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Math problems and solutions on integers.

Problems related to integer numbers in mathematics are presented along with their solutions.

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How to Solve Integers and Their Properties

Last Updated: April 6, 2024

This article was reviewed by Joseph Meyer . Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 29,623 times.

An integer is a set of natural numbers, their negatives, and zero. However, some integers are natural numbers, including 1, 2, 3, and so on. Their negative values are, -1, -2, -3, and so on. So integers are the set of numbers including (…-3, -2, -1, 0, 1, 2, 3,…). An integer is never a fraction, decimal, or percentage, it can only be a whole number. To solve integers and use their properties, learn to use addition and subtraction properties and use multiplication properties.

Using Addition and Subtraction Properties

• a + b = c (where both a and b are positive numbers the sum c is also positive)
• For example: 2 + 2 = 4

• -a + -b = -c (where both a and b are negative, you get the absolute value of the numbers then you proceed to add, and use the negative sign for the sum)
• For example: -2+ (-2)=-4

• a + (-b) = c (when your terms are of different signs, determine the larger number's value, then get the absolute value of both terms and subtract the lesser value from the larger value. Use the sign of the larger number for the answer.)
• For example: 5 + (-1) = 4

• -a +b = c (get the absolute value of the numbers and again, proceed to subtract the lesser value from the larger value and assume the sign of the larger value)
• For example: -5 + 2 = -3

• An example of the additive identity is: a + 0 = a
• Mathematically, the additive identity looks like: 2 + 0 = 2 or 6 + 0 = 6

• The additive inverse is when a number is added to the negative equivalent of itself.
• For example: a + (-b) = 0, where b is equal to a
• Mathematically, the additive inverse looks like: 5 + -5 = 0

• For example: (5+3) +1 = 9 has the same sum as 5+ (3+1) = 9

Using Multiplication Properties

• When a and b are positive numbers and not equal to zero: +a * + b = +c
• When a and b are both negative numbers and not equal to zero: -a*-b = +c

• However, understand that any number multiplied by zero, equals zero.

• For example: a(b+c) = ab + ac
• Mathematically, this looks like: 5(2+3) = 5(2) + 5(3)
• Note that there is no inverse property for multiplication because the inverse of a whole number is a fraction, and fractions are not an element of integer.

Joseph Meyer

The distributive property helps you avoid repetitive calculations. You can use the distributive property to solve equations where you must multiply a number by a sum or difference. It simplifies calculations, enables expression manipulation (like factoring), and forms the basis for solving many equations.

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Integer Word Problems

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• Word Problems On Integers

Integers: Word Problems On Integers

An arithmetic operation is an elementary branch of mathematics. Arithmetical operations include addition, subtraction, multiplication and division. Arithmetic operations are applicable to different types of numbers including integers.

Integers are a special group of numbers that do not have a fractional or a decimal part. It includes positive numbers, negative numbers and zero.  Arithmetic operations on integers are similar to that of whole numbers. Since integers can be positive or negative numbers i.e. as these numbers are preceded either by a positive (+) or a negative sign (-), it makes them a little confusing concept. Therefore, they are different from whole numbers . Let us now see how various arithmetical operations can be performed on integers with the help of a few word problems. Solve the following word problems using various rules of operations of integers.

Word problems on integers Examples:

Example 1: Shyak has overdrawn his checking account by Rs.38.  The bank debited him Rs.20 for an overdraft fee.  Later, he deposited Rs.150.  What is his current balance?

Solution:  Given,

Total amount deposited= Rs. 150

Amount overdrew by Shyak= Rs. 38

Amount charged by bank= Rs. 20

⇒ Debit amount= -20

Total amount debited = (-38) + (-20) = -58

Current balance= Total deposit +Total Debit

Hence, the current balance is Rs. 92.

Example 2: Anna is a microbiology student. She was doing research on optimum temperature for the survival of different strains of bacteria. Studies showed that bacteria X need optimum temperature of -31˚C while bacteria Y need optimum temperature of -56˚C. What is the temperature difference?

