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• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Scientific Notation Arithmetics
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## How to Solve Math Problems

Last Updated: May 16, 2023 Fact Checked

This article was co-authored by Daron Cam . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 575,028 times.

Although math problems may be solved in different ways, there is a general method of visualizing, approaching and solving math problems that may help you to solve even the most difficult problem. Using these strategies can also help you to improve your math skills overall. Keep reading to learn about some of these math problem solving strategies.

## Understanding the Problem

• Draw a Venn diagram. A Venn diagram shows the relationships among the numbers in your problem. Venn diagrams can be especially helpful with word problems.
• Draw a graph or chart.
• Arrange the components of the problem on a line.
• Draw simple shapes to represent more complex features of the problem.

## Expert Q&A

• Seek help from your teacher or a math tutor if you get stuck or if you have tried multiple strategies without success. Your teacher or a math tutor may be able to easily identify what is wrong and help you to understand how to correct it. Thanks Helpful 1 Not Helpful 1
• Keep practicing sums and diagrams. Go through the concept your class notes regularly. Write down your understanding of the methods and utilize it. Thanks Helpful 1 Not Helpful 0

## You Might Also Like

• ↑ Daron Cam. Math Tutor. Expert Interview. 29 May 2020.
• ↑ http://tutorial.math.lamar.edu/Extras/StudyMath/ProblemSolving.aspx
• ↑ https://math.berkeley.edu/~gmelvin/polya.pdf

To solve a math problem, try rewriting the problem in your own words so it's easier to solve. You can also make a drawing of the problem to help you figure out what it's asking you to do. If you're still completely stuck, try solving a different problem that's similar but easier and then use the same steps to solve the harder problem. Even if you can't figure out how to solve it, try to make an educated guess instead of leaving the question blank. To learn how to come up with a solid plan to use to help you solve a math problem, scroll down! Did this summary help you? Yes No

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## Unit 2: Solving equations & inequalities

• Why we do the same thing to both sides: Variable on both sides (Opens a modal)
• Intro to equations with variables on both sides (Opens a modal)
• Equations with variables on both sides: 20-7x=6x-6 (Opens a modal)
• Equation with variables on both sides: fractions (Opens a modal)
• Equation with the variable in the denominator (Opens a modal)
• Equations with variables on both sides Get 3 of 4 questions to level up!
• Equations with variables on both sides: decimals & fractions Get 3 of 4 questions to level up!

## Linear equations with parentheses

• Equations with parentheses (Opens a modal)
• Reasoning with linear equations (Opens a modal)
• Multi-step equations review (Opens a modal)
• Equations with parentheses Get 3 of 4 questions to level up!
• Equations with parentheses: decimals & fractions Get 3 of 4 questions to level up!
• Reasoning with linear equations Get 3 of 4 questions to level up!

## Analyzing the number of solutions to linear equations

• Number of solutions to equations (Opens a modal)
• Worked example: number of solutions to equations (Opens a modal)
• Creating an equation with no solutions (Opens a modal)
• Creating an equation with infinitely many solutions (Opens a modal)
• Number of solutions to equations Get 3 of 4 questions to level up!
• Number of solutions to equations challenge Get 3 of 4 questions to level up!

## Linear equations with unknown coefficients

• Linear equations with unknown coefficients (Opens a modal)
• Why is algebra important to learn? (Opens a modal)
• Linear equations with unknown coefficients Get 3 of 4 questions to level up!

## Multi-step inequalities

• Inequalities with variables on both sides (Opens a modal)
• Inequalities with variables on both sides (with parentheses) (Opens a modal)
• Multi-step inequalities (Opens a modal)
• Using inequalities to solve problems (Opens a modal)
• Multi-step linear inequalities Get 3 of 4 questions to level up!
• Using inequalities to solve problems Get 3 of 4 questions to level up!

## Compound inequalities

• Compound inequalities: OR (Opens a modal)
• Compound inequalities: AND (Opens a modal)
• A compound inequality with no solution (Opens a modal)
• Double inequalities (Opens a modal)
• Compound inequalities examples (Opens a modal)
• Compound inequalities review (Opens a modal)
• Solving equations & inequalities: FAQ (Opens a modal)
• Compound inequalities Get 3 of 4 questions to level up!

## Over 5 Billion Problems Solved

Step-by-step examples.

