 5 Whys provides everything you need to truly discover the cause of an issue.

The 5 Whys   #hyperisland   #innovation   This simple and powerful method is useful for getting to the core of a problem or challenge. As the title suggests, the group defines a problems, then asks the question “why” five times, often using the resulting explanation as a starting point for creative problem solving.

5. World Cafe

World Cafe is a simple but powerful facilitation technique to help bigger groups to focus their energy and attention on solving complex problems.

World Cafe enables this approach by creating a relaxed atmosphere where participants are able to self-organize and explore topics relevant and important to them which are themed around a central problem-solving purpose. Create the right atmosphere by modeling your space after a cafe and after guiding the group through the method, let them take the lead!

Making problem-solving a part of your organization’s culture in the long term can be a difficult undertaking. More approachable formats like World Cafe can be especially effective in bringing people unfamiliar with workshops into the fold. 

World Cafe   #hyperisland   #innovation   #issue analysis   World Café is a simple yet powerful method, originated by Juanita Brown, for enabling meaningful conversations driven completely by participants and the topics that are relevant and important to them. Facilitators create a cafe-style space and provide simple guidelines. Participants then self-organize and explore a set of relevant topics or questions for conversation.

6. Discovery & Action Dialogue (DAD)

One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions.

With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so. It’s great at helping remove resistance to change and can help get buy-in at every level too!

This process of enabling frontline ownership is great in ensuring follow-through and is one of the methods you will want in your toolbox as a facilitator.

Discovery & Action Dialogue (DAD)   #idea generation   #liberating structures   #action   #issue analysis   #remote-friendly   DADs make it easy for a group or community to discover practices and behaviors that enable some individuals (without access to special resources and facing the same constraints) to find better solutions than their peers to common problems. These are called positive deviant (PD) behaviors and practices. DADs make it possible for people in the group, unit, or community to discover by themselves these PD practices. DADs also create favorable conditions for stimulating participants’ creativity in spaces where they can feel safe to invent new and more effective practices. Resistance to change evaporates as participants are unleashed to choose freely which practices they will adopt or try and which problems they will tackle. DADs make it possible to achieve frontline ownership of solutions.

7. Design Sprint 2.0

Want to see how a team can solve big problems and move forward with prototyping and testing solutions in a few days? The Design Sprint 2.0 template from Jake Knapp, author of Sprint, is a complete agenda for a with proven results.

Developing the right agenda can involve difficult but necessary planning. Ensuring all the correct steps are followed can also be stressful or time-consuming depending on your level of experience.

Use this complete 4-day workshop template if you are finding there is no obvious solution to your challenge and want to focus your team around a specific problem that might require a shortcut to launching a minimum viable product or waiting for the organization-wide implementation of a solution.

8. Open space technology

Open space technology- developed by Harrison Owen – creates a space where large groups are invited to take ownership of their problem solving and lead individual sessions. Open space technology is a great format when you have a great deal of expertise and insight in the room and want to allow for different takes and approaches on a particular theme or problem you need to be solved.

Start by bringing your participants together to align around a central theme and focus their efforts. Explain the ground rules to help guide the problem-solving process and then invite members to identify any issue connecting to the central theme that they are interested in and are prepared to take responsibility for.

Once participants have decided on their approach to the core theme, they write their issue on a piece of paper, announce it to the group, pick a session time and place, and post the paper on the wall. As the wall fills up with sessions, the group is then invited to join the sessions that interest them the most and which they can contribute to, then you’re ready to begin!

Everyone joins the problem-solving group they’ve signed up to, record the discussion and if appropriate, findings can then be shared with the rest of the group afterward.

Open Space Technology   #action plan   #idea generation   #problem solving   #issue analysis   #large group   #online   #remote-friendly   Open Space is a methodology for large groups to create their agenda discerning important topics for discussion, suitable for conferences, community gatherings and whole system facilitation

Techniques to identify and analyze problems

Using a problem-solving method to help a team identify and analyze a problem can be a quick and effective addition to any workshop or meeting.

While further actions are always necessary, you can generate momentum and alignment easily, and these activities are a great place to get started.

We’ve put together this list of techniques to help you and your team with problem identification, analysis, and discussion that sets the foundation for developing effective solutions.

Let’s take a look!

• The Creativity Dice
• Fishbone Analysis
• Problem Tree
• SWOT Analysis
• Agreement-Certainty Matrix
• The Journalistic Six
• LEGO Challenge
• What, So What, Now What?
• Journalists

Individual and group perspectives are incredibly important, but what happens if people are set in their minds and need a change of perspective in order to approach a problem more effectively?

Flip It is a method we love because it is both simple to understand and run, and allows groups to understand how their perspectives and biases are formed. 

Participants in Flip It are first invited to consider concerns, issues, or problems from a perspective of fear and write them on a flip chart. Then, the group is asked to consider those same issues from a perspective of hope and flip their understanding.  

No problem and solution is free from existing bias and by changing perspectives with Flip It, you can then develop a problem solving model quickly and effectively.

Flip It!   #gamestorming   #problem solving   #action   Often, a change in a problem or situation comes simply from a change in our perspectives. Flip It! is a quick game designed to show players that perspectives are made, not born.

10. The Creativity Dice

One of the most useful problem solving skills you can teach your team is of approaching challenges with creativity, flexibility, and openness. Games like The Creativity Dice allow teams to overcome the potential hurdle of too much linear thinking and approach the process with a sense of fun and speed. 

In The Creativity Dice, participants are organized around a topic and roll a dice to determine what they will work on for a period of 3 minutes at a time. They might roll a 3 and work on investigating factual information on the chosen topic. They might roll a 1 and work on identifying the specific goals, standards, or criteria for the session.

Encouraging rapid work and iteration while asking participants to be flexible are great skills to cultivate. Having a stage for idea incubation in this game is also important. Moments of pause can help ensure the ideas that are put forward are the most suitable. 

The Creativity Dice   #creativity   #problem solving   #thiagi   #issue analysis   Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

11. Fishbone Analysis

Organizational or team challenges are rarely simple, and it’s important to remember that one problem can be an indication of something that goes deeper and may require further consideration to be solved.

Fishbone Analysis helps groups to dig deeper and understand the origins of a problem. It’s a great example of a root cause analysis method that is simple for everyone on a team to get their head around. 

Participants in this activity are asked to annotate a diagram of a fish, first adding the problem or issue to be worked on at the head of a fish before then brainstorming the root causes of the problem and adding them as bones on the fish. 

Using abstractions such as a diagram of a fish can really help a team break out of their regular thinking and develop a creative approach.

Fishbone Analysis   #problem solving   ##root cause analysis   #decision making   #online facilitation   A process to help identify and understand the origins of problems, issues or observations.

12. Problem Tree 

Encouraging visual thinking can be an essential part of many strategies. By simply reframing and clarifying problems, a group can move towards developing a problem solving model that works for them. 

In Problem Tree, groups are asked to first brainstorm a list of problems – these can be design problems, team problems or larger business problems – and then organize them into a hierarchy. The hierarchy could be from most important to least important or abstract to practical, though the key thing with problem solving games that involve this aspect is that your group has some way of managing and sorting all the issues that are raised.

Once you have a list of problems that need to be solved and have organized them accordingly, you’re then well-positioned for the next problem solving steps.

Problem tree   #define intentions   #create   #design   #issue analysis   A problem tree is a tool to clarify the hierarchy of problems addressed by the team within a design project; it represents high level problems or related sublevel problems.

13. SWOT Analysis

Chances are you’ve heard of the SWOT Analysis before. This problem-solving method focuses on identifying strengths, weaknesses, opportunities, and threats is a tried and tested method for both individuals and teams.

Start by creating a desired end state or outcome and bare this in mind – any process solving model is made more effective by knowing what you are moving towards. Create a quadrant made up of the four categories of a SWOT analysis and ask participants to generate ideas based on each of those quadrants.

Once you have those ideas assembled in their quadrants, cluster them together based on their affinity with other ideas. These clusters are then used to facilitate group conversations and move things forward. 

SWOT analysis   #gamestorming   #problem solving   #action   #meeting facilitation   The SWOT Analysis is a long-standing technique of looking at what we have, with respect to the desired end state, as well as what we could improve on. It gives us an opportunity to gauge approaching opportunities and dangers, and assess the seriousness of the conditions that affect our future. When we understand those conditions, we can influence what comes next.

14. Agreement-Certainty Matrix

Not every problem-solving approach is right for every challenge, and deciding on the right method for the challenge at hand is a key part of being an effective team.

The Agreement Certainty matrix helps teams align on the nature of the challenges facing them. By sorting problems from simple to chaotic, your team can understand what methods are suitable for each problem and what they can do to ensure effective results. 

If you are already using Liberating Structures techniques as part of your problem-solving strategy, the Agreement-Certainty Matrix can be an invaluable addition to your process. We’ve found it particularly if you are having issues with recurring problems in your organization and want to go deeper in understanding the root cause. 

Agreement-Certainty Matrix   #issue analysis   #liberating structures   #problem solving   You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic .  A problem is simple when it can be solved reliably with practices that are easy to duplicate.  It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably.  A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail.  Chaotic is when the context is too turbulent to identify a path forward.  A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.”  The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Organizing and charting a team’s progress can be important in ensuring its success. SQUID (Sequential Question and Insight Diagram) is a great model that allows a team to effectively switch between giving questions and answers and develop the skills they need to stay on track throughout the process. 

Begin with two different colored sticky notes – one for questions and one for answers – and with your central topic (the head of the squid) on the board. Ask the group to first come up with a series of questions connected to their best guess of how to approach the topic. Ask the group to come up with answers to those questions, fix them to the board and connect them with a line. After some discussion, go back to question mode by responding to the generated answers or other points on the board.

It’s rewarding to see a diagram grow throughout the exercise, and a completed SQUID can provide a visual resource for future effort and as an example for other teams.

SQUID   #gamestorming   #project planning   #issue analysis   #problem solving   When exploring an information space, it’s important for a group to know where they are at any given time. By using SQUID, a group charts out the territory as they go and can navigate accordingly. SQUID stands for Sequential Question and Insight Diagram.

16. Speed Boat

To continue with our nautical theme, Speed Boat is a short and sweet activity that can help a team quickly identify what employees, clients or service users might have a problem with and analyze what might be standing in the way of achieving a solution.

Methods that allow for a group to make observations, have insights and obtain those eureka moments quickly are invaluable when trying to solve complex problems.

In Speed Boat, the approach is to first consider what anchors and challenges might be holding an organization (or boat) back. Bonus points if you are able to identify any sharks in the water and develop ideas that can also deal with competitors!   

Speed Boat   #gamestorming   #problem solving   #action   Speedboat is a short and sweet way to identify what your employees or clients don’t like about your product/service or what’s standing in the way of a desired goal.

17. The Journalistic Six

Some of the most effective ways of solving problems is by encouraging teams to be more inclusive and diverse in their thinking.

Based on the six key questions journalism students are taught to answer in articles and news stories, The Journalistic Six helps create teams to see the whole picture. By using who, what, when, where, why, and how to facilitate the conversation and encourage creative thinking, your team can make sure that the problem identification and problem analysis stages of the are covered exhaustively and thoughtfully. Reporter’s notebook and dictaphone optional.

The Journalistic Six – Who What When Where Why How   #idea generation   #issue analysis   #problem solving   #online   #creative thinking   #remote-friendly   A questioning method for generating, explaining, investigating ideas.

18. LEGO Challenge

Now for an activity that is a little out of the (toy) box. LEGO Serious Play is a facilitation methodology that can be used to improve creative thinking and problem-solving skills. 

The LEGO Challenge includes giving each member of the team an assignment that is hidden from the rest of the group while they create a structure without speaking.

What the LEGO challenge brings to the table is a fun working example of working with stakeholders who might not be on the same page to solve problems. Also, it’s LEGO! Who doesn’t love LEGO! 

LEGO Challenge   #hyperisland   #team   A team-building activity in which groups must work together to build a structure out of LEGO, but each individual has a secret “assignment” which makes the collaborative process more challenging. It emphasizes group communication, leadership dynamics, conflict, cooperation, patience and problem solving strategy.

19. What, So What, Now What?

If not carefully managed, the problem identification and problem analysis stages of the problem-solving process can actually create more problems and misunderstandings.

The What, So What, Now What? problem-solving activity is designed to help collect insights and move forward while also eliminating the possibility of disagreement when it comes to identifying, clarifying, and analyzing organizational or work problems. 

Facilitation is all about bringing groups together so that might work on a shared goal and the best problem-solving strategies ensure that teams are aligned in purpose, if not initially in opinion or insight.

Throughout the three steps of this game, you give everyone on a team to reflect on a problem by asking what happened, why it is important, and what actions should then be taken. 

This can be a great activity for bringing our individual perceptions about a problem or challenge and contextualizing it in a larger group setting. This is one of the most important problem-solving skills you can bring to your organization.