Solution: Given,

Optimum temperature for bacteria X = -31˚C

Optimum temperature for bacteria Y= -56˚C

Temperature difference= Optimum temperature for bacteria X – Optimum temperature for bacteria Y

⇒ (-31) – (-56)

Hence, temperature difference is 25˚C.

Example 3: A submarine submerges at the rate of 5 m/min. If it descends from 20 m above the sea level, how long will it take to reach 250 m below sea level?

Initial position = 20 m    (above sea level)

Final position = 250 m    (below sea level)

Total depth it submerged = (250+20) = 270 m

Thus, the submarine travelled 270 m below sea level.

Time taken to submerge 1 meter = 1/5 minutes

Time taken to submerge 270 m = 270 (1/5) = 54 min

Hence, the submarine will reach 250 m below sea level in 54 minutes.

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Integer Word Problems

Welcome to the fascinating world of integer word problems! Don’t let the fancy name scare you off; these problems might be easier and more fun than you think. Simplifying them is handy in daily life, and they’ll reappear in various forms throughout your academic journey. Let’s dive into the fundamental components.

What are Integer Word Problems?

In essence, integer word problems are mathematical problems involving number-related questions in the form of a story or practical situation. Specifically, these problems use integers — whole numbers that can be positive, negative, or zero. For instance, you might be asked how many more books Mike read than Sarah if Mike reads 15 and Sarah reads 7. Since you’re subtracting 7 from 15, you’re dealing with an integer word problem.

Importance of Solving Integer Word Problems

Mastering integer word problems plays a significant role in building your mathematical expertise. They help improve your problem-solving skills and enhance your ability to think logically and critically. Moreover, these problems are a cornerstone of real-world situations. Whether you are calculating the distance between two cities, determining profit and loss in business, or even figuring out temperature changes, integers and their problems come into play.

How to Solve Integer Word Problems

Are you ready to tackle integer word problems? Here are a few steps:

• Understand the Problem:  Breaking the problem into smaller parts makes it less daunting. Take your time to understand what the problem is about.
• Identify the Key Information:  Highlight or underline important facts or figures in the problem. Look for clues indicating whether you’re dealing with addition, subtraction, or a combination.
• Formulate a Plan:  Write down your actions to arrive at the solution.
• Execute Your Plan:  Apply the actions you’ve mapped out to solve the problem.
• Verify Your Answer:  Always double-check your outcome. Does it make sense in terms of the problem?

In dealing with integer word problems, practice is critical. The more problems you tackle, the more proficient you become. Happy problem-solving!

Basic Concepts of Integers

As a student or math enthusiast, knowing and mastering the basic concepts of integers will help you understand and tackle integer word problems better. In this section, we’ll delve into the definitions of integers, further distinguishing between positive and negative integers.

Defining Integers

Integers  are a number category that includes all the whole numbers, their opposites (negative counterparts), and zero. They are distinct from fractions, decimals, and percents. An integer can be a zero, a positive, or a negative whole number. The set of integers is denoted mathematically as {…, -3, -2, -1, 0, 1, 2, 3}. These numbers form the backbone of many mathematical operations and concepts, especially in algebra.

Positive and Negative Integers

Positive and negative integers make understanding and calculating many real-world situations better and more efficient.

Positive integers , often natural numbers , are numbers greater than zero. They are frequently used to denote weight, distance, or money values. However, not all situations can be expressed with positive numbers; sometimes, we must resort to negative ones.

Negative integers are the opposites of natural numbers, excluding zero, and fall below zero on the number line. They are typically used when something is decreased, removed, or lost. An excellent example of using negative integers is in banking, where they represent debt. Or in meteorology, where they represent temperatures below zero.

Understanding the concept of positive and negative integers is paramount because they are central to successfully dealing with integer word problems. In the next segment, we will dive deeper into strategies for solving these problems, so tighten your seatbelts as we explore a fun section of the mathematical world.

Addition and Subtraction Word Problems

When it comes to integers, understanding how to add and subtract these numbers is crucial, taking center stage in everyday mathematical operations. While learning, students begin grappling with word problems – mathematical problems presented in the form of a narrative or story – which include real-world scenarios. These serve as a bridge for children and adults to apply theoretical knowledge practically.