• Long Subtraction
• Long Multiplication
• Long Division
• Dividing Using Partial Quotients Division
• Converting Regular to Scientific Notation
• Arranging a List in Order
• Expanded Notation
• Prime or Composite
• Comparing Expressions
• Converting to a Percentage
• Finding the Multiplicative Inverse
• Reducing Fractions
• Finding the Reciprocal
• Converting to a Decimal
• Converting to a Mixed Number
• Subtracting Fractions
• Multiplying Fractions
• Dividing Fractions
• Converting Ratios to Fractions
• Converting Percents to Decimal
• Converting Percents to Fractions
• Converting the Percent Grade to Degree
• Converting the Degree to Percent Grade
• Finding the Area of a Rectangle
• Finding the Perimeter of a Rectangle
• Finding the Area of a Square
• Finding the Perimeter of a Square
• Finding the Area of a Circle
• Finding the Circumference of a Circle
• Finding the Area of a Triangle
• Finding the Area of a Trapezoid
• Finding the Volume of a Box
• Finding the Volume of a Cylinder
• Finding the Volume of a Cone
• Finding the Volume of a Pyramid
• Finding the Volume of a Sphere
• Finding the Surface Area of a Box
• Finding the Surface Area of a Cylinder
• Finding the Surface Area of a Cone
• Finding the Surface Area of a Pyramid
• Converting to a Fraction
• Simple Exponents
• Prime Factorizations
• Finding the Factors
• Simplifying Fractions
• Converting Grams to Kilograms
• Converting Grams to Pounds
• Converting Grams to Ounces
• Converting Feet to Inches
• Converting to Meters
• Converting Feet to Miles
• Converting Feet to Yards
• Converting to Feet
• Converting to Yards
• Converting Miles to Feet
• Converting Miles to Kilometers
• Converting Miles to Yards
• Converting Kilometers to Miles
• Converting Kilometers to Meters
• Converting Meters to Feet
• Converting Meters to Inches
• Converting Ounces to Grams
• Converting Ounces to Pounds
• Converting Ounces to Tons
• Converting Pounds to Grams
• Converting Pounds to Ounces
• Converting Pounds to Tons
• Converting Yards to Feet
• Converting Yards to Millimeters
• Converting Yards to Inches
• Converting Yards to Miles
• Converting Yards to Meters
• Converting Fahrenheit to Celsius
• Converting Celsius to Fahrenheit
• Finding the Median
• Finding the Mean (Arithmetic)
• Finding the Mode
• Finding the Minimum
• Finding the Maximum
• Finding the Lower or First Quartile
• Finding the Upper or Third Quartile
• Finding the Five Number Summary
• Finding the Midpoint of a Line Segment
• Distance Formula
• Arithmetic Operations
• Combining Like Terms
• Determining if the Expression is a Polynomial
• Distributive Property
• Simplifying
• Multiplication
• Polynomial Subtraction
• Polynomial Multiplication
• Polynomial Division
• Simplifying Expressions
• Evaluate the Expression Using the Given Values
• Multiplying Polynomials Using FOIL
• Identifying Degree
• Operations on Polynomials
• Negative Exponents
• Solving by Multiplying/Dividing
• Solving Containing Decimals
• Solving for a Variable
• Solving Linear Equations
• Solving Linear Inequalities
• Finding the Quadratic Constant of Variation
• Converting the Percent Grade to Slope
• Converting the Slope to Percent Grade
• Finding Equations Using Slope-Intercept
• Finding the Slope
• Finding the y Intercept
• Calculating Slope and y-Intercept
• Rewriting in Slope-Intercept Form
• Finding Equations Using the Slope-Intercept Formula
• Finding Equations Using Two Points
• Finding a Perpendicular Line Containing a Given Point
• Finding a Parallel Line Containing a Given Point
• Finding a Parallel Line to the Given Line
• Finding a Perpendicular Line to the Given Line
• Finding Ordered Pair Solutions
• Using a Table of Values to Graph an Equation
• Finding the Equation Using Point-Slope Form
• Finding the Surface Area of a Sphere
• Solving by Graphing
• Finding the LCM of a List of Expressions
• Finding the LCD of a List of Expressions
• Determining if the Number is a Perfect Square
• Finding the Domain
• Evaluating the Difference Quotient
• Solving Using the Square Root Property
• Determining if True
• Finding the Holes in a Graph
• Finding the Common Factors
• Expand a Trinomial with the Trinomial Theorem