W³ – What, So What, Now What?   #issue analysis   #innovation   #liberating structures   You can help groups reflect on a shared experience in a way that builds understanding and spurs coordinated action while avoiding unproductive conflict. It is possible for every voice to be heard while simultaneously sifting for insights and shaping new direction. Progressing in stages makes this practical—from collecting facts about What Happened to making sense of these facts with So What and finally to what actions logically follow with Now What . The shared progression eliminates most of the misunderstandings that otherwise fuel disagreements about what to do. Voila!

20. Journalists  

Problem analysis can be one of the most important and decisive stages of all problem-solving tools. Sometimes, a team can become bogged down in the details and are unable to move forward.

Journalists is an activity that can avoid a group from getting stuck in the problem identification or problem analysis stages of the process.

In Journalists, the group is invited to draft the front page of a fictional newspaper and figure out what stories deserve to be on the cover and what headlines those stories will have. By reframing how your problems and challenges are approached, you can help a team move productively through the process and be better prepared for the steps to follow.

Journalists   #vision   #big picture   #issue analysis   #remote-friendly   This is an exercise to use when the group gets stuck in details and struggles to see the big picture. Also good for defining a vision.

Problem-solving techniques for developing solutions 

The success of any problem-solving process can be measured by the solutions it produces. After you’ve defined the issue, explored existing ideas, and ideated, it’s time to narrow down to the correct solution.

Use these problem-solving techniques when you want to help your team find consensus, compare possible solutions, and move towards taking action on a particular problem.

• Improved Solutions
• Four-Step Sketch
• 15% Solutions
• How-Now-Wow matrix
• Impact Effort Matrix

21. Mindspin  

Brainstorming is part of the bread and butter of the problem-solving process and all problem-solving strategies benefit from getting ideas out and challenging a team to generate solutions quickly. 

With Mindspin, participants are encouraged not only to generate ideas but to do so under time constraints and by slamming down cards and passing them on. By doing multiple rounds, your team can begin with a free generation of possible solutions before moving on to developing those solutions and encouraging further ideation. 

This is one of our favorite problem-solving activities and can be great for keeping the energy up throughout the workshop. Remember the importance of helping people become engaged in the process – energizing problem-solving techniques like Mindspin can help ensure your team stays engaged and happy, even when the problems they’re coming together to solve are complex. 

MindSpin   #teampedia   #idea generation   #problem solving   #action   A fast and loud method to enhance brainstorming within a team. Since this activity has more than round ideas that are repetitive can be ruled out leaving more creative and innovative answers to the challenge.

22. Improved Solutions

After a team has successfully identified a problem and come up with a few solutions, it can be tempting to call the work of the problem-solving process complete. That said, the first solution is not necessarily the best, and by including a further review and reflection activity into your problem-solving model, you can ensure your group reaches the best possible result. 

One of a number of problem-solving games from Thiagi Group, Improved Solutions helps you go the extra mile and develop suggested solutions with close consideration and peer review. By supporting the discussion of several problems at once and by shifting team roles throughout, this problem-solving technique is a dynamic way of finding the best solution. 

Improved Solutions   #creativity   #thiagi   #problem solving   #action   #team   You can improve any solution by objectively reviewing its strengths and weaknesses and making suitable adjustments. In this creativity framegame, you improve the solutions to several problems. To maintain objective detachment, you deal with a different problem during each of six rounds and assume different roles (problem owner, consultant, basher, booster, enhancer, and evaluator) during each round. At the conclusion of the activity, each player ends up with two solutions to her problem.

23. Four Step Sketch

Creative thinking and visual ideation does not need to be confined to the opening stages of your problem-solving strategies. Exercises that include sketching and prototyping on paper can be effective at the solution finding and development stage of the process, and can be great for keeping a team engaged. 

By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.

Four-Step Sketch   #design sprint   #innovation   #idea generation   #remote-friendly   The four-step sketch is an exercise that helps people to create well-formed concepts through a structured process that includes: Review key information Start design work on paper,  Consider multiple variations , Create a detailed solution . This exercise is preceded by a set of other activities allowing the group to clarify the challenge they want to solve. See how the Four Step Sketch exercise fits into a Design Sprint

24. 15% Solutions

Some problems are simpler than others and with the right problem-solving activities, you can empower people to take immediate actions that can help create organizational change. 

Part of the liberating structures toolkit, 15% solutions is a problem-solving technique that focuses on finding and implementing solutions quickly. A process of iterating and making small changes quickly can help generate momentum and an appetite for solving complex problems.

Problem-solving strategies can live and die on whether people are onboard. Getting some quick wins is a great way of getting people behind the process.   

It can be extremely empowering for a team to realize that problem-solving techniques can be deployed quickly and easily and delineate between things they can positively impact and those things they cannot change. 

15% Solutions   #action   #liberating structures   #remote-friendly   You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference.  15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change.  With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

25. How-Now-Wow Matrix

The problem-solving process is often creative, as complex problems usually require a change of thinking and creative response in order to find the best solutions. While it’s common for the first stages to encourage creative thinking, groups can often gravitate to familiar solutions when it comes to the end of the process. 

When selecting solutions, you don’t want to lose your creative energy! The How-Now-Wow Matrix from Gamestorming is a great problem-solving activity that enables a group to stay creative and think out of the box when it comes to selecting the right solution for a given problem.

Problem-solving techniques that encourage creative thinking and the ideation and selection of new solutions can be the most effective in organisational change. Give the How-Now-Wow Matrix a go, and not just for how pleasant it is to say out loud. 

How-Now-Wow Matrix   #gamestorming   #idea generation   #remote-friendly   When people want to develop new ideas, they most often think out of the box in the brainstorming or divergent phase. However, when it comes to convergence, people often end up picking ideas that are most familiar to them. This is called a ‘creative paradox’ or a ‘creadox’. The How-Now-Wow matrix is an idea selection tool that breaks the creadox by forcing people to weigh each idea on 2 parameters.

26. Impact and Effort Matrix

All problem-solving techniques hope to not only find solutions to a given problem or challenge but to find the best solution. When it comes to finding a solution, groups are invited to put on their decision-making hats and really think about how a proposed idea would work in practice. 

The Impact and Effort Matrix is one of the problem-solving techniques that fall into this camp, empowering participants to first generate ideas and then categorize them into a 2×2 matrix based on impact and effort.

Activities that invite critical thinking while remaining simple are invaluable. Use the Impact and Effort Matrix to move from ideation and towards evaluating potential solutions before then committing to them. 

Impact and Effort Matrix   #gamestorming   #decision making   #action   #remote-friendly   In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

27. Dotmocracy

If you’ve followed each of the problem-solving steps with your group successfully, you should move towards the end of your process with heaps of possible solutions developed with a specific problem in mind. But how do you help a group go from ideation to putting a solution into action? 

Dotmocracy – or Dot Voting -is a tried and tested method of helping a team in the problem-solving process make decisions and put actions in place with a degree of oversight and consensus. 

One of the problem-solving techniques that should be in every facilitator’s toolbox, Dot Voting is fast and effective and can help identify the most popular and best solutions and help bring a group to a decision effectively. 

Dotmocracy   #action   #decision making   #group prioritization   #hyperisland   #remote-friendly   Dotmocracy is a simple method for group prioritization or decision-making. It is not an activity on its own, but a method to use in processes where prioritization or decision-making is the aim. The method supports a group to quickly see which options are most popular or relevant. The options or ideas are written on post-its and stuck up on a wall for the whole group to see. Each person votes for the options they think are the strongest, and that information is used to inform a decision.

All facilitators know that warm-ups and icebreakers are useful for any workshop or group process. Problem-solving workshops are no different.

Use these problem-solving techniques to warm up a group and prepare them for the rest of the process. Activating your group by tapping into some of the top problem-solving skills can be one of the best ways to see great outcomes from your session.

• Check-in/Check-out
• Doodling Together
• Show and Tell
• Constellations
• Draw a Tree

28. Check-in / Check-out

Solid processes are planned from beginning to end, and the best facilitators know that setting the tone and establishing a safe, open environment can be integral to a successful problem-solving process.

Check-in / Check-out is a great way to begin and/or bookend a problem-solving workshop. Checking in to a session emphasizes that everyone will be seen, heard, and expected to contribute. 

If you are running a series of meetings, setting a consistent pattern of checking in and checking out can really help your team get into a groove. We recommend this opening-closing activity for small to medium-sized groups though it can work with large groups if they’re disciplined!

Check-in / Check-out   #team   #opening   #closing   #hyperisland   #remote-friendly   Either checking-in or checking-out is a simple way for a team to open or close a process, symbolically and in a collaborative way. Checking-in/out invites each member in a group to be present, seen and heard, and to express a reflection or a feeling. Checking-in emphasizes presence, focus and group commitment; checking-out emphasizes reflection and symbolic closure.

29. Doodling Together  

Thinking creatively and not being afraid to make suggestions are important problem-solving skills for any group or team, and warming up by encouraging these behaviors is a great way to start. 

Doodling Together is one of our favorite creative ice breaker games – it’s quick, effective, and fun and can make all following problem-solving steps easier by encouraging a group to collaborate visually. By passing cards and adding additional items as they go, the workshop group gets into a groove of co-creation and idea development that is crucial to finding solutions to problems. 

Doodling Together   #collaboration   #creativity   #teamwork   #fun   #team   #visual methods   #energiser   #icebreaker   #remote-friendly   Create wild, weird and often funny postcards together & establish a group’s creative confidence.

30. Show and Tell

You might remember some version of Show and Tell from being a kid in school and it’s a great problem-solving activity to kick off a session.

Asking participants to prepare a little something before a workshop by bringing an object for show and tell can help them warm up before the session has even begun! Games that include a physical object can also help encourage early engagement before moving onto more big-picture thinking.

By asking your participants to tell stories about why they chose to bring a particular item to the group, you can help teams see things from new perspectives and see both differences and similarities in the way they approach a topic. Great groundwork for approaching a problem-solving process as a team! 

Show and Tell   #gamestorming   #action   #opening   #meeting facilitation   Show and Tell taps into the power of metaphors to reveal players’ underlying assumptions and associations around a topic The aim of the game is to get a deeper understanding of stakeholders’ perspectives on anything—a new project, an organizational restructuring, a shift in the company’s vision or team dynamic.

31. Constellations

Who doesn’t love stars? Constellations is a great warm-up activity for any workshop as it gets people up off their feet, energized, and ready to engage in new ways with established topics. It’s also great for showing existing beliefs, biases, and patterns that can come into play as part of your session.

Using warm-up games that help build trust and connection while also allowing for non-verbal responses can be great for easing people into the problem-solving process and encouraging engagement from everyone in the group. Constellations is great in large spaces that allow for movement and is definitely a practical exercise to allow the group to see patterns that are otherwise invisible. 

Constellations   #trust   #connection   #opening   #coaching   #patterns   #system   Individuals express their response to a statement or idea by standing closer or further from a central object. Used with teams to reveal system, hidden patterns, perspectives.

32. Draw a Tree

Problem-solving games that help raise group awareness through a central, unifying metaphor can be effective ways to warm-up a group in any problem-solving model.

Draw a Tree is a simple warm-up activity you can use in any group and which can provide a quick jolt of energy. Start by asking your participants to draw a tree in just 45 seconds – they can choose whether it will be abstract or realistic. 

Once the timer is up, ask the group how many people included the roots of the tree and use this as a means to discuss how we can ignore important parts of any system simply because they are not visible.

All problem-solving strategies are made more effective by thinking of problems critically and by exposing things that may not normally come to light. Warm-up games like Draw a Tree are great in that they quickly demonstrate some key problem-solving skills in an accessible and effective way.

Draw a Tree   #thiagi   #opening   #perspectives   #remote-friendly   With this game you can raise awarness about being more mindful, and aware of the environment we live in.

Each step of the problem-solving workshop benefits from an intelligent deployment of activities, games, and techniques. Bringing your session to an effective close helps ensure that solutions are followed through on and that you also celebrate what has been achieved.

Here are some problem-solving activities you can use to effectively close a workshop or meeting and ensure the great work you’ve done can continue afterward.

• One Breath Feedback
• Who What When Matrix
• Response Cards

How do I conclude a problem-solving process?

All good things must come to an end. With the bulk of the work done, it can be tempting to conclude your workshop swiftly and without a moment to debrief and align. This can be problematic in that it doesn’t allow your team to fully process the results or reflect on the process.

At the end of an effective session, your team will have gone through a process that, while productive, can be exhausting. It’s important to give your group a moment to take a breath, ensure that they are clear on future actions, and provide short feedback before leaving the space. 

The primary purpose of any problem-solving method is to generate solutions and then implement them. Be sure to take the opportunity to ensure everyone is aligned and ready to effectively implement the solutions you produced in the workshop.

Remember that every process can be improved and by giving a short moment to collect feedback in the session, you can further refine your problem-solving methods and see further success in the future too.

33. One Breath Feedback

Maintaining attention and focus during the closing stages of a problem-solving workshop can be tricky and so being concise when giving feedback can be important. It’s easy to incur “death by feedback” should some team members go on for too long sharing their perspectives in a quick feedback round. 