Adding and Subtracting Integers

In terms of  adding integers , there are a few rules to remember. If the integers have the same sign, add their absolute values and keep the standard sign. On the flip side, when the integers have different signs, subtract the smaller absolute value from the larger one and give the solution the sign of the number with the more considerable absolute value.

Subtracting integers , however, involves an additional step. More specifically, any subtraction can be reinterpreted as an addition. To subtract an integer, add its opposite. For example, to subtract -3 from 5 (5 – -3), we add 3 to 5 (5 + 3), with the sum coming to 8.

Real-life Examples of Addition and Subtraction Word Problems

Let’s explore a few word problems that imitate daily life scenarios. Suppose a child has £5 and they want to buy a toy that costs £10. How many more pounds do they need? The problem here is 10 – 5, which equals 5. Thus, the child needs five more pounds.

In another situation, imagine the temperature was 5 degrees Celsius in the morning and dropped 3 degrees by the afternoon. What’s the temperature now? Here, we have 5 – 3 = 2. The answer is 2 degrees Celsius.

These examples illustrate how adding and subtracting integers can help us solve practical problems and better understand the world. We encourage you to find your examples and practice to enhance your understanding and mastery of this fundamental mathematical skill.

Multiplication and Division Word Problems

As the journey of discovery with integers continues, multiplication and division of these numbers become an integral part of our everyday mathematical activities. Understanding how to tackle word problems – mathematical problems in narrative form – becomes critical. Specifically, multiplication and division integer word problems provide the groundwork for applying knowledge practically in real-world situations.

Multiplying and Dividing Integers

Multiplying integers might initially seem complex , but it becomes straightforward once you grasp the core concept. When multiplying two integers, the result will be positive if the signs are the same (positive or negative). However, if the signs are different (positive and negative), the result will be a negative integer.

Dividing integers  follows a similar concept. If the integers have the same sign, the quotient is positive, and if they have different signs, it is negative.

Application of Multiplication and Division Word Problems

Now, let’s see how these concepts apply in real-world scenarios. Suppose a person has $20 and wants to buy as many chocolates as possible, with each chocolate bar costing $4. In this case, they’d need to divide 20 by 4. The question boils down to 20 ÷ 4, which equals 5. So, they can buy five chocolate bars.

Considering multiplication, imagine a scenario where a store sells packages of bottled drinking water. Each package contains six bottles, and the store has twenty packages. To calculate the total number of bottles, you would multiply 6 (bottles per package) by 20 (number of packages), getting 6 x 20 = 120. So, the store has 120 bottled water.

These real-world examples show how multiplication and division word problems offer practical ways to understand and apply mathematical knowledge. Engaging with these problems enhances understanding of fundamental math concepts and promotes problem-solving skills crucial for daily life.

Multi-Step Word Problems

In a journey through mathematics, we commonly encounter complex multi-step word problems. These problems often involve multiple operations using integers , such as addition, subtraction, multiplication, and division. Solving these tasks enhances problem-solving skills, logical thinking, and mathematical proficiency. This part will delve into complex integer word problems and introduce strategies for solving multi-step problems.

Complex Integer Word Problems

Complex integer word problems  involve more than one mathematical operation, often requiring a systematic approach to reach the solution. For instance, imagine a scenario where a garden filled with 120 roses and petunias is being prepared for a garden show. There are twice as many roses as there are petunias. The question is, “How many petunias are there?”

Here, the problem will be solved in two steps. First, understanding that the number of roses is twice that of petunias. That means, if we denote the number of petunias as ‘p,’ then the number of roses is ‘2p’. The total quantity of flowers (120) is the sum of roses and petunias, leading to the equation 2p + p = 120. Solving this equation provides the number of petunias. Since multi-step word problems rely heavily on integers, understanding their operation rules is essential.