• Finding the Start Point Given the Mid and End Points
• Finding the End Point Given the Start and Mid Points
• Finding the Slope and y-Intercept
• Finding the Equation of the Parabola
• Finding the Average Rate of Change
• Finding the Slope of the Perpendicular Line to the Line Through the Two Points
• Rewriting Using Negative Exponents
• Synthetic Division
• Maximum Number of Real Roots/Zeros
• Finding All Possible Roots/Zeros (RRT)
• Finding All Roots with Rational Root Test (RRT)
• Finding the Remainder
• Finding the Remainder Using Long Polynomial Division
• Reordering the Polynomial in Ascending Order
• Reordering the Polynomial in Descending Order
• Finding the GCF of a Polynomial
• Factoring Out Greatest Common Factor (GCF)
• Identifying the Common Factors
• Cancelling the Common Factors
• Finding the LCM using GCF
• Finding the GCF
• Factoring Trinomials
• Trinomial Squares
• Factoring Using Any Method
• Factoring a Difference of Squares
• Factoring a Sum of Cubes
• Factoring by Grouping
• Factoring a Difference of Cubes
• Determine if an Expression is a Factor
• Determining if Factor Using Synthetic Division
• Find the Factors Using the Factor Theorem
• Determining if Polynomial is Prime
• Determining if the Polynomial is a Perfect Square
• Expand using the Binomial Theorem
• Factoring over the Complex Numbers
• Finding All Integers k Such That the Trinomial Can Be Factored
• Determining if Linear
• Rewriting in Standard Form
• Finding x and y Intercepts
• Finding Equations Using the Point Slope Formula
• Finding Equations Given Point and y-Intercept
• Finding the Constant Using Slope
• Finding the Slope of a Parallel Line
• Finding the Slope of a Perpendicular Line
• Simplifying Absolute Value Expressions
• Solving with Absolute Values
• Finding the Vertex for the Absolute Value
• Rewriting the Absolute Value as Piecewise
• Calculating the Square Root
• Rewriting with Rational (Fractional) Exponents
• Finding the Square Root End Point
• Operations on Rational Expressions
• Determining if the Point is a Solution
• Solving over the Interval
• Finding the Range
• Finding the Domain and Range
• Solving Rational Equations
• Subtracting Rational Expressions
• Multiplying Rational Expressions
• Finding the Equation Given the Roots
• Finding the Asymptotes
• Finding the Constant of Variation
• Finding the Equation of Variation
• Substitution Method
• Graphing Method
• Determining Parallel Lines
• Determining Perpendicular Lines
• Dependent, Independent, and Inconsistent Systems
• Finding the Intersection (and)
• Using the Simplex Method for Constraint Maximization
• Using the Simplex Method for Constraint Minimization
• Finding the Union (or)
• Finding the Equation with Real Coefficients
• Solving in Terms of the Arbitrary Variable
• Finding a Direct Variation Equation
• Finding the Slope for Every Equation
• Finding a Variable Using the Constant of Variation
• Solving by Factoring
• Solve by Completing the Square
• Finding the Perfect Square Trinomial
• Finding the Quadratic Equation Given the Solution Set
• Finding a,b, and c in the Standard Form
• Finding the Discriminant
• Finding the Zeros by Completing the Square
• Rational Inequalities
• Converting from Interval to Inequality
• Converting to Interval Notation
• Rewriting as a Single Interval
• Determining if the Relation is a Function
• Finding the Domain and Range of the Relation
• Finding the Inverse of the Relation
• Finding the Inverse
• Determining if One Relation is the Inverse of Another
• Determining if Surjective (Onto)
• Determining if Bijective (One-to-One)
• Determining if Injective (One to One)
• Rewriting as an Equation
• Rewriting as y=mx+b
• Solving Function Systems
• Find the Behavior (Leading Coefficient Test)
• Determining Odd and Even Functions
• Describing the Transformation
• Finding the Symmetry
• Arithmetic of Functions
• Domain of Composite Functions
• Finding Roots Using the Factor Theorem
• Determine if Injective (One to One)
• Determine if Surjective (Onto)
• Finding the Vertex
• Finding the Sum
• Finding the Difference
• Finding the Product
• Finding the Quotient
• Finding the Domain of the Sum of the Functions