One Breath Feedback is a great closing activity for workshops. You give everyone an opportunity to provide feedback on what they’ve done but only in the space of a single breath. This keeps feedback short and to the point and means that everyone is encouraged to provide the most important piece of feedback to them. 

One breath feedback   #closing   #feedback   #action   This is a feedback round in just one breath that excels in maintaining attention: each participants is able to speak during just one breath … for most people that’s around 20 to 25 seconds … unless of course you’ve been a deep sea diver in which case you’ll be able to do it for longer.

34. Who What When Matrix 

Matrices feature as part of many effective problem-solving strategies and with good reason. They are easily recognizable, simple to use, and generate results.

The Who What When Matrix is a great tool to use when closing your problem-solving session by attributing a who, what and when to the actions and solutions you have decided upon. The resulting matrix is a simple, easy-to-follow way of ensuring your team can move forward. 

Great solutions can’t be enacted without action and ownership. Your problem-solving process should include a stage for allocating tasks to individuals or teams and creating a realistic timeframe for those solutions to be implemented or checked out. Use this method to keep the solution implementation process clear and simple for all involved. 

Who/What/When Matrix   #gamestorming   #action   #project planning   With Who/What/When matrix, you can connect people with clear actions they have defined and have committed to.

35. Response cards

Group discussion can comprise the bulk of most problem-solving activities and by the end of the process, you might find that your team is talked out! 

Providing a means for your team to give feedback with short written notes can ensure everyone is head and can contribute without the need to stand up and talk. Depending on the needs of the group, giving an alternative can help ensure everyone can contribute to your problem-solving model in the way that makes the most sense for them.

Response Cards is a great way to close a workshop if you are looking for a gentle warm-down and want to get some swift discussion around some of the feedback that is raised. 

Response Cards   #debriefing   #closing   #structured sharing   #questions and answers   #thiagi   #action   It can be hard to involve everyone during a closing of a session. Some might stay in the background or get unheard because of louder participants. However, with the use of Response Cards, everyone will be involved in providing feedback or clarify questions at the end of a session.

Save time and effort discovering the right solutions

A structured problem solving process is a surefire way of solving tough problems, discovering creative solutions and driving organizational change. But how can you design for successful outcomes?

With SessionLab, it’s easy to design engaging workshops that deliver results. Drag, drop and reorder blocks  to build your agenda. When you make changes or update your agenda, your session  timing   adjusts automatically , saving you time on manual adjustments.

Collaborating with stakeholders or clients? Share your agenda with a single click and collaborate in real-time. No more sending documents back and forth over email.

Explore  how to use SessionLab  to design effective problem solving workshops or  watch this five minute video  to see the planner in action!

Over to you

The problem-solving process can often be as complicated and multifaceted as the problems they are set-up to solve. With the right problem-solving techniques and a mix of creative exercises designed to guide discussion and generate purposeful ideas, we hope we’ve given you the tools to find the best solutions as simply and easily as possible.

Is there a problem-solving technique that you are missing here? Do you have a favorite activity or method you use when facilitating? Let us know in the comments below, we’d love to hear from you! 

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

Leave a Comment Cancel reply

Your email address will not be published. Required fields are marked *

Going from a mere idea to a workshop that delivers results for your clients can feel like a daunting task. In this piece, we will shine a light on all the work behind the scenes and help you learn how to plan a workshop from start to finish. On a good day, facilitation can feel like effortless magic, but that is mostly the result of backstage work, foresight, and a lot of careful planning. Read on to learn a step-by-step approach to breaking the process of planning a workshop into small, manageable chunks.  The flow starts with the first meeting with a client to define the purposes of a workshop.…

How does learning work? A clever 9-year-old once told me: “I know I am learning something new when I am surprised.” The science of adult learning tells us that, in order to learn new skills (which, unsurprisingly, is harder for adults to do than kids) grown-ups need to first get into a specific headspace.  In a business, this approach is often employed in a training session where employees learn new skills or work on professional development. But how do you ensure your training is effective? In this guide, we'll explore how to create an effective training session plan and run engaging training sessions. As team leader, project manager, or consultant,…

Effective online tools are a necessity for smooth and engaging virtual workshops and meetings. But how do you choose the right ones? Do you sometimes feel that the good old pen and paper or MS Office toolkit and email leaves you struggling to stay on top of managing and delivering your workshop? Fortunately, there are plenty of online tools to make your life easier when you need to facilitate a meeting and lead workshops. In this post, we’ll share our favorite online tools you can use to make your job as a facilitator easier. In fact, there are plenty of free online workshop tools and meeting facilitation software you can…

Design your next workshop with SessionLab

Join the 150,000 facilitators using SessionLab

Sign up for free

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

9.4: Applications of Statics, Including Problem-Solving Strategies

• Last updated
• Save as PDF
• Page ID 1548

Learning Objectives

By the end of this section, you will be able to:

• Discuss the applications of Statics in real life.
• State and discuss various problem-solving strategies in Statics.

Statics can be applied to a variety of situations, ranging from raising a drawbridge to bad posture and back strain. We begin with a discussion of problem-solving strategies specifically used for statics. Since statics is a special case of Newton’s laws, both the general problem-solving strategies and the special strategies for Newton’s laws, discussed in Problem-Solving Strategies , still apply.

Problem-Solving Strategy: Static Equilibrium Situations

• The first step is to determine whether or not the system is in static equilibrium . This condition is always the case when the acceleration of the system is zero and accelerated rotation does not occur .
• It is particularly important to draw a free body diagram for the system of interest . Carefully label all forces, and note their relative magnitudes, directions, and points of application whenever these are known.
• Solve the problem by applying either or both of the conditions for equilibrium (represented by the equations \(net \, F = 0\) and \(net \, \tau = 0 \), depending on the list of known and unknown factors. If the second condition is involved, choose the pivot point to simplify the solution . Any pivot point can be chosen, but the most useful ones cause torques by unknown forces to be zero. (Torque is zero if the force is applied at the pivot (then \(r = 0\)), or along a line through the pivot point (then \\theta = 0\)). Always choose a convenient coordinate system for projecting forces.
• Check the solution to see if it is reasonable by examining the magnitude, direction, and units of the answer. The importance of this last step never diminishes, although in unfamiliar applications, it is usually more difficult to judge reasonableness. These judgments become progressively easier with experience.

Now let us apply this problem-solving strategy for the pole vaulter shown in the three figures below. The pole is uniform and has a mass of 5.00 kg. In Figure \(\PageIndex{1}\), the pole’s cg lies halfway between the vaulter’s hands. It seems reasonable that the force exerted by each hand is equal to half the weight of the pole, or 24.5 N. This obviously satisfies the first condition for equilibrium \((net \, F = 0)\). The second condition \((net \, \tau = 0)\) is also satisfied, as we can see by choosing the cg to be the pivot point. The weight exerts no torque about a pivot point located at the cg, since it is applied at that point and its lever arm is zero. The equal forces exerted by the hands are equidistant from the chosen pivot, and so they exert equal and opposite torques.

Similar arguments hold for other systems where supporting forces are exerted symmetrically about the cg. For example, the four legs of a uniform table each support one-fourth of its weight. In Figure \(\PageIndex{2}\), a pole vaulter holding a pole with its cg halfway between his hands is shown. Each hand exerts a force equal to half the weight of the pole, \(F_r = F_L = w/2\). (b) The pole vaulter moves the pole to his left, and the forces that the hands exert are no longer equal. See Figure . If the pole is held with its cg to the left of the person, then he must push down with his right hand and up with his left. The forces he exerts are larger here because they are in opposite directions and the cg is at a long distance from either hand.

Similar observations can be made using a meter stick held at different locations along its length.

If the pole vaulter holds the pole as shown in Figure , the situation is not as simple. The total force he exerts is still equal to the weight of the pole, but it is not evenly divided between his hands. (If \(F_L = F_R \), then the torques about the cg would not be equal since the lever arms are different.) Logically, the right hand should support more weight, since it is closer to the cg. In fact, if the right hand is moved directly under the cg, it will support all the weight. This situation is exactly analogous to two people carrying a load; the one closer to the cg carries more of its weight. Finding the forces \(F_L\) and \(F_R\) is straightforward, as the next example shows.

If the pole vaulter holds the pole from near the end of the pole ( Figure ), the direction of the force applied by the right hand of the vaulter reverses its direction.

Example \(\PageIndex{1}\): What Force Is Needed to Support a Weight Held Near Its CG?

For the situation shown in Figure , calculate: (a) \(F_R\), the force exerted by the right hand, and (b) \(F_L\), the force exerted by the left hand. The hands are 0.900 m apart, and the cg of the pole is 0.600 m from the left hand.

Figure includes a free body diagram for the pole, the system of interest. There is not enough information to use the first condition for equilibrium \((net \, F = 0)\), since two of the three forces are unknown and the hand forces cannot be assumed to be equal in this case. There is enough information to use the second condition for equilibrium \((net \, \tau = 0)\) if the pivot point is chosen to be at either hand, thereby making the torque from that hand zero. We choose to locate the pivot at the left hand in this part of the problem, to eliminate the torque from the left hand.

Solution for (a)

There are now only two nonzero torques, those from the gravitational force \((\tau_W)\) and from the push or pull of the right hand \((\tau_R)\). Stating the second condition in terms of clockwise and counterclockwise torques,

\[net \, \tau_{CW} = -net \, \tau_{CCW}.\]

or the algebraic sum of the torques is zero.

Here this is \[\tau_R = -\tau_W\]

since the weight of the pole creates a counterclockwise torque and the right hand counters with a clockwise toque. Using the definition of torque, \(\tau = rF \, sin \, \theta\), noting that \(\theta = 90^o\), and substituting known values, we obtain

\[(0.900 \, m)(F_R) = (0.600 \, m)(mg).\] Thus,

\[F_R = (0.667)(5.00 \, kg)(9.80 \, m/s^2)\]\[=32.7 \, N.\]

Solution for (b)

The first condition for equilibrium is based on the free body diagram in the figure. This implies that by Newton’s second law:

\[F_L + F_R - mg = 0\]

From this we can conclude \[F_L + F_R = w = mg\]

Solving for \(F_L\), we obtain

\[F_L = mg - F_R\]

\]= mg - 43.7 \, N\]

\[= (5.00 \, kg)(9.80 \, m/s^2) - 32.7 \, N\]

\[= 16.3 \, N\]

\(F_L\) is seen to be exactly half of \(F_R\), as we might have guessed, since \(F_L\) is applied twice as far from for cg as \(F_R\).

If the pole vaulter holds the pole as he might at the start of a run, shown in Figure , the forces change again. Both are considerably greater, and one force reverses direction.

Take-Home Experiment

• This is an experiment to perform while standing in a bus or a train. Stand facing sideways. How do you move your body to readjust the distribution of your mass as the bus accelerates and decelerates? Now stand facing forward. How do you move your body to readjust the distribution of your mass as the bus accelerates and decelerates? Why is it easier and safer to stand facing sideways rather than forward? Note: For your safety (and those around you), make sure you are holding onto something while you carry out this activity!

PhET Explorations: Balancing Act

Play with objects on a teeter totter to learn about balance. Test what you've learned by trying the Balance Challenge game.

Figure \(\PageIndex{4}\): Balancing Act

• Statics can be applied to a variety of situations, ranging from raising a drawbridge to bad posture and back strain. We have discussed the problem-solving strategies specifically useful for statics. Statics is a special case of Newton’s laws, both the general problem-solving strategies and the special strategies for Newton’s laws, discussed in Problem-Solving Strategies , still apply.

Interview Questions

Comprehensive Interview Guide: 60+ Professions Explored in Detail

26 Good Examples of Problem Solving (Interview Answers)

By Biron Clark

Published: November 15, 2023

Employers like to hire people who can solve problems and work well under pressure. A job rarely goes 100% according to plan, so hiring managers will be more likely to hire you if you seem like you can handle unexpected challenges while staying calm and logical in your approach.

But how do they measure this?

They’re going to ask you interview questions about these problem solving skills, and they might also look for examples of problem solving on your resume and cover letter. So coming up, I’m going to share a list of examples of problem solving, whether you’re an experienced job seeker or recent graduate.

Then I’ll share sample interview answers to, “Give an example of a time you used logic to solve a problem?”

Problem-Solving Defined

It is the ability to identify the problem, prioritize based on gravity and urgency, analyze the root cause, gather relevant information, develop and evaluate viable solutions, decide on the most effective and logical solution, and plan and execute implementation. 

Problem-solving also involves critical thinking, communication, listening, creativity, research, data gathering, risk assessment, continuous learning, decision-making, and other soft and technical skills.

Solving problems not only prevent losses or damages but also boosts self-confidence and reputation when you successfully execute it. The spotlight shines on you when people see you handle issues with ease and savvy despite the challenges. Your ability and potential to be a future leader that can take on more significant roles and tackle bigger setbacks shine through. Problem-solving is a skill you can master by learning from others and acquiring wisdom from their and your own experiences. 

It takes a village to come up with solutions, but a good problem solver can steer the team towards the best choice and implement it to achieve the desired result.