Strategies for Solving Multi-Step Word Problems

Solving multi-step word problems  can seem daunting, but a systematic approach simplifies the task. Below are vital strategies:

• Understand the Problem:  Read the problem carefully, ensure you understand what it’s asking, and identify the operations needed.
• Develop a Plan:  Break down the problem into smaller, manageable steps. Form equations if needed.
• Solve:  Carry out each operation. Ensure your calculations are correct at each step.
• Check Your Answer:  Review your solution, ensuring you answered the initial question correctly. Doing this validates that your solution aligns with the problem’s conditions.

Remember, practice significantly improves problem-solving skills and the ability to tackle complex multi-step word problems involving integers. Happy problem-solving!

Common Mistakes and Tips for Success

In particular, integer word problems can sometimes throw you off course. Like every journey, it is customary to make mistakes along the way. However, understanding and learning from these common errors can help you avoid detours and get you on the fast track to mastery.

Common Errors in Solving Integer Word Problems

Misinterpretation  is one of the most common mistakes when handling integer word problems. Often, students need to understand the operations required or interpret the relationship between the integers presented in the problem.

Inaccurate Calculations  – Integers include both positive and negative numbers, and it is easy to miscalculate when it comes to subtraction, addition, or other operations involving such numbers. For example, subtracting a negative integer leads to an addition instead.

Helpful Tips and Tricks for Solving Integer Word Problems

Once you’re aware of common pitfalls, arm yourself with the right strategies to navigate your way through complex integer word problems adeptly.

Thorough Understanding:  Read the integer word problem carefully and understand what is being asked. It can be helpful to jot down essential information or even draw diagrams to visualize the problem.

Plan:  Make a plan. Break the problem down into smaller, solvable parts and create equations representing each step of the problem.

Check Your Work:  After solving, double-check your calculations to ensure accuracy. Compare your answer with the original question to see if it makes sense.

Practice:  Just like anything, practice makes perfect. The more problems you solve, the more comfortable you become with integers and their operations.

Always remember making mistakes is part of the learning process. By staying aware and utilizing strategies, you’ll soon find yourself an expert at solving integer word problems. Happy Practicing!

Practice Exercises

Knowing the common errors and tips for solving integer word problems, it is time to put that knowledge into practice. With the right amount of practice, anyone can enhance their skills in solving such problems. With that in mind, let’s tackle some practice exercises to understand integer word problems further.

Practice Problems for Integer Word Problems

Here are some various types of integer word problems. Remember to read carefully, understand what’s asked, and plan your solution before jumping into the problem.

• Maria has $15 in her pocket. She spends $7 on a movie and $6 on snacks. Write an integer to represent Maria’s money situation and calculate how much she has left.
• At the start of the week, the temperature is 5 degrees. The temperature then drops by 7 degrees the next day. What is the temperature now?
• A company lost $2000 this year, 3 times the amount they lost last year. How much did the company lose last year?

Step-by-Step Solutions for Practice Exercises

Let’s walk through the solutions together to help you understand how these problems are solved.

• Maria has $15. She spent $7 and $6. This expenditure is a loss, so we represent it with negative integers. So, the situation becomes: 15 + (-7) + (-6) = 2. Maria has $2 left.
• The temperature is 5 degrees initially. Then, it drops by 7 degrees (a decrease is a negative operation). So, the situation is 5 + (-7) = -2 degrees. The temperature is now -2 degrees.
• Let’s denote the amount of money the company lost last year as x. We know that 3x = $2000. So, x = $2000 / 3 = $666.67. The company lost around $666.67 last year.

Do more exercises and get comfortable with solving integer word problems. It may take some time, but you will get there with consistent practice. Remember, avoiding rushing and breaking the problem into smaller parts can be very helpful. Practicing will make you better at solving integer word problems effectively and efficiently. Happy learning!

Emerging victorious in integer word problems opens up an exciting facet of mathematical knowledge. After all, these problems translate mathematical concepts into real-world scenarios, thereby cultivating critical thinking skills. Let’s explore the benefits of mastering integer word problems and round off with a few parting thoughts.

Benefits of Mastering Integer Word Problems

Boosts Problem-solving Skills:  Integer word problems are an ideal way to sharpen problem-solving skills. They compel one to think logically and systematically about how to apply mathematical operations accurately.