• Finding the Domain of the Difference of the Functions
• Finding the Domain of the Product of the Functions
• Finding the Domain of the Quotient of the Functions
• Finding Roots (Zeros)
• Identifying Zeros and Their Multiplicities
• Finding the Bounds of the Zeros
• Proving a Root is on the Interval
• Finding Maximum Number of Real Roots
• Function Composition
• Rewriting as a Function
• Determining if a Function is Rational
• Determining if a Function is Proper or Improper
• Finding All Complex Number Solutions
• Rationalizing with Complex Conjugates
• Vector Arithmetic
• Finding the Complex Conjugate
• Finding the Magnitude of a Complex Number
• Simplifying Logarithmic Expressions
• Expanding Logarithmic Expressions
• Evaluating Logarithms
• Rewriting in Exponential Form
• Converting to Logarithmic Form
• Exponential Expressions
• Exponential Equations
• Find the Nth Root of the Given Value
• Simplifying Matrices
• Finding the Variables
• Solving the System of Equations Using an Inverse Matrix
• Finding the Dimensions
• Multiplication by a Scalar
• Subtraction
• Finding the Determinant of the Resulting Matrix
• Finding the Inverse of the Resulting Matrix
• Finding the Identity Matrix
• Finding the Scalar multiplied by the Identity Matrix
• Simplifying the Matrix Operation
• Finding the Determinant of a 2x2 Matrix
• Finding the Determinant of a 3x3 Matrix
• Finding the Determinant of Large Matrices
• Inverse of a 2x2 Matrix
• Inverse of an nxn Matrix
• Finding Reduced Row Echelon Form
• Finding the Transpose
• Finding the Cofactor Matrix
• Finding the Pivot Positions and Pivot Columns
• Finding the Basis and Dimension for the Row Space of the Matrix
• Finding the Basis and Dimension for the Column Space of the Matrix
• Finding the LU Decomposition of a Matrix
• Identifying Conic Sections
• Identifying Circles
• Finding a Circle Using the Center and Another Point
• Finding a Circle by the Diameter End Points
• Finding the Parabola Equation Using the Vertex and Another Point
• Finding the Properties of the Parabola
• Finding the Vertex Form of the Parabola
• Finding the Vertex Form of an Ellipse
• Finding the Vertex Form of a Circle
• Finding the Vertex Form of a Hyperbola
• Finding the Standard Form of a Parabola
• Finding the Expanded Form of an Ellipse
• Finding the Expanded Form of a Circle
• Finding the Expanded Form of a Hyperbola
• Vector Subtraction
• Vector Multiplication by a Scalar
• Finding the Length
• Finding the Position Vector
• Determining Column Spaces
• Finding an Orthonormal Basis by Gram-Schmidt Method
• Rewrite the System as a Vector Equality
• Finding the Rank
• Finding the Nullity
• Finding the Distance
• Finding the Plane Parallel to a Line Given four 3d Points
• Finding the Intersection of the Line Perpendicular to Plane 1 Through the Origin and Plane 2
• Finding the Eigenvalues
• Finding the Characteristic Equation
• Finding the Eigenvectors/Eigenspace of a Matrix
• Proving a Transformation is Linear
• Finding the Kernel of a Transformation
• Projecting Using a Transformation
• Finding the Pre-Image
• Finding the Intersection of Sets
• Finding the Union of Number Sets
• Determining if a Set is a Subset of Another Set
• Determining if Two Sets are Mutually Exclusive
• Finding the Set Complement of Two Sets
• Finding the Power Set
• Finding the Cardinality
• Finding the Cartesian Product of Two Sets
• Determining if a Set is a Proper Subset of Another Set
• Finding the Function Rule
• Finding the Square or Rectangle Area Given Four Points
• Finding the Square or Rectangle Perimeter Given Four Points
• Finding the Square or Rectangle Area Given Three Points
• Finding the Square or Rectangle Perimeter Given Three Points
• Finding the Equation of a Circle
• Finding the Equation of a Hyperbola
• Finding the Equation of an Ellipse
• Partial Fraction Decomposition
• Finding an Angle Using another Angle
• Pythagorean Theorem
• Finding the Sine
• Finding the Cosine
• Finding the Tangent
• Finding the Trig Value
• Converting to Degrees, Minutes, and Seconds
• Finding Trig Functions Using Identities
• Finding Trig Functions Using the Right Triangle
• Finding a Reference Angle