Watch: 26 Good Examples of Problem Solving

Examples of problem solving scenarios in the workplace.

• Correcting a mistake at work, whether it was made by you or someone else
• Overcoming a delay at work through problem solving and communication
• Resolving an issue with a difficult or upset customer
• Overcoming issues related to a limited budget, and still delivering good work through the use of creative problem solving
• Overcoming a scheduling/staffing shortage in the department to still deliver excellent work
• Troubleshooting and resolving technical issues
• Handling and resolving a conflict with a coworker
• Solving any problems related to money, customer billing, accounting and bookkeeping, etc.
• Taking initiative when another team member overlooked or missed something important
• Taking initiative to meet with your superior to discuss a problem before it became potentially worse
• Solving a safety issue at work or reporting the issue to those who could solve it
• Using problem solving abilities to reduce/eliminate a company expense
• Finding a way to make the company more profitable through new service or product offerings, new pricing ideas, promotion and sale ideas, etc.
• Changing how a process, team, or task is organized to make it more efficient
• Using creative thinking to come up with a solution that the company hasn’t used before
• Performing research to collect data and information to find a new solution to a problem
• Boosting a company or team’s performance by improving some aspect of communication among employees
• Finding a new piece of data that can guide a company’s decisions or strategy better in a certain area

Problem Solving Examples for Recent Grads/Entry Level Job Seekers

• Coordinating work between team members in a class project
• Reassigning a missing team member’s work to other group members in a class project
• Adjusting your workflow on a project to accommodate a tight deadline
• Speaking to your professor to get help when you were struggling or unsure about a project
• Asking classmates, peers, or professors for help in an area of struggle
• Talking to your academic advisor to brainstorm solutions to a problem you were facing
• Researching solutions to an academic problem online, via Google or other methods
• Using problem solving and creative thinking to obtain an internship or other work opportunity during school after struggling at first

You can share all of the examples above when you’re asked questions about problem solving in your interview. As you can see, even if you have no professional work experience, it’s possible to think back to problems and unexpected challenges that you faced in your studies and discuss how you solved them.

Interview Answers to “Give an Example of an Occasion When You Used Logic to Solve a Problem”

Now, let’s look at some sample interview answers to, “Give me an example of a time you used logic to solve a problem,” since you’re likely to hear this interview question in all sorts of industries.

Example Answer 1:

At my current job, I recently solved a problem where a client was upset about our software pricing. They had misunderstood the sales representative who explained pricing originally, and when their package renewed for its second month, they called to complain about the invoice. I apologized for the confusion and then spoke to our billing team to see what type of solution we could come up with. We decided that the best course of action was to offer a long-term pricing package that would provide a discount. This not only solved the problem but got the customer to agree to a longer-term contract, which means we’ll keep their business for at least one year now, and they’re happy with the pricing. I feel I got the best possible outcome and the way I chose to solve the problem was effective.

Example Answer 2:

In my last job, I had to do quite a bit of problem solving related to our shift scheduling. We had four people quit within a week and the department was severely understaffed. I coordinated a ramp-up of our hiring efforts, I got approval from the department head to offer bonuses for overtime work, and then I found eight employees who were willing to do overtime this month. I think the key problem solving skills here were taking initiative, communicating clearly, and reacting quickly to solve this problem before it became an even bigger issue.

Example Answer 3:

In my current marketing role, my manager asked me to come up with a solution to our declining social media engagement. I assessed our current strategy and recent results, analyzed what some of our top competitors were doing, and then came up with an exact blueprint we could follow this year to emulate our best competitors but also stand out and develop a unique voice as a brand. I feel this is a good example of using logic to solve a problem because it was based on analysis and observation of competitors, rather than guessing or quickly reacting to the situation without reliable data. I always use logic and data to solve problems when possible. The project turned out to be a success and we increased our social media engagement by an average of 82% by the end of the year.

Answering Questions About Problem Solving with the STAR Method

When you answer interview questions about problem solving scenarios, or if you decide to demonstrate your problem solving skills in a cover letter (which is a good idea any time the job description mention problem solving as a necessary skill), I recommend using the STAR method to tell your story.

STAR stands for:

It’s a simple way of walking the listener or reader through the story in a way that will make sense to them. So before jumping in and talking about the problem that needed solving, make sure to describe the general situation. What job/company were you working at? When was this? Then, you can describe the task at hand and the problem that needed solving. After this, describe the course of action you chose and why. Ideally, show that you evaluated all the information you could given the time you had, and made a decision based on logic and fact.

Finally, describe a positive result you got.

Whether you’re answering interview questions about problem solving or writing a cover letter, you should only choose examples where you got a positive result and successfully solved the issue.

Example answer:

Situation : We had an irate client who was a social media influencer and had impossible delivery time demands we could not meet. She spoke negatively about us in her vlog and asked her followers to boycott our products. (Task : To develop an official statement to explain our company’s side, clarify the issue, and prevent it from getting out of hand). Action : I drafted a statement that balanced empathy, understanding, and utmost customer service with facts, logic, and fairness. It was direct, simple, succinct, and phrased to highlight our brand values while addressing the issue in a logical yet sensitive way.   We also tapped our influencer partners to subtly and indirectly share their positive experiences with our brand so we could counter the negative content being shared online.  Result : We got the results we worked for through proper communication and a positive and strategic campaign. The irate client agreed to have a dialogue with us. She apologized to us, and we reaffirmed our commitment to delivering quality service to all. We assured her that she can reach out to us anytime regarding her purchases and that we’d gladly accommodate her requests whenever possible. She also retracted her negative statements in her vlog and urged her followers to keep supporting our brand.

What Are Good Outcomes of Problem Solving?

Whenever you answer interview questions about problem solving or share examples of problem solving in a cover letter, you want to be sure you’re sharing a positive outcome.

Below are good outcomes of problem solving:

• Saving the company time or money
• Making the company money
• Pleasing/keeping a customer
• Obtaining new customers
• Solving a safety issue
• Solving a staffing/scheduling issue
• Solving a logistical issue
• Solving a company hiring issue
• Solving a technical/software issue
• Making a process more efficient and faster for the company
• Creating a new business process to make the company more profitable
• Improving the company’s brand/image/reputation
• Getting the company positive reviews from customers/clients

Every employer wants to make more money, save money, and save time. If you can assess your problem solving experience and think about how you’ve helped past employers in those three areas, then that’s a great start. That’s where I recommend you begin looking for stories of times you had to solve problems.

Tips to Improve Your Problem Solving Skills

Throughout your career, you’re going to get hired for better jobs and earn more money if you can show employers that you’re a problem solver. So to improve your problem solving skills, I recommend always analyzing a problem and situation before acting. When discussing problem solving with employers, you never want to sound like you rush or make impulsive decisions. They want to see fact-based or data-based decisions when you solve problems.

Next, to get better at solving problems, analyze the outcomes of past solutions you came up with. You can recognize what works and what doesn’t. Think about how you can get better at researching and analyzing a situation, but also how you can get better at communicating, deciding the right people in the organization to talk to and “pull in” to help you if needed, etc.

Finally, practice staying calm even in stressful situations. Take a few minutes to walk outside if needed. Step away from your phone and computer to clear your head. A work problem is rarely so urgent that you cannot take five minutes to think (with the possible exception of safety problems), and you’ll get better outcomes if you solve problems by acting logically instead of rushing to react in a panic.

You can use all of the ideas above to describe your problem solving skills when asked interview questions about the topic. If you say that you do the things above, employers will be impressed when they assess your problem solving ability.

If you practice the tips above, you’ll be ready to share detailed, impressive stories and problem solving examples that will make hiring managers want to offer you the job. Every employer appreciates a problem solver, whether solving problems is a requirement listed on the job description or not. And you never know which hiring manager or interviewer will ask you about a time you solved a problem, so you should always be ready to discuss this when applying for a job.

Related interview questions & answers:

• How do you handle stress?
• How do you handle conflict?
• Tell me about a time when you failed

About the Author

Read more articles by Biron Clark

Continue Reading

15 Most Common Pharmacist Interview Questions and Answers

15 most common paralegal interview questions and answers, top 30+ funny interview questions and answers, 60 hardest interview questions and answers, 100+ best ice breaker questions to ask candidates, top 20 situational interview questions (& sample answers), 15 most common physical therapist interview questions and answers, 15 most common project manager interview questions and answers.

Login Get started

Math Review of Solving Application Problems

• July 24, 2014
• No Comments

Solving application problems is a process that includes understanding the problem, translating it into an equation, solving the equation, checking the answer, and answering the question. This process can be used to solve many different types of problems.

Understanding and Setting up The Application Problem

The first part of the process involves understanding what is being asked. This includes noticing any key words that refer to operations and any quantities that are in relationship to one another. It is important to have an idea of what sort of quantity will represent a solution. Suppose a problem asks how many points the first-place winner had. That would require just one answer. If the problem described that the first-place winner had 10 points more than the second-place winner, and 17 points more than the third-place winner, all three point values would be necessary to completely answer the question.

Figure 1: Setting up the application problem involves finding quantities in relationship to one another.

Translating the Problem into an Equation

The second part of the problem involves putting the problem in symbol form. This includes choosing a letter to represent a variable, and writing down exactly what the variable stands for. Suppose that the point values of the three winners totaled 114. Let the point value of the first-place winner be x, the second-place winner x – 10, and the third-place winner x – 17. Then, x + x – 10 + x – 17 = 114.

Figure 2: Translating the problem into an equation involves putting the problem into symbol form.

Solving and Checking the Equation

If x + x – 10 + x – 17 = 114, then 3x = 114 + 27, or 3x = 141. So, x = 47, x – 10 = 37, and x -17 = 30. To check, 47 + 37 + 30 = 114. The answer makes sense, and fits the parameters of the problem. In this case, the original equation was x + x -10 + x – 17 = 114.

Answer the Question Asked

If the work is done to understand the problem before it is set up and solved, it is easier to answer the appropriate question. In this case, all three point values were necessary to completely answer the question. Suppose that more information were added to the problem. During the same contest last year, the point values of the first-place, second-place and third-place winners totaled 100, but the second place winner had 14 points less than the first-place winner, and the third-place winner had 18 points less than the first place winner. Which year did the third-place winner earn more points, and what was the difference? Last year, the equation was x + x – 14 + x – 18 = 100, so 3x = 132. The first-place winner earned 44 points, the second-place winner earned 44-14 or 30 points, and the third-place winner earned 28 points. However, the question asked has 2 parts. Last year, the third-place winner earned 28 points, and this year, the third-place winner earned 30 points. This year, the third-place winner earned more points, and the difference was a gain of 2 points.

Figure 3: Answering the question asked brings all the parts of the problem into balance.

Interested in algebra tutoring services ? Learn more about how we are assisting thousands of students each academic year.

SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Lenexa, KS: visit Tutoring in Lenexa, KS

Math Review of Changing Application Problems into Equations

Math review of factoring a monomial from a polynomial.

Module 2: General Problem Solving

Discussion: problem solving application.

Pick a unique real problem and try to solve it using the general problem solving strategies from this module. Present the problem and the solution to the rest of the class. View the problems posted by your classmates and respond to at least two.

This assignment will help you practice applying what you have learned about general problem solving strategies to real problems or situations, and communicating about mathematical ideas with others.

Pick a problem:

Option 1:   Pick a real problem that you need to solve and use the different general problem solving strategies to try to find a solution.   For example, perhaps you need to figure out how to pack an awkward item in your car trunk, or decide what to spend the most time studying for an exam in another class, or plan the assignments and schedule for a charity clean-up project you are organizing.

Option 2:   Pick a real problem that someone else has solved and describe the general strategies they used.   This could be a friend or relative who solved a problem like the ones described in Option 1, or it could be a historical person, like George Washington Carver, who invented peanut butter and many other things, or Mary Anderson, who invented the windshield wiper.

Option 3:   Pick a currently unsolved problem, like curing cancer or finding better energy sources, and analyze how people are using the different problem solving strategies to find a solution.

Solve the problem or analyze the solution:

If you are solving your own problem, use the different general problem solving strategies that you learned about in this unit to try to solve it.   If you are analyzing a problem someone else solved, or is trying to solve, identify the different general problem solving strategies used.

Present the problem, the general problem solving strategies used, and the solution to the others in the class:

Post a message in the application discussion forum for this unit.   In your message, describe the problem and how you solved it.   Use the equation editor as necessary to show any mathematical operations.   The better you communicate, the more points you will earn.   If you enjoy and know how to use multimedia, such as video, audio, and graphics, you may use those as well, but this is not required.

View and respond to the application problems submitted by your classmates.

Pick two of your classmates’ applications that were particularly helpful to you.   Write a response to each, explaining in a paragraph or two why their applications helped you better understand the mathematics for this unit or better understand how the mathematics for this unit could be used outside of class.

Background: This Saturday is the annual chocolate chip cookie bake sale event. At each of the last 2 bake sale events, there were 600 people who attended. Every annual bake sale must receive 600 attendees. Every attendee eats 5 cookies each.