Enhances Numerical Literacy: With a firm grasp of integers, people can better comprehend numerical information daily. For instance, understanding debt and assets or gain and loss in finance becomes clearer.

Encourages Diversity of Thought:  Integer word problems offer multiple ways to find a solution, fostering creativity. It encourages diverse approaches to problem-solving.

Promotes Practical Application:  Integers have ubiquitous applications in diverse fields, including science, engineering, and information technology. Being comfortable with integer word problems equips one with skills applicable to these areas.

Final Thoughts on Integer Word Problems

Integer word problems seem daunting initially, but their mastery is a matter of regular practice and strategy. Break down the problem, identify what operation is warranted, and then move towards a solution progressively. Remember to cross-check the answer, as it ensures correctness.

Remember, it’s perfectly fine to make mistakes while learning. They are merely stepping stones to success. So, stay patient, persist in your efforts, and remember the tips shared. You will soon gain a commendable prowess over integer word problems. The confidence and skills you gain here will be beneficial throughout your mathematical journey.

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Integer Word Problems

An integer is defined as a number that can be written without a fractional component. For example, 11, 8, 0, and −1908 are integers whereas √5, Π are not integers. The set of integers consists of zero, the positive natural numbers, and their additive inverses. Integers are closed under the operations of addition and multiplication.

We use integers in our day-to-day life like measuring temperature, sea level, and speed limit. Translating verbal descriptions into expressions is an essential initial step in solving word problems. Deposits are normally represented by a positive sign and withdrawals are denoted by a negative sign. Negative numbers are used in weather forecasting to show the temperature of a region. Solving these integers word problems will help us relate the concept with practical applications.

Adding Integers 1. Rearrange the terms so that integers with the same sign are next to each other. 2. Add integers with like signs together. 3. Subtract the absolute values of integers with different signs. 4. The sign of the solution will be the sign of the larger integer.

Subtracting Integers 1. Rewrite the problem by changing the second term to its additive inverse. 2. Add the values.

Multiplying Integers 1. Multiply the absolute values of the integers. 2. If the two factors have the same sign, the product is positive. 3. If the two factors have different signs, the product is negative.

The price of one share of stock fell 4 dollars each day for 8 days. How much value did one share of the stock lose?

The stock price fell, so it is represented by -4. This happened for 8 days. -4 x 8 = -32 The stock’s value dropped by $32.

Practice Integer Word Problems

Practice Problem 1

Practice Problem 2

Practice Problem 3

Practice Problem 4

Integer – whole numbers and their opposites {…, -3, -2, -1, 0, 1, 2, 3, …}

Negative integer – any integer less than zero.

Positive integer – any integer greater than zero.

Absolute value – the positive distance that a number is from 0 on a number line.

Zero Pair – a yellow counter and a red counter that represents zero.

Opposites – two integers that are the same distance from 0 on a number line but in opposite directions, like -5 and 5.

Additive inverses – two integers that are opposites.

Pre-requisite Skills Multi-Step Word Problems II Order of Operations

Related Skills Evaluating Algebraic Expressions Evaluate Algebraic Expressions Add Linear Expressions Subtract Linear Expressions Solve Complex Equations

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Integer Equations - Stars and Bars

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Guided training for mathematical problem solving at the level of the AMC 10 and 12.

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A frequently occurring problem in combinatorics arises when counting the number of ways to group identical objects, such as placing indistinguishable balls into labelled urns. We discuss a combinatorial counting technique known as stars and bars or balls and urns to solve these problems, where the indistinguishable objects are represented by stars and the separation into groups is represented by bars.

This allows us to transform the set to be counted into another, which is easier to count. As we have a bijection , these sets have the same size.

Introduction

Stars and bars theorem, problem solving.

Consider the equation \(a+b+c+d=12\) where \(a,b,c,d\) are non-negative integers. We're looking for the number of solutions this equation has. At first, it's not exactly obvious how we can approach this problem. One way is brute force: fixing possibilities for one variable, and analyzing the result for other variables. But we want something nicer, something really elegant. We use the above-noted strategy: transforming a set to another by showing a bijection so that the second set is easier to count.