• Finding a Supplement
• Finding a Complement
• Converting RPM to Radians per Second
• Finding the Quadrant of the Angle
• Graphing Sine & Cosine Functions
• Graphing Other Trigonometric Functions
• Amplitude, Period, and Phase Shift
• Finding the Other Trig Values in a Quadrant
• Finding the Exact Value
• Finding the Value Using the Unit Circle
• Expanding Trigonometric Expressions
• Expanding Using Double-Angle Formulas
• Expanding Using Triple-Angle Formulas
• Expanding Using Sum/Difference Formulas
• Simplify Using Pythagorean Identities
• Simplify by Converting to Sine/Cosine
• Inverting Trigonometric Expressions
• Finding the Trig Value of an Angle
• Expanding Using De Moivre's Theorem
• Verifying Trigonometric Identities
• Using Fundamental Identities
• Solving Standard Angle Equations
• Complex Trigonometric Equations
• Solving the Triangle
• Find the Roots of a Complex Number
• Complex Operations
• Trigonometric Form of a Complex Number
• Converting to Polar Coordinates
• Identifying and Graphing Circles
• Identifying and Graphing Limacons
• Identifying and Graphing Roses
• Identifying and Graphing Cardioids
• Difference Quotient
• Finding Upper and Lower Bounds
• Evaluating Functions
• Right Triangle Trigonometry
• Arithmetic Sequences/Progressions
• Geometric Sequences/Progressions
• Finding the Next Term of the Sequence
• Finding the nth Term Given a List of Numbers
• Finding the nth Term
• Finding the Sum of First n Terms
• Expanding Series Notation
• Finding the Sum of the Series
• Finding the Sum of the Infinite Geometric Series
• Converting to Rectangular Coordinates
• Evaluating Limits Approaching a Value
• Evaluating Limits Approaching Infinity
• Finding the Angle Between the Vectors
• Determining if the Point is on the Graph
• Finding the Antiderivative
• Checking if Continuous Over an Interval
• Determining if a Series is Divergent
• Using the Integral Test for Convergence
• Determining if an Infinite Series is Convergent Using Cauchy's Root Test
• Using the Limit Definition to Find the Tangent Line at a Given Point
• Finding the nth Derivative
• Finding the Derivative Using Product Rule
• Finding the Derivative Using Quotient Rule
• Finding the Derivative Using Chain Rule
• Use Logarithmic Differentiation to Find the Derivative
• Finding the Derivative
• Implicit Differentiation
• Using the Limit Definition to Find the Derivative
• Evaluating the Derivative
• Finding Where dy/dx is Equal to Zero
• Finding the Linearization
• Finding a Tangent Line to a Curve
• Checking if Differentiable Over an Interval
• The Mean Value Theorem
• Finding the Inflection Points
• Find Where the Function Increases/Decreases
• Finding the Critical Points of a Function
• Find Horizontal Tangent Line
• Evaluating Limits with L'Hospital Rule
• Local Maxima and Minima
• Finding the Absolute Maximum and Minimum on the Given Interval
• Finding Concavity using the Second Derivative
• Finding the Derivative using the Fundamental Theorem of Calculus
• Find the Turning Points
• Finding the Integral
• Evaluating Definite Integrals
• Evaluating Indefinite Integrals
• Substitution Rule
• Finding the Arc Length
• Finding the Average Value of the Derivative
• Finding the Average Value of the Equation
• Finding Area Between Curves
• Finding the Volume
• Finding the Average Value of the Function
• Finding the Root Mean Square
• Integration by Parts
• Trigonometric Integrals
• Trigonometric Substitution
• Integration by Partial Fractions
• Eliminating the Parameter from the Function
• Verify the Solution of a Differential Equation
• Solve for a Constant Given an Initial Condition
• Find an Exact Solution to the Differential Equation
• Verify the Existence and Uniqueness of Solutions for the Differential Equation
• Solve for a Constant in a Given Solution
• Solve the Bernoulli Differential Equation
• Solve the Linear Differential Equation
• Solve the Homogeneous Differential Equation
• Solve the Exact Differential Equation
• Approximate a Differential Equation Using Euler's Method
• Finding Elasticity of Demand
• Finding the Consumer Surplus
• Finding the Producer Surplus
• Finding the Gini Index