Problem: We must raise $1,500 for our daughter’s team travel expenses. How many cookies must we make and how much money must each cookie sell for in order to raise the appropriate money?

Solution: 1) The amount of cookies to make are 600 (attendees) times 5 (cookies eaten per person) equals 3,000 cookies to make. 2) 3,000 (cookies made) divided by $1,500 (money needed) equal 50 cents per cookie.

Other Topics:   shipping textbooks, pattern recognition, MPG, budgeting, transportation, metric/English conversions, lottery, time management, retail sales, volume of dirt on a softball field, tipping at a restaurant, yard sales, auto part manufacturing, room design, rate of change, estimate the square root of 3 to two decimal places with a calculator without square root key

• Mathematics for the Liberal Arts I. Provided by : Extended Learning Institute of Northern Virginia Community College. Located at : http://eli.nvcc.edu/ . License : CC BY: Attribution

Polytomous Effectiveness Indicators in Complex Problem-Solving Tasks and Their Applications in Developing Measurement Model

• Theory & Methods
• Published: 09 April 2024

Cite this article

• Pujue Wang   ORCID: orcid.org/0000-0001-6931-6829 1 , 2 &
• Hongyun Liu   ORCID: orcid.org/0000-0002-3472-9102 1 , 2  

10 Accesses

Explore all metrics

Recent years have witnessed the emergence of measurement models for analyzing action sequences in computer-based problem-solving interactive tasks. The cutting-edge psychometrics process models require pre-specification of the effectiveness of state transitions often simplifying them into dichotomous indicators. However, the dichotomous effectiveness becomes impractical when dealing with complex tasks that involve multiple optimal paths and numerous state transitions. Building on the concept of problem-solving, we introduce polytomous indicators to assess the effectiveness of problem states \(d_{s}\) and state-to-state transitions \({\mathrm {\Delta }d}_{\mathrm {s\rightarrow s'}}\) . The three-step evaluation method for these two types of indicators is proposed and illustrated across two real problem-solving tasks. We further present a novel psychometrics process model, the sequential response model with polytomous effectiveness indicators (SRM-PEI), which is tailored to encompass a broader range of problem-solving tasks. Monte Carlo simulations indicated that SRM-PEI performed well in the estimation of latent ability and transition tendency parameters across different conditions. Empirical studies conducted on two real tasks supported the better fit of SRM-PEI over previous models such as SRM and SRMM, providing rational and interpretable estimates of latent abilities and transition tendencies through effectiveness indicators. The paper concludes by outlining potential avenues for the further application and enhancement of polytomous effectiveness indicators and SRM-PEI.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price includes VAT (Russian Federation)

Instant access to the full article PDF.

Rent this article via DeepDyve

Institutional subscriptions

Data Availability

The data analyzed in the empirical example of this study are available on this project’s Open Science Framework (OSF) page: https://osf.io/fw82q/ .

Anderson, J. R., Funke, J., & Plata, G. (Eds.). (2007). Cognitive psychologic (6 Aufl.). Spektrum Akademischer Verlag http://www.gbv.de/dms/bs/toc/529836963.pdf

Arieli-Attali, M., Ou, L., & Simmering, V. R. (2019). Understanding test takers’ choices in a self-adapted test: A hidden Markov modeling of process data. Frontiers in Psychology, 10 , 83. https://doi.org/10.3389/fpsyg.2019.00083

Article   PubMed   PubMed Central   Google Scholar  

Bergner, Y., & von Davier, A. A. (2019). Process data in NAEP: Past, present, and future. Journal of Educational and Behavioral Statistics, 44 (6), 706–732.

Article   Google Scholar  

Bock, R. D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37 (1), 29–51. https://doi.org/10.1007/BF02291411

Brooks, S. P., & Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7 , 434–455.

Google Scholar  

Buchner, A., & Funke, J. (1993). Finite-state automata: Dynamic task environments in problem-solving research. The Quarterly Journal of Experimental Psychology, 46 (1), 83–118.

Chen, Y. (2020). A continuous-time dynamic choice measurement model for problem-solving process data. Psychometrika, 85 (4), 1052–1075. https://doi.org/10.1007/s11336-020-09734-1

Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2022). Introduction to algorithms (4th ed., pp. 563–572). Cambridge: MIT Press.

Fu, Y., Zhan, P., Chen, Q., & Jiao, H. (2023). Joint modeling of action sequences and action time in computer-based interactive tasks. Behav Res Methods . https://doi.org/10.3758/s13428-023-02178-2

Article   PubMed   Google Scholar  

Funke, J. (2001). Dynamic systems as tools for analysing human judgement. Think Reason, 7 , 69–89.

Geisser, S., & Eddy, W. F. (1979). A predictive approach to model selection. J Am Stat Assoc, 74 , 153–160.

Gelfand, A. E., & Dey, D. K. (1994). Bayesian model choice: Asymptotics and exact calculations. Journal of the Royal Statistical Society Series B, 56 , 501–514.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2014). Bayesian data analysis (3rd ed.). Boca Raton: Chapman & Hall/CRC Press.

Gelman, A., Meng, X.-L., & Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies. Statistica Sinica, 6 , 733–760.

Gelman, A., & Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7 (4), 457–472. https://doi.org/10.1214/ss/1177011136

Griffin, P., & Care, E. (2014). Assessment and teaching of 21st century skills: Methods and approach . New York, NY: Springer.

Guttman, I. (1967). The use of the concept of a future observation in goodness-of-fit problems. Journal of the Royal Statistical Society: Series B (Methodological), 29 (1), 83–100. https://doi.org/10.1111/j.2517-6161.1967.tb00676.x

Han, Y., Liu, H., & Ji, F. (2022). A sequential response model for analyzing process data on technology-based problem-solving tasks. Multivariate Behavioral Research, 57 (6), 960–977. https://doi.org/10.1080/00273171.2021.1932403

Han, Y., & Wilson, M. (2022). Analyzing student response processes to evaluate success on a technology-based problem-solving task. Applied Measurement in Education, 35 (1), 33–45.

He, Q., & von Davier, M. (2015). Identifying feature sequences from process data in problem-solving items with n-grams. In L. A. van der Ark, D. M. Bolt, W.-C. Wang, J. A. Douglas, & S.-M. Chow (Eds.), Quantitative psychology research (pp. 173–190). Berlin: Springer. https://doi.org/10.1007/978-3-319-19977-1_13

Chapter   Google Scholar  

He, Q., & von Davier, M. (2016). Analyzing process data from problem-solving items with N-Grams: Insights from a computer-based large-scale assessment. In Y. Rosen, S. Ferrara, & M. Mosharraf (Eds.), Handbook of research on technology tools for real-world skill development (pp. 750–777). Pennsylvania: IGI Global.

Koehler, E., Brown, E., & Haneuse, J. P. A. (2009). On the assessment of monte carlo error in simulation-based statistical analyses. The American Statistician, 63 (2), 155–162. https://doi.org/10.1198/tast.2009.0030

LaMar, M. M. (2018). Markov decision process measurement model. Psychometrika, 83 (1), 67–88. https://doi.org/10.1007/s11336-017-9570-0

Levy, R., & Mislevy, R. J. (2016). Bayesian psychometric modeling . Cambridge: CRC Press.

Li, M., Liu, H., Cai, M., & Yuan, J. (2023). Estimation of individuals’ collaborative problem solving ability in computer-based assessment. Education and Information Technologies . https://doi.org/10.1007/s10639-023-12271-w

Liu, H., Liu, Y., & Li, M. (2018). Analysis of process data of PISA 2012 computer-based problem solving: Application of the modified multilevel mixture IRT model. Frontiers in Psychology, 9 , 1372.

Mayer, R. E., & Wittrock, M. C. (2006). Problem solving. In P. A. Alexander & P. H. Winne (Eds.), Handbook of educational psychology (2nd ed., pp. 287–304). Mahwah: Erlbaum.

Newell, A., & Simon, H. A. (1972). Human problem solving (Vol. 104). Englewood Cliffs: Prentice-Hall.

OECD (2014). PISA 2012 results: Creative problem solving: Students ’ skills in tackling real-life problems (Vol. V). OECD.

OECD. (2016). PISA 2015 Assessment and analytical framework: Science . Reading, mathematic and financial literacy: PISA. OECD Publishing. https://doi.org/10.1787/9789264255425-en

OECD. (2018). The future of education and skills: Education 2030 . Paris: OECD Publishing.

Patz, R. J., & Junker, B. W. (1999). Applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses. Journal of Educational and Behavioral Statistics, 24 (4), 342–366. https://doi.org/10.3102/10769986024004342

Patz, R. J., & Junker, B. W. (1999). A straightforward approach to Markov chain Monte Carlo methods for item response models. Journal of Educational and Behavioral Statistics, 24 (2), 146–178. https://doi.org/10.3102/10769986024002146

Rubin, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for the applies statistician. The Annals of Statistics, 12 , 1151–1172. https://doi.org/10.2307/2240995

Shu, Z., Bergner, Y., Zhu, M., Hao, J., & von Davier, A. A. (2017). An item response theory analysis of problem-solving processes in scenario-based tasks. Psychological Test and Assessment Modeling, 59 (1), 109–131.

Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & Van der Linde, A. (1998). Bayesian deviance, the effective number of parameters, and the comparison of arbitrarily complex models . MRC Biostatistics Unit: Technical report.

Tang, X. (2023). A latent hidden Markov model for process data. Psychometrika . https://doi.org/10.1007/s11336-023-09938-1

Tang, X., Wang, Z., He, Q., Liu, J., & Ying, Z. (2020). Latent feature extraction for process data via multidimensional scaling. Psychometrika, 85 (2), 378–397.

von Davier, A. A. (2017). Computational psychometrics in support of collaborative educational assessments. Journal of Educational Measurement, 54 (1), 3–11.

Xiao, Y., He, Q., Veldkamp, B., & Liu, H. (2021). Exploring latent states of problem-solving competence using hidden Markov model on process data. Journal of Computer Assisted Learning, 37 (5), 1232–1247.

Xiao, Y., & Liu, H. (2023). A state response measurement model for problem-solving process data. Behavior Research Methods, 1–20 . https://doi.org/10.3758/s13428-022-02042-9

Yuan, J., Xiao, Y., & Liu, H. (2019). Assessment of collaborative problem solving based on process stream data: A new paradigm for extracting indicators and modeling dyad data. Frontiers in Psychology, 10 , 369. https://doi.org/10.3389/fpsyg.2019.0036

Download references

No funding was received to assist with the preparation of this manuscript. The authors have no competing interests to declare that are relevant to the content of this article.

Author information

Authors and affiliations.

Beijing Key Laboratory of Applied Experimental Psychology, National Demonstration Center for Experimental Psychology Education (Beijing Normal University), Beijing Normal University, No. 19 Xin Jie Kou Wai Street, Beijing, 100875, China

Pujue Wang & Hongyun Liu

Beijing Key Laboratory of Applied Experimental Psychology, National Demonstration Center for Experimental Psychology Education (Beijing Normal University), Faculty of Psychology, Beijing Normal University, No. 19, Xin Jie Kou Wai St., Hai Dian District, Beijing, 100875, People’s Republic of China

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Hongyun Liu .

Ethics declarations

Code availability.

The codes are available on this project’s Open Science Framework (OSF) page: https://osf.io/fw82q/ .

Additional information

Publisher's note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (pdf 1032 KB)

Appendix a. algorithm for automatically calculating state effectiveness in the balance beam task.

The interface of the initial state in the Chinese version of the Balance Beam task.

The diagram for the four types of transitions that can occur when a weight moves among ten possible positions in the Balance Beam task.

In the Balance Beam task, the ten potential positions for each weight are categorized into four groups: (1) Positions 1–4: Positioned on side A of the beam; (2) Position 5: Not suspended on side A; (3) Position 6: Not suspended on side B; (4) Positions 7–10: Positioned on side B of the beam. Figure 5 illustrates the transition of each weight among ten positions through four types of operations: (1) removing a weight from the beam; (2) hanging an unhung weight; (3) passing a weight to the other student; and (4) shifting the position of a weight on the same side. Each arrow represents an operation that can lead to a transition. Through this figure, we can easily find the minimum number of transitions between any two positions for one weight. Since an operation can only alter the position of one weight once, the shortest distance between states s and \(s'\) equals the sum of the minimum number of operations required for each of the four weights to change its position from state s to \(s'\) . Then, we can quickly and accurately calculate the shortest distance \(d_{s}^{(k)}\) between a state s and the target state \(s_{target}^{(k)}\) using the state code and rules to change the position according to Fig. 6 . Finally, we select the minimum distance \(d_{s}=\min \left( d_{s}^{(1)}, d_{s}^{(2)}, \ldots , d_{s}^{(k)}\right) \) as the effectiveness indicator \(d_{s}\) of the state s

During the process of programming the calculations mentioned above, the position of each weight can be assigned a unique number from one to ten. Therefore, any given state in the Balance Beam task can be encoded by a sequence of four numbers, a representation we refer to as the state code. For one weight, calculating the shortest distance between any two positions can be simplified by several rules. The R code for evaluating the effectiveness of states for the Balance Beam task that requires the use of two weights to achieve balance is available at https://osf.io/fw82q/ .