Suppose we have \(15\) places, where we put \(12\) stars and \(3\) bars, one item per place. The key idea is that this configuration stands for a solution to our equation. For example, \(\{*|*****|****|**\}\) stands for the solution \(1+5+4+2=12\). Because we have \(1\) star, then a bar (standing for a plus sign), then \(5\) stars, again a bar, and similarly \(4\) and \(2\) stars follow. Similarly, \(\{|*****|***|****\}\) denotes the solution \(0+5+3+4=12\) because we have no star at first, then a bar, and similar reasoning like the previous.

We see that any such configuration stands for a solution to the equation, and any solution to the equation can be converted to such a stars-bars series. So we've established a bijection between the solutions to our equation and the configurations of \(12\) stars and \(3\) bars. So our problem reduces to "in how many ways can we place \(12\) stars and \(3\) bars in \(15\) places?" This is the same as fixing \(3\) places out of \(15\) places and filling the rest with stars. We can do this in, of course, \(\dbinom{15}{3}\) ways. So the number of solutions to our equation is \[\dbinom{15}{3}=455.\]

The stars and bars/balls and urns technique is as stated below.

The number of ways to place \(n\) indistinguishable balls into \(k\) labelled urns is \[ \binom{n+k-1}{n} = \binom{n+k-1}{k-1}. \ _\square \]

Here is the proof the above theorem.

We represent the \(n\) balls by \(n\) adjacent stars and consider inserting \(k-1\) bars in between stars to separate the bars into \(k\) groups. For example, for \(n=12\) and \(k=5\), the following is a representation of a grouping of \(12\) indistinguishable balls in 5 urns, where the size of urns 1, 2, 3, 4, and 5 are 2, 4, 0, 3, and 3, respectively: \[ * * | * * * * | \, | * * * | * * * \] Note that in the grouping, there may be empty urns. There are a total of \(n+k-1\) positions, of which \(n\) are stars and \(k-1\) are bars. Thus, the number of ways to place \(n\) indistinguishable balls into \(k\) labelled urns is the same as the number of ways of choosing \(n\) positions among \(n+k-1\) spaces for the stars, with all remaining positions taken as bars. The number of ways this can be done is \( \binom{n+k-1}{n}. \) \(_\square\) Note: \( \binom{n+k-1}{n} = \binom{n+k-1}{k-1}\) can be interpreted as the number of ways to instead choose the positions for \(k-1\) bars and take all remaining positions to be stars.

Check out the following example:

How many ordered sets of non-negative integers \( (a, b, c, d) \) are there such that \[ a + b + c + d = 10 ?\] We first create a bijection between the solutions to \( a+b+c +d = 10\) and the sequences of length 13 consisting of 10 \( 1\)'s and 3 \( 0\)'s. In other words, we will associate each solution with a unique sequence, and vice versa. Given a set of 4 integers \( (a, b, c, d) \), we create the sequence that starts with \( a\) \( 1\)'s, then has a \( 0\), then has \( b\) \( 1\)'s, then has a \( 0\), then has \( c\) \( 1\)'s, then has a \( 0\), then has \( d\) \( 1\)'s. For example, if \( (a, b, c, d) = (1, 4, 0, 2) \), then the associated sequence is \( 1 0 1 1 1 1 0 0 1 1 \). Now, if we add the restriction that \( a + b + c + d = 10 \), the associated sequence will consist of 10 \( 1\)'s (from \( a, b, c, d\)) and 3 \( 0\)'s (from our manual insert), and thus has total length 13. Conversely, given a sequence of length 13 that consists of 10 \( 1\)'s and 3 \( 0\)'s, let \( a\) be the length of the initial string of \( 1\)'s (before the first \( 0\)), let \( b\) be the length of the next string of 1's (between the first and second \( 0\)), let \( c\) be the length of the third string of \( 1\)'s (between the second and third \( 0\)), and let \( d\) be the length of the last string of \( 1\)'s (after the third \( 0\)). These values give a solution to the equation \( a + b + c + d = 10\). This construction associates each solution with a unique sequence, and vice versa, and hence gives a bijection. Now that we have a bijection, the problem is equivalent to counting the number of sequences of length 13 that consist of 10 \( 1\)'s and 3 \( 0\)'s, which we count using the stars and bars technique. There are \(13\) positions from which we choose \(10\) positions as 1's and let the remaining positions be 0's. By stars and bars, there are \( {13 \choose 10} = {13 \choose 3} = 286 \) different choices. \(_\square\) Note: Another approach for solving this problem is the method of generating functions .