• Finding the Geometric Mean
• Finding the Quadratic Mean (RMS)
• Find the Mean Absolute Deviation
• Finding the Mid-Range (Mid-Extreme)
• Finding the Interquartile Range (H-Spread)
• Finding the Midhinge
• Finding the Standard Deviation
• Finding the Skew of a Data Set
• Finding the Range of a Data Set
• Finding the Variance of a Data Set
• Finding the Class Width
• Solving Combinations
• Solving Permutations
• Finding the Probability of Both Independent Events
• Finding the Probability of Both Dependent Events
• Finding the Probability for Both Mutually Exclusive Events
• Finding the Conditional Probability for Independent Events
• Determining if Given Events are Independent/Dependent Events
• Determining if Given Events are Mutually Exclusive Events
• Finding the Probability of Both not Mutually Exclusive Events
• Finding the Conditional Probability Using Bayes' Theorem
• Finding the Probability of the Complement
• Describing Distribution's Two Properties
• Finding the Expectation
• Finding the Variance
• Finding the Probability of a Binomial Distribution
• Finding the Probability of the Binomial Event
• Finding the Mean
• Finding the Relative Frequency
• Finding the Percentage Frequency
• Finding the Upper and Lower Class Limits of the Frequency Table
• Finding the Class Boundaries of the Frequency Table
• Finding the Class Width of the Frequency Table
• Finding the Midpoints of the Frequency Table
• Finding the Mean of the Frequency Table
• Finding the Variance of the Frequency Table
• Finding the Standard Deviation of the Frequency Table
• Finding the Cumulative Frequency of the Frequency Table
• Finding the Relative Frequency of the Frequency Table
• Finding the Median Class Interval of the Frequency Table
• Finding the Modal Class of the Frequency Table
• Creating a Grouped Frequency Distribution Table
• Finding the Data Range
• Finding a z-Score for a Normal Distribution
• Approximating Using Normal Distribution
• Finding the Probability of the z-Score Range
• Finding the Probability of a Range in a Nonstandard Normal Distribution
• Finding the z-Score Using the Table
• Finding the z-Score
• Testing the Claim
• Finding a t-Value for a Confidence Level
• Finding the Critical t-Value
• Setting the Alternative Hypothesis
• Setting the Null Hypothesis
• Determining if Left, Right, or Two Tailed Test Given the Null Hypothesis
• Determining if Left, Right, or Two Tailed Test Given the Alternative Hypothesis
• Finding Standard Error
• Finding the Linear Correlation Coefficient
• Determining if the Correlation is Significant
• Finding a Regression Line
• Cramer's Rule
• Solving using Matrices by Elimination
• Solving using Matrices by Row Operations
• Solving using an Augmented Matrix
• Finding the Simple Interest Received
• Finding the Present Value with Compound Interest
• Finding the Simple Interest Future Value
• Finding the Future Value with Continuous Interest
• Finding the Norm in Real Vector Space
• Finding the Direction Angle of the Vector
• Finding the Cross Product of Vectors
• Finding the Dot Product of Vectors
• Determining if Vectors are Orthogonal
• Finding the Distance Between the Vectors
• Finding a Unit Vector in the Same Direction as the Given Vector
• Finding the Angle Between Two Vectors Using the Cross Product
• Finding the Angle Between Two Vectors Using the Dot Product
• Finding the Projection of One Vector Onto another Vector
• Matrices Subtraction
• Matrices Multiplication
• Finding the Trace
• Finding the Basis
• Matrix Dimension
• Convert to a Linear System
• Diagonalizing a Matrix
• Determining the value of k for which the system has no solutions
• Linear Independence of Real Vector Spaces
• Finding the Null Space
• Determining if the Vector is in the Span of the Set
• Finding the Number of Protons
• Finding the Number of Electrons
• Finding the Number of Neutrons
• Finding the Mass of a Single Atom
• Finding the Electron Configuration
• Finding the Atomic Mass
• Finding the Atomic Number
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Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

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## How to Solve Math Problems