In the example of the code, the four positions for hanging weights on the balance beam on student A’s side are coded as 1 to 4, and the four positions on student B’s side are coded as \(-1\) to \(-4\) . The unhung weights are coded as 0.5 when in student A’s hand and \(-\) 0.5 when in student B’s hand. In the initial state, all four weights are in the hand of A, and the state code is (0.5, 0.5, 0.5, 0.5). The effectiveness of the initial state is equal to 3, which means that the balance state using two weights can be achieved after a minimum of three transitions. Another example is that Student B holds the 50 g and 100 g weights and Student A has hung the 300 g weight at position 1 and the 500 g weight at position 2. This state is at a minimum distance of 2 from the balance state.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Wang, P., Liu, H. Polytomous Effectiveness Indicators in Complex Problem-Solving Tasks and Their Applications in Developing Measurement Model. Psychometrika (2024). https://doi.org/10.1007/s11336-024-09963-8

Download citation

Received : 05 August 2023

Published : 09 April 2024

DOI : https://doi.org/10.1007/s11336-024-09963-8

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• polytomous effectiveness indicators
• process data
• problem-solving
• Find a journal
• Publish with us
• Track your research

ORIGINAL RESEARCH article

Integrating inquiry and mathematical modeling when teaching a common topic in lower secondary school: an istem approach provisionally accepted.

• 1 Hong Kong Baptist University, Hong Kong, SAR China

The final, formatted version of the article will be published soon.

The world has been increasingly shaped by Science, Technology, Engineering and Mathematics (STEM). This has resulted in educational systems across the globe implementing STEM education. To reap maximum benefits, researchers are now advocating for the integration of STEM domains. In recent studies, the integration of science and mathematics has become increasingly popular. The domains are much more suitable for integration because of their fields of application and their mutual approach towards problem-solving. However, there is little empirical evidence to drive the development of a practical model for classroom implementation. This study aims to cover that gap through integrating mathematics and science concepts when teaching a common topic to two classes of Form 1 (13-14 years) students. A mathematics and a science teacher went through two cycles of lesson study, integrating and teaching the concept of density. Results show a strong synergy between the BSCS 5E instructional model of inquiry and mathematical modeling; hence the methodological approaches can be used to integrate common topics like density. Further, teacher collaboration, teacher immersion in the iSTEM practices, teacher's knowledge, and skills of the other subject and an in-depth understanding of a problem and its contextualization, are variables that can be capitalized on to enhance the teacher's capacity to implement innovative and integrated STEM programs effectively.

Keywords: iSTEM1, Inquiry2, mathematical modeling3, Integration model4, Science5. Mathematics6, Density7

Received: 26 Jan 2024; Accepted: 10 Apr 2024.

Copyright: © 2024 Manunure and Leung. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Mr. Kevin Manunure, Hong Kong Baptist University, Kowloon, Hong Kong, SAR China

People also looked at

Help | Advanced Search

Computer Science > Neural and Evolutionary Computing

Title: solving the food-energy-water nexus problem via intelligent optimization algorithms.

Abstract: The application of evolutionary algorithms (EAs) to multi-objective optimization problems has been widespread. However, the EA research community has not paid much attention to large-scale multi-objective optimization problems arising from real-world applications. Especially, Food-Energy-Water systems are intricately linked among food, energy and water that impact each other. They usually involve a huge number of decision variables and many conflicting objectives to be optimized. Solving their related optimization problems is essentially important to sustain the high-quality life of human beings. Their solution space size expands exponentially with the number of decision variables. Searching in such a vast space is challenging because of such large numbers of decision variables and objective functions. In recent years, a number of large-scale many-objectives optimization evolutionary algorithms have been proposed. In this paper, we solve a Food-Energy-Water optimization problem by using the state-of-art intelligent optimization methods and compare their performance. Our results conclude that the algorithm based on an inverse model outperforms the others. This work should be highly useful for practitioners to select the most suitable method for their particular large-scale engineering optimization problems.

Submission history

Access paper:.

• Other Formats

References & Citations

• Google Scholar
• Semantic Scholar

BibTeX formatted citation

Bibliographic and Citation Tools

Code, data and media associated with this article, recommenders and search tools.

• Institution

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

Study record managers: refer to the Data Element Definitions if submitting registration or results information.

Search for terms

• Advanced Search
• See Studies by Topic
• See Studies on Map
• How to Search
• How to Use Search Results
• How to Find Results of Studies
• How to Read a Study Record

• Learn About Studies
• Other Sites About Studies
• Glossary of Common Site Terms

• Submit Studies to ClinicalTrials.gov PRS
• Why Should I Register and Submit Results?
• FDAAA 801 and the Final Rule
• How to Apply for a PRS Account
• How to Register Your Study
• How to Edit Your Study Record
• How to Submit Your Results
• Frequently Asked Questions
• Support Materials
• Training Materials

• Selected Publications
• Clinical Alerts and Advisories
• Trends, Charts, and Maps
• Downloading Content for Analysis

• ClinicalTrials.gov Background
• About the Results Database
• History, Policies, and Laws
• ClinicalTrials.gov Modernization
• Media/Press Resources
• Linking to This Site
• Terms and Conditions
• Search Results
• Study Record Detail

Efficacy of Digital Problem Solving Application in Reduction of Anxiety, Depression and Substance Use Disorder Symptoms (Efficacy)

• Study Details
• Tabular View
• No Results Posted

Inclusion Criteria:

• Diagnosis of anxiety, depression or substance use disorder. Resident on or near Turtle Mountain or Spirit Lake reservation. Enrolled in federally recognized tribe.

Exclusion Criteria:

• Literacy below fourth-grade level. Diagnosis of personality disorder or schizophrenia, or other impaired perception of reality.
• For Patients and Families
• For Researchers
• For Study Record Managers
• Customer Support
• Accessibility
• Viewers and Players
• Freedom of Information Act
• HHS Vulnerability Disclosure
• U.S. National Library of Medicine
• U.S. National Institutes of Health
• U.S. Department of Health and Human Services

• Today's news
• Reviews and deals
• Climate change
• 2024 election
• Fall allergies
• Health news
• Mental health
• Sexual health
• Family health
• So mini ways
• Unapologetically
• Buying guides

Entertainment

• How to Watch
• My watchlist
• Stock market
• Biden economy
• Personal finance
• Stocks: most active
• Stocks: gainers
• Stocks: losers
• Trending tickers
• World indices
• US Treasury bonds
• Top mutual funds
• Highest open interest
• Highest implied volatility
• Currency converter
• Basic materials
• Communication services
• Consumer cyclical
• Consumer defensive
• Financial services
• Industrials
• Real estate
• Mutual funds
• Credit cards
• Credit card rates
• Balance transfer credit cards
• Business credit cards
• Cash back credit cards
• Rewards credit cards
• Travel credit cards
• Checking accounts
• Online checking accounts
• High-yield savings accounts
• Money market accounts
• Personal loans
• Student loans
• Car insurance
• Home buying
• Options pit
• Investment ideas
• Research reports
• Fantasy football
• Pro Pick 'Em
• College Pick 'Em
• Fantasy baseball
• Fantasy hockey
• Fantasy basketball
• Download the app
• Daily fantasy
• Scores and schedules
• GameChannel
• World Baseball Classic
• Premier League
• CONCACAF League
• Champions League
• Motorsports
• Horse racing
• Newsletters

New on Yahoo

• Privacy Dashboard

Applications for Albuquerque FBI’s Teen Academy now open

ALBUQUERQUE, N.M.  (KRQE) – Summer applications are now open for the Albuquerque FBI’s Teen Academy program. The academy explores a wide variety of topics designed to improve the student’s eye for details and problem-solving. Those who apply for the competitive program will undergo a background check.

Albuquerque Isotopes are back in town, having first Mariachis game

The academy will be held for three days in July. To apply for the academy, click this link .

For the latest news, weather, sports, and streaming video, head to KRQE NEWS 13 - Breaking News, Albuquerque News, New Mexico News, Weather, and Videos.

Recommended Stories

Last chance to land a free 65-inch tv (samsung's bogo deal ends today) — plus other sweet savings we found.

Also on our cheat sheet: No. 1 bestselling Fiskars pruners for just $10, a George Foreman grill for $14 and more!

Mercedes’ new EQS looks a lot more like an S-Class

The 2025 Mercedes-Benz EQS will look more like the S-Class than its predecessor does

New Balance CEO details the 'next big revolution' in sneakers

New Balance CEO Preston may not be a molecular biophysicist, but he knows that footwear innovation is the product of scientific research.

For women who stay single, the pressure is higher to save more for retirement

The number of single adult women in the US is at its highest level ever — and for many of them, the risk of running out of money in retirement is a serious concern.

iPhone users in 92 countries received a spyware attack warning from Apple

Apple notified individuals in 92 countries of possible mercenary spyware attacks on their systems.

Mortgage rates today, April 11, 2024: It could be a while before rates fall

Here are today's mortgage rates. Rates are staying pretty high as inflation increases. It might not be worth waiting for rates to drop. Lock in your rate today.

Why the Fed risks relearning the painful inflation lessons of the 1970s

The Fed is aiming to avoid a repeat of the double inflation episode that rocked the 1970s and early 1980s.

2024 Cadillac Lyriq Sport AWD Road Test: Old-school drive with new-school looks

Our road test of the 2024 Cadillac Lyriq in Sport 3 AWD form where we tell you all about how it drives and everything else you want to know.

Why big Medicare Advantage insurers may root for Biden to lose in 2024

Insurance giants have a bigger stake in this year’s presidential election after recent moves by the Biden administration cut into the profitability of Medicare Advantage plans.

Overwatch 2 introduces harsher punishments for players who leave mid-match

Leaving 10 out of 20 competitive games will get you banned for the rest of the season.

Alternative browsers report uplift after EU's DMA choice screen mandate

A flagship European Union digital market regulation appears to be shaking up competition in the mobile browser market. It's been a little over a month since the Digital Markets Act (DMA) came into application and there are early signs it's having an impact by forcing phone makers to show browser choice screens to users. On Wednesday, Reuters reported growth data shared by Cyprus-based web browser Aloha and others that it said suggests the new law is stirring the competitive pot and helping smaller browser makers gain share or at least grab more attention than they were.

Shohei Ohtani's ex-interpreter, Ippei Mizuhara, reportedly in negotiations to plead guilty in gambling scandal

Ohtani has been interviewed by the feds as part of the investigation, per the New York Times.

Caitlin Clark's No. 22 to be retired by Iowa Hawkeyes, school announces

Clark will be the third player in program history to receive the honor.

2024 NFL Draft: Favorite players outside the top 50

These aren't household names, but there are plenty of quality, if imperfect, prospects that should be available on Day 2 of the NFL Draft and beyond.

Police reportedly issue warrant for Chiefs WR Rashee Rice after multi-vehicle crash in Dallas

Rashee Rice is facing eight charges in connection with the hit-and-run crash in Texas late last month.

Varo Believe Card review: Build credit with 0% APR and no fees

Varo Bank's Believe Card has zero fees — not even interest charges. See how it could help improve your credit.

'Cute, stretchy and comfortable': These No. 1 bestselling no-show socks actually stay in place, shoppers say

'I love them so much that I wear them in the house after taking off my shoes,' says one fan.

Trump legal news brief: 3rd appeals court judge in 3 days rejects Trump motion to delay hush money trial

An appeals court judge rejects former President Donald Trump’s latest attempt to delay the start of the hush money trial on the grounds that presidential immunity protects him from being prosecuted.

Report: Alex Rodriguez and Marc Lore planned significant payroll cuts with Timberwolves

Alex Rodriguez and Marc Lore reportedly planned to bring the Timberwolves below the projected luxury tax threshold next season, which left Glen Taylor very concerned.

Save 30% on sunny spring and summer fashion with these Target Circle Week deals

Load up on the retailer's famously stylish clothing, shoes and accessories for less — here are our picks.

Wisconsin has prison problems. Government isn't doing much to solve them, advocates say.

MADISON - Two months after Gov. Tony Evers issued a statement vowing action on bleak circumstances in some of the state's prisons, little appears to have changed.

In two of the state's prisons, inmates remain largely confined to their cells as they have been for months with limited access to recreation, showers and visits from loved ones.

Advocates for inmates and their families say there's blame to go around — including to state lawmakers who ended their session last month without acting on legislation aimed at improving prison conditions — but they in particular point their finger at inaction from the administration of Gov. Tony Evers.

"I'm just shocked and dismayed and that it's not getting the kind of response that it needs, either from the Legislature or frankly, from the administration. I mean, this is everybody getting it wrong," said David Liners, the executive director of WISDOM, an advocacy organization aimed at supporting incarcerated individuals.

"People in prison are out of sight, out of mind, unless someone reminds people."

More: 'Time to hold somebody accountable': Family sues DOC over suicide at Waupun prison

Specifically, advocates say, Evers could act on his own without the Legislature and point to actions taken during the coronavirus pandemic that lowered prison populations by not incarcerating people who committed technical violations of their terms of supervision, in addition to sending newly sentenced people to county jails instead of crowded prisons.