This section contains examples followed by problems to try.

Find the number of non-negative integer solutions to \[a+b+c+d+e+f=23.\] We have \(6\) variables, thus \(5\) plus signs. So by stars and bars, the answer is \[\dbinom{23+5}{5}=\dbinom{28}{5}=98280. \ _\square\]

How many ways are there to choose a 5-letter word from the 26-letter English alphabet with replacement, where words that are anagrams are considered the same? Observe that since anagrams are considered the same, the feature of interest is how many times each letter appears in the word (ignoring the order in which the letters appear). To translate this into a stars and bars problem, we consider writing 5 as a sum of 26 integers \(c_A, c_B, \ldots c_Y,\) and \(c_Z,\) where \(c_A\) is the number of times letter \(A\) is chosen, \(c_B\) is the number of times letter \(B\) is chosen, etc. Thus, \(n\) = 5 and \(k\) = 26. Then by stars and bars, the number of 5-letter words is \[ \binom{26 +5 -1}{5} = \binom{30}{25} = 142506. \ _\square\]

For some problems, the stars and bars technique does not apply immediately. In these instances, the solutions to the problem must first be mapped to solutions of another problem which can then be solved by stars and bars. We illustrate one such problem in the following example:

How many ordered sets of positive integers \( (a_1, a_2, a_3, a_4, a_5, a_6) \) are there such that \(a_i \geq i\) for \(i = 1,2, \ldots, 6\) and \[ a_1 + a_2 + a_3 + a_4 + a_5 + a_6 \leq 100 ?\] Because of the inequality, this problem does not map directly to the stars and bars framework. To proceed, consider a bijection between the integers \( (a_1, a_2, a_3, a_4, a_5, a_6) \) satisfying the conditions and the integers \( (a_1, a_2, a_3, a_4, a_5, a_6, c) \) satisfying \( a_i \geq i, c \geq 0,\) and \[ a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + c = 100 .\] Now, by setting \(b_i= a_i-i\) for \(i = 1,2, \ldots, 6\), we would like to find the set of integers \( (b_1, b_2, b_3, b_4, b_5, b_6, c) \) such that \(b_i \geq 0, c \geq 0,\) and \[ b_1 + b_2 + b_3 + b_4 + b_5 + b_6 + c = 100 - (1 + 2 + 3 + 4 + 5 + 6) = 79.\] By stars and bars, this is equal to \( \binom{79+7-1}{79} = \binom{85}{79} \). \(_\square\)

Try the following problems:

Find the number of ordered triples of positive integers \((a,b,c)\) such that \(a+b+c=8\).

Find the number of non-negative integer solutions of

\[ 3x +y + z = 24.\]

Find the number of positive integer solutions of the equation

\[x + y + z = 12.\]

Find the number of non-negative integers \(x_1,x_2,\ldots,x_5\) satisfying

\[\large{x_1 + x_2 + x_3 + x_4 + x_5 = 17.}\]

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13.6: Integer Solutions of Linear Programming Problems

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• Page ID 97954

• Mitchel T. Keller & William T. Trotter
• Georgia Tech & Morningside College

A linear programming problem is an optimization problem that can be stated in the following form: Find the maximum value of a linear function

\(c_1x_1+c_2x_2+c_3x_3+ \cdot \cdot \cdot +c_nx_n\)

subject to \(m\) constraints \(C_1\), \(C_2\),…,\(C_m\), where each constraint \(C_i\) is a linear equation of the form:

\(C_i\): \(a_{i1}x_1+a_{i2}x_2+a_{i3}x_3+ \cdot \cdot \cdot +a_{in}x_n=b_i\)

where all coefficients and constants are real numbers.