"They can change some of these practices. Matter of fact, they did back during COVID,” Liners said. "In one fell swoop the Secretary of Corrections changed the policy and it reduced the prison population by 1,200 people.”

As of Dec. 31, 2023, 4,330 individuals were incarcerated due to revocation in Wisconsin, according to the DOC. Revocation means that the person is placed in prison because they violated a rule or condition of their supervision but does not necessarily mean that the person engaged in criminal behavior, DOC Communications Director Beth Hardtke said.

More: 'The screams were blood-curdling': Before Cameron Williams died at Waupun, prisoners say he begged staff for help

Evers said in an interview Monday that he visited Waupun recently as educational classes were being restarted. He said those incarcerated were "excited" about the programming.

When asked about the ongoing complaints about the conditions inside some of the state's prisons, Evers said they'll work to get bills over the finish line that address issues, noting that the composition of the Legislature could change after November's elections.

"We're working on it right now," he said. "We're going to have a new Legislature next time, and hopefully it will be helpful."

More: National Guard won't be sent to Wisconsin prisons, despite ongoing staffing shortages

Republicans who control the state Legislature also did not take action on proposed measures to address conditions inside Wisconsin's prisons, including the state's largest facility that is under federal investigation and has been on lockdown for months.

Advocates and family members of incarcerated individuals have protested the lack of attention paid to prisons, showing up outside with signs, music and microphones outside of many of the embattled corrections institutions. Rep. Ryan Clancy, D-Milwaukee, said he's heard from constituents about the events and their desire for change.

"Collectively, both the Legislature and the governor are making it worse," he said. "We should be making conditions in our jails better."

The inaction comes after former GOP Gov. Scott Walker and the Republican-led Legislature were criticized for years for doing little to address similarly dangerous conditions at the state's youth prison. A bill to close the prison wasn't passed until Walker ran for re-election in 2018 — a race Evers won after making Walker's inaction on prison conditions a campaign focus.

While campaigning in 2018, Evers also said he supported eventually reducing the state's prison population through the release of inmates for good behavior, expanding or creating court diversion programs, and treating 17-year-old offenders as juveniles instead of adults.

He has pursued the changes through the Republican-controlled state Legislature but GOP lawmakers have blocked the efforts.

Liners said the state Capitol is not conducive for solutions with Democrats and Republicans not seeing eye to eye on how best to reduce crime.

More: Green Bay prison is overrun with mice, inmate advocates say

"I'd love to see more bipartisan work," he said. "I think it's disingenuous to not do anything at all and then blame the people who actually do try to introduce the topic for introducing it wrong."

Evers has taken some steps recently to address issues in prisons, particularly the restriction of movement by inmates. Compensation for correctional officers was also raised, drawing in more applicants and reducing the levels of understaffing that contributed to the restricted conditions within prisons.

But even with those actions, Liners said, Evers could be doing more.

"He made a lot of promises when he first ran for governor, about reducing the prison population, and he's just stepped away from all that," Liners said.

"The governor has a responsibility for people in prison that is unique. He doesn't speak on criminal legal system issues hardly ever, and almost never speaks on people in prison. And he certainly isn't offering a lot of leadership on the issue and we find that really disappointing."

When asked about that campaign promise, a spokeswoman for Evers declined to answer questions, instead pointing to the governor's past budgets.

There has been an outcry over state prisons, run by the DOC, with several deaths, allegations of mistreatment and months-long lockdowns that have largely confined those incarcerated to their cells.

"The conditions in some of our prison institutions are unacceptable," Sen. Kelda Roys, D-Madison, said during an event in February.

"There is no excuse for the way people are being treated and there is no reason to incarcerate the vast number of people that we do."

Wisconsin had more than 22,000 people incarcerated as of March 29 in all of its facilities , a number much larger than neighboring Minnesota, which had 8,275 people incarcerated as of Jan. 1, the latest data available .

James Wilbur, a criminal justice reform advocate, said in early February after Evers announced that DOC would be examining the lockdowns, that there hadn't been much movement and that lack of action was hurting those incarcerated.

"The men in these facilities are continuing to be subjected to nothing less than vicious conditions. I really need to emphasize that word, that conditions that these men are being subjected to are absolutely vicious," he said.

"Think about what that kind of living situation would be like when you're subjected to a six by eight cell for up to 23 and one-half-hours a day. With no access to recreation, no access to non-artificial light, no programming. Your ability to communicate with your family and your friends is virtually unknown because you don't know if you're going to get a phone call."

Waupun, one of the state's five maximum-security prisons, has seen four in-custody deaths in the last eight months , with the first confirmed as a suicide, the second due to an overdose and the third as a result of a stroke. Cameron Williams, 24, was found dead in his cell in October after begging and yelling for help for days, only to be ignored by correctional officers, according to prisoners who spoke with the Journal Sentinel.

Waupun has been in a state of modified movement for nearly a year, meaning that those incarcerated are largely confined to their cells, and have complained about a lack of access to shower facilities, and little to no opportunities to interact with loved ones. The building itself has been a source of complaints due to its age.

Green Bay Correctional Institution has also been a focal point , due to ongoing restrictions there, as well as the deterioration of the building itself. Incarcerated individuals have reported an infestation of mice , a lack of access to natural daylight, a lack of access to showers and confinement to cells so small they barely have space to move around their cellmate.

Hardtke, the DOC spokesperson, said in an email that the department "continues to assess modified movement at the two facilities."

And DOC Secretary Kevin Carr announced his retirement in early March , leaving the embattled agency without leadership as it navigates the growing number of issues being raised in its prisons.

Legislation could create change within the prisons, improving conditions and providing more resources to those who need them. But for the most part, GOP lawmakers who control the state Legislature have avoided the issue, or allowed bills introduced to languish without a hearing.

And now that the Legislature has finished for 2024, some are wondering if more could have been done to help.

'They were misled'

Seventeen bills aimed at addressing the conditions inside Wisconsin's prisons were introduced last year, spearheaded by Rep. Darrin Madison, D-Milwaukee, and Clancy, but none came to fruition.

The bills covered a host of issues inside state prisons and would have raised the minimum wage for jobs done by those incarcerated, required facilities to allow people to bathe more frequently, required more visitations, more recreational time, and required that those in solitary confinement be provided with paper, pen, envelopes and their address book, as well as a toothbrush, toothpaste, shampoo and deodorant.

More: Feds investigating alleged smuggling of drugs, cell phones at Waupun Correctional

Advocates alongside the bill authors called the bills "a step forward," and expressed hope that they would bring about change.

John McCray Jones, a policy analyst with the Wisconsin ACLU, said at a November news conference that Wisconsin needs to work to reduce the number of people it incarcerates but the legislation was a move to help people.

"These bills respect the human dignity of incarcerated people," he said. "They improve public safety by fostering an environment that leads to rehabilitation."

None of the bills received a hearing, let alone a vote.

Clancy said the lack of movement on the bills was frustrating.

"It is unfortunate that we have yet to see even a hearing for any of those 17 bills, because we all agree that there are problems," he said. "And yet we're not even giving a hearing for it, for any of the potential solutions."

Rep. Michael Schraa, a Republican from Oshkosh and the chair of the Assembly Committee on Corrections, has been working for years on improving the corrections system, including co-authoring the push to close the Lincoln Hills School for Boys and Copper Lake School for Girls youth prison when problems were brought to light in 2015.

More: Gov. Evers says independent investigation into Wisconsin prison deaths possible

While he said some of the bills introduced by Democrats could have made it over the finish line, there was no effort to work in a bipartisan manner by the authors of the package and the way they were announced was a "publicity stunt."

"There were a couple (bills) in there that I think we could have worked on some things together, but not when they're introduced by the minority party on their own," he said. "I think there were a lot of people that were counting on the possibility of some of those bills actually getting hearings, and they were really misled."

Clancy disputed that claim, saying his office had several appointments scheduled with Schraa's that were all canceled.

"It's just the opposite. I mean, we spent months and months reaching out to stakeholders reaching out for Republicans," he said.

Clancy said loved ones of incarcerated people met with Republicans, too, asking for them to support the bills.

"Formerly incarcerated people and their advocates, they've been waiting for some of this legislation for decades. And they are they were overjoyed that it was even put out there," Clancy said. "But I think we owe it to them to take the next step, give them hearings and actually pass these in the law."

Some bills have received bipartisan support and were recently signed by Evers.

One focuses on treating prisoners struggling with serious mental illness and another provides an alternative to solitary confinement for Wisconsin prisons' sickest inmates ― the only bill that addresses conditions inside prisons this session.

The new law follows the death of Dean Hoffman, whose suicide last year at Waupun came after his family said correctional officers didn't take his mental health needs seriously, despite his documented challenges, including bipolar disorder and suicidal thoughts.

More: Senators question DOC secretary Kevin Carr over prison staffing challenges, problems at Waupun and Green Bay

Another newly signed law aims to help people released from prison obtain jobs by providing a state-run hotline and website to help employers hire individuals with previous conviction records.

More is needed, advocates say.

"The problem is we have too many people in the prisons. We've got some facilities that just shouldn't be there," said Liners, the advocate. "If we just did the common sense thing of getting the people out of prisons that can be gotten out safely, with no danger to anybody, we could solve a lot of these problems quickly."

Laura Schulte can be reached at [email protected] and on X at  @SchulteLaura .

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

5.6: Application Problems with Exponential and Logarithmic Functions

• Last updated
• Save as PDF
• Page ID 38600

• Rupinder Sekhon and Roberta Bloom
• De Anza College

Learning Objectives

In this section, you will:

• review strategies for solving equations arising from exponential formulas
• solve application problems involving exponential functions and logarithmic functions

STRATEGIES FOR SOLVING EQUATIONS THAT CONTAIN EXPONENTS

When solving application problems that involve exponential and logarithmic functions, we need to pay close attention to the position of the variable in the equation to determine the proper way solve the equation we investigate solving equations that contain exponents.

Suppose we have an equation in the form : value = coefficient(base) exponent

We consider four strategies for solving the equation:

STRATEGY A : If the coefficient, base, and exponent are all known, we only need to evaluate the expression for coefficient(base) exponent to evaluate its value.

STRATEGY B: If the variable is the coefficient, evaluate the expression for (base) exponent . Then it becomes a linear equation which we solve by dividing to isolate the variable.

STRATEGY C : If the variable is in the exponent, use logarithms to solve the equation.

STRATEGY D: If the variable is not in the exponent, but is in the base, use roots to solve the equation.

Below we examine each strategy with one or two examples of its use.

Example \(\PageIndex{1}\)

Suppose that a stock’s price is rising at the rate of 7% per year, and that it continues to increase at this rate. If the value of one share of this stock is $43 now, find the value of one share of this stock three years from now.

The problem tells us that \(a\) = 43 and \(r\) = 0.07, so \(b = 1+ r = 1+ 0.07 = 1.07\)

Therefore, function is \(y = 43(1.07)^t\).

In this case we know that \(t\) = 3 years, and we need to evaluate \(y\) when \(t\) = 3.

At the end of 3 years, the value of this one share of this stock will be

\[y=43(1.07)^{3}=\$ 52.68 \nonumber \]

Example \(\PageIndex{2}\)

The value of a new car depreciates (decreases) after it is purchased. Suppose that the value of the car depreciates according to an exponential decay model. Suppose that the value of the car is $12000 at the end of 5 years and that its value has been decreasing at the rate of 9% per year. Find the value of the car when it was new.

The function is \(y = a(0.91)^t\)

In this case we know that when \(t\) = 5, then \(y\) = 12000; substituting these values gives

\[12000 = a(0.91)^5 \nonumber \]

We need to solve for the initial value a, the purchase price of the car when new.

First evaluate (0.91) 5 ; then solve the resulting linear equation to find \(a\).

\[ 1200 = a(0.624) \nonumber \]

\(a=\frac{12000}{0.624} = \$ 19,230.77\); The car's value was $19,230.77 when it was new.

Example \(\PageIndex{3}\)

A national park has a population of 5000 deer in the year 2016. Conservationists are concerned because the deer population is decreasing at the rate of 7% per year. If the population continues to decrease at this rate, how long will it take until the population is only 3000 deer?

\(r\) = -0.07 and \(b = 1+r = 1+(-0.07) = 0.93\) and the initial population is \(a\) = 5000

The exponential decay function is \(y = 5000(0.93)^t\)

To find when the population will be 3000, substitute \(y\) = 3000

\[ 3000 = 5000(0.93)^t \nonumber \]

Next, divide both sides by 5000 to isolate the exponential expression

\[\begin{array}{l} \frac{3000}{5000}=\frac{5000}{5000}(0.93)^{2} \\ 0.6=0.93^{t} \end{array} \nonumber \]

Rewrite the equation in logarithmic form; then use the change of base formula to evaluate.

\[t=\log _{0.93}(0.6) \nonumber \]

\(t = \frac{\ln(0.6)}{\ln(0.93)}=7.039\) years; After 7.039 years, there are 3000 deer.

Note: In Example \(\PageIndex{3}\), we needed to state the answer to several decimal places of precision to remain accurate. Evaluating the original function using a rounded value of \(t\) = 7 years gives a value that is close to 3000, but not exactly 3000.

\[y=5000(0.93)^{7}=3008.5 \text { deer } \nonumber \]

However using \(t\) = 7.039 years produces a value of 3000 for the population of deer

\[ y=5000(0.93)^{7.039}=3000.0016 \approx 3000 \text { deer } \nonumber \]

Example \(\PageIndex{4}\)

A video posted on YouTube initially had 80 views as soon as it was posted. The total number of views to date has been increasing exponentially according to the exponential growth function \(y = 80e^{0.2t}\), where \(t\) represents time measured in days since the video was posted. How many days does it take until 2500 people have viewed this video?

Let \(y\) be the total number of views \(t\) days after the video is initially posted. We are given that the exponential growth function is \(y = 80e^{0.2t}\) and we want to find the value of \(t\) for which \(y\) = 2500. Substitute \(y\) = 2500 into the equation and use natural log to solve for \(t\).

\[2500 = 80e^{0.12t} \nonumber \]

Divide both sides by the coefficient, 80, to isolate the exponential expression.

\[\begin{array}{c} \frac{2500}{80}=\frac{80}{80} e^{0.12 t} \\ 31.25=e^{0.12 t} \end{array} \nonumber \]

Rewrite the equation in logarithmic form

\[ 0.12t = \ln(31.25) \nonumber \]

Divide both sides by 0.04 to isolate \(t\); then use your calculator and its natural log function to evaluate the expression and solve for \(t\).

\[\begin{array}{l} \mathrm{t}=\frac{\ln (31.25)}{0.12} \\ \mathrm{t}=\frac{3.442}{0.12} \\ \mathrm{t} \approx 28.7 \text { days } \end{array} \nonumber \]

This video will have 2500 total views approximately 28.7 days after it was posted.

STRATEGY D: If the variable is not in the exponent, but is in the base, we use roots to solve the equation. It is important to remember that we only use logarithms when the variable is in the exponent.

Example \(\PageIndex{5}\)

A statistician creates a website to analyze sports statistics. His business plan states that his goal is to accumulate 50,000 followers by the end of 2 years (24 months from now). He hopes that if he achieves this goal his site will be purchased by a sports news outlet. The initial user base of people signed up as a result of pre-launch advertising is 400 people. Find the monthly growth rate needed if the user base is to accumulate to 50,000 users at the end of 24 months.

Let \(y\) be the total user base \(t\) months after the site is launched.

The growth function for this site is \(y = 400(1+r)^t\);

We don’t know the growth rate \(r\). We do know that when \(t\) = 24 months, then \(y\) = 50000.

Substitute the values of \(y\) and \(t\); then we need to solve for \(r\).

\[5000 = 400(1+r)^{24} \nonumber \]

Divide both sides by 400 to isolate (1+r) 24 on one side of the equation

\[\begin{array}{l} \frac{50000}{400}=\frac{400}{400}(1+r)^{24} \\ 125=(1+r)^{24} \end{array} \nonumber \]

Because the variable in this equation is in the base, we use roots:

\[\begin{array}{l} \sqrt[24]{125}=1+r \\ 125^{1 / 24}=1+r \\ 1.2228 \approx 1+r \\ 0.2228 \approx r \end{array} \nonumber \]

The website’s user base needs to increase at the rate of 22.28% per month in order to accumulate 50,000 users by the end of 24 months.

Example \(\PageIndex{6}\)

A fact sheet on caffeine dependence from Johns Hopkins Medical Center states that the half life of caffeine in the body is between 4 and 6 hours. Assuming that the typical half life of caffeine in the body is 5 hours for the average person and that a typical cup of coffee has 120 mg of caffeine.

• Write the decay function.
• Find the hourly rate at which caffeine leaves the body.
• How long does it take until only 20 mg of caffiene is still in the body? www.hopkinsmedicine.org/psyc...fact_sheet.pdf

a. Let \(y\) be the total amount of caffeine in the body \(t\) hours after drinking the coffee.

Exponential decay function \(y = ab^t\) models this situation.

The initial amount of caffeine is \(a\) = 120.

We don’t know \(b\) or \(r\), but we know that the half- life of caffeine in the body is 5 hours. This tells us that when \(t\) = 5, then there is half the initial amount of caffeine remaining in the body.

\[\begin{array}{l} y=120 b^{t} \\ \frac{1}{2}(120)=120 b^{5} \\ 60=120 b^{5} \end{array} \nonumber \]

Divide both sides by 120 to isolate the expression \(b^5\) that contains the variable.

\[\begin{array}{l} \frac{60}{120}=\frac{120}{120} \mathrm{b}^{5} \\ 0.5=\mathrm{b}^{5} \end{array} \nonumber \]

The variable is in the base and the exponent is a number. Use roots to solve for \(b\):

\[\begin{array}{l} \sqrt[5]{0.5}=\mathrm{b} \\ 0.5^{1 / 5}=\mathrm{b} \\ 0.87=\mathrm{b} \end{array} \nonumber \]

We can now write the decay function for the amount of caffeine (in mg.) remaining in the body \(t\) hours after drinking a cup of coffee with 120 mg of caffeine

\[y=f(t)=120(0.87)^{t} \nonumber \]

b. Use \(b = 1 + r\) to find the decay rate \(r\). Because \(b = 0.87 < 1\) and the amount of caffeine in the body is decreasing over time, the value of \(r\) will be negative.

\[\begin{array}{l} 0.87=1+r \\ r=-0.13 \end{array} \nonumber \]

The decay rate is 13%; the amount of caffeine in the body decreases by 13% per hour.

c. To find the time at which only 20 mg of caffeine remains in the body, substitute \(y\) = 20 and solve for the corresponding value of \(t\).

\[\begin{array}{l} y=120(.87)^{t} \\ 20=120(.87)^{t} \end{array} \nonumber \]

Divide both sides by 120 to isolate the exponential expression.

\[\begin{array}{l} \frac{20}{120}=\frac{120}{120}\left(0.87^{t}\right) \\ 0.1667=0.87^{t} \end{array} \nonumber \]

Rewrite the expression in logarithmic form and use the change of base formula

\[\begin{array}{l} t=\log _{0.87}(0.1667) \\ t=\frac{\ln (0.1667)}{\ln (0.87)} \approx 12.9 \text { hours } \end{array} \nonumber \]

After 12.9 hours, 20 mg of caffeine remains in the body.

EXPRESSING EXPONENTIAL FUNCTIONS IN THE FORMS y = ab t and y = ae kt

Now that we’ve developed our equation solving skills, we revisit the question of expressing exponential functions equivalently in the forms \(y = ab^t\) and \(y = ae^{kt}\)

We’ve already determined that if given the form \(y = ae^{kt}\), it is straightforward to find \(b\).

Example \(\PageIndex{7}\)

For the following examples, assume \(t\) is measured in years.

• Express \(y = 3500e^{0.25t}\) in form \(y = ab^t\) and find the annual percentage growth rate.
• Express \(y = 28000e^{-0.32t}\) in form \(y = ab^t\) and find the annual percentage decay rate.

a. Express \(y = 3500e^{0.25t}\) in the form \(y = ab^t\)

\[\begin{array}{l} y=a e^{k t}=a b^{t} \\ a\left(e^{k}\right)^{t}=a b^{t} \end{array} \nonumber \]

Thus \(e^k=b\)

In this example \(b=e^{0.25} \approx 1.284\)

We rewrite the growth function as y = 3500(1.284 t )

To find \(r\), recall that \(b = 1+r\) \[\begin{aligned} &1.284=1+r\\ &0.284=\mathrm{r} \end{aligned} \nonumber \]

The continuous growth rate is \(k\) = 0.25 and the annual percentage growth rate is 28.4% per year.

b. Express \(y = 28000e^{-0.32t}\) in the form \(y = ab^t\)

In this example \(\mathrm{b}=e^{-0.32} \approx 0.7261\)

We rewrite the growth function as y = 28000(0.7261 t )

To find \(r\), recall that \(b = 1+r\) \[\begin{array}{l} 0.7261=1+r \\ 0.2739=r \end{array} \nonumber \]

The continuous decay rate is \(k\) = -0.32 and the annual percentage decay rate is 27.39% per year.

In the sentence, we omit the negative sign when stating the annual percentage decay rate because we have used the word “decay” to indicate that r is negative.

Example \(\PageIndex{8}\)

• Express \(y = 4200 (1.078)^t\) in the form \(y =ae^{kt}\)
• Express \(y = 150 (0.73)^t\) in the form \(y =ae^{kt}\)

a. Express \(y = 4200 (1.078)^t\) in the form \(y =ae^{kt}\)

\[\begin{array}{l} \mathrm{y}=\mathrm{a} e^{\mathrm{k} t}=\mathrm{ab}^{\mathrm{t}} \\ \mathrm{a}\left(e^{\mathrm{k}}\right)^{\mathrm{t}}=\mathrm{ab}^{\mathrm{t}} \\ e^{\mathrm{k}}=\mathrm{b} \\ e^{k}=1.078 \end{array} \nonumber \]

Therefore \(\mathrm{k}=\ln 1.078 \approx 0.0751\)

We rewrite the growth function as \(y = 3500e^{0.0751t}\)

b. Express \(y =150 (0.73)^t\) in the form \(y = ae^{kt}\)

\[\begin{array}{l} y=a e^{k t}=a b^{t} \\ a\left(e^{k}\right)^{t}=a b^{t} \\ e^{k}=b \\ e^{k}=0.73 \end{array} \nonumber \]

Therefore \(\mathrm{k}=\ln 0.73 \approx-0.3147\)

We rewrite the growth function as \(y = 150e^{-0.3147t}\)

AN APPLICATION OF A LOGARITHMIC FUNCTON

Suppose we invest $10,000 today and want to know how long it will take to accumulate to a specified amount, such as $15,000. The time \(t\) needed to reach a future value \(y\) is a logarithmic function of the future value: \(t = g(y)\)

Example \(\PageIndex{9}\)

Suppose that Vinh invests $10000 in an investment earning 5% per year. He wants to know how long it would take his investment to accumulate to $12000, and how long it would take to accumulate to $15000.

We start by writing the exponential growth function that models the value of this investment as a function of the time since the $10000 is initially invested

\[y=10000(1.05)^{t} \nonumber \]

We divide both sides by 10000 to isolate the exponential expression on one side.

\[\frac{y}{10000}=1.05^{t} \nonumber \]

Next we rewrite this in logarithmic form to express time as a function of the accumulated future value. We’ll use function notation and call this function \(g(y)\).

\[\mathrm{t}=\mathrm{g}(\mathrm{y})=\log _{1.05}\left(\frac{\mathrm{y}}{10000}\right) \nonumber \]

Use the change of base formula to express \(t\) as a function of \(y\) using natural logarithm:

\[\mathrm{t}=\mathrm{g}(\mathrm{y})=\frac{\ln \left(\frac{\mathrm{y}}{10000}\right)}{\ln (1.05)} \nonumber \]

We can now use this function to answer Vinh’s questions.

To find the number of years until the value of this investment is $12,000, we substitute \(y\) = $12,000 into function \(g\) and evaluate \(t\):

\[\mathrm{t}=\mathrm{g}(12000)=\frac{\ln \left(\frac{12000}{10000}\right)}{\ln (1.05)}=\frac{\ln (1.2)}{\ln (1.05)}=3.74 \text { years } \nonumber \]

To find the number of years until the value of this investment is $15,000, we substitute \(y\) = $15,000 into function \(g\) and evaluate \(t\):

\[\mathrm{t}=\mathrm{g}(15000)=\frac{\ln \left(\frac{15000}{10000}\right)}{\ln (1.05)}=\frac{\ln (1.5)}{\ln (1.05)}=8.31 \text { years } \nonumber \]

Before ending this section, we investigate the graph of the function \(\mathrm{t}=\mathrm{g}(\mathrm{y})=\frac{\ln \left(\frac{\mathrm{y}}{10000}\right)}{\ln (1.05)}\). We see that the function has the general shape of logarithmic functions that we examined in section 5.5. From the points plotted on the graph, we see that function \(g\) is an increasing function but it increases very slowly.

If we consider just the function \(\mathrm{t}=\mathrm{g}(\mathrm{y})=\frac{\ln \left(\frac{\mathrm{y}}{10000}\right)}{\ln (1.05)}\), then the domain of function would be \(y > 0\), all positive real numbers, and the range for \(t\) would be all real numbers.

In the context of this investment problem, the initial investment at time \(t\) = 0 is \(y\) =$10,000. Negative values for time do not make sense. Values of the investment that are lower than the initial amount of $10,000 also do not make sense for an investment that is increasing in value.

Therefore the function and graph as it pertains to this problem concerning investments has domain \(y ≥ 10,000\) and range \(t ≥ 0\).

The graph below is restricted to the domain and range that make practical sense for the investment in this problem.