While the general subject of linear programming is far too broad for this course, we would be remiss if we didn't point out that:

• Linear programming problems are a very important class of optimization problems and they have many applications in engineering, science, and industrial settings.
• There are relatively efficient algorithms for finding solutions to linear programming problems.
• A linear programming problem posed with rational coefficients and constants has an optimal solution with rational values—if it has an optimal solution at all.
• A linear programming problem posed with integer coefficients and constants need not have an optimal solution with integer values—even when it has an optimal solution with rational values.
• A very important theme in operations research is to determine when a linear programming problem posed in integers has an optimal solution with integer values. This is a subtle and often very difficult problem.
• The problem of finding a maximum flow in a network is a special case of a linear programming problem.
• A network flow problem in which all capacities are integers has a maximum flow in which the flow on every edge is an integer. The Ford-Fulkerson labeling algorithm guarantees this!
• In general, linear programming algorithms are not used on networks. Instead, special purpose algorithms, such as Ford-Fulkerson, have proven to be more efficient in practice.

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Why the Influencer Industry Needs Guardrails

The author argues for an industry in which unethical behavior is punished; professional expectations, pay, and desired outcomes are standardized; and creators are given the same rights and protections as other professional marketers.

And how to professionalize a maturing practice

Idea in Brief

The problem.

Influencer marketing is a global force with huge potential for both positive and negative social impact. Influencers, brands, and social media companies that mislead the public could ruin an industry reliant on credibility.

The industry is built on precarity, with little professional cohesion and inconsistent consequences for unfair play.

The Solution

Marketers must build teams of trustworthy professionals. They must commission work that prioritizes quality and integrity over virality. And the industry as a whole should develop trade organizations and unions to protect influencers, marketers, and the public.

Over the past 20 years the social media influencer industry has grown from nothing into a pervasive global force that has completely rearranged the way information and culture are conceived, produced, marketed, and shared. Commercial sectors such as fashion, beauty, and travel led the way, but nonprofits, government services, and political campaigns are increasingly joining in, hoping to harness the seemingly more authentic medium of influencer marketing.

Stars are using their influencer status to launch their own products and capture more profits for themselves.

• Emily Hund is the author of The Influencer Industry: The Quest for Authenticity on Social Media and a research affiliate in the Center on Digital Culture and Society at the University of Pennsylvania’s Annenberg School for Communication.

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Smudges on your TV? Make your own DIY screen cleaner with just two items

Many of us spend a lot of time looking at our television. They are the vessels that bring us binge-worthy content like "Bridgerton" or "Real Housewives," inform us about what's happening in our communities and the world around us and stream YouTube content about how to keep indoor plants alive.

But something is standing between you and the best picture possible: dust, dirt and a few fingerprints.

Your TV might receive a quick wipe or dusting when you're cleaning, but sometimes it needs more than that. And with such an expensive piece of technology, you want to be careful about what you're using to clean off all those pesky fingerprints.

This DIY solution will have your TV clean and your picture as clear as the day you bought it.

Watch this video to see how you can easily wipe away fingerprints and grime with this easy-to-make, do-it-yourself cleaning solution.  

What can you use to clean a TV screen?

You want to avoid chemical cleaners and products containing alcohol or ammonia when cleaning your TV screen. 

So, put down that bottle of Windex and grab these four items to make your own TV cleaning solution: 

• Distilled water
• Spray bottle
• Microfiber cloth

A microfiber cloth is essential because other cloths or towels can scratch your screen.

To make this simple cleaning solution: 

• Mix one part distilled water with one part vinegar in a spray bottle.
• Shake well.  
• Spray the solution onto your microfiber cloth. Never spray the cleaning solution directly onto the screen.
• Gently wipe your TV to remove any smudges and dirt on your screen. 

This homemade solution is also safe to use on your computer screen.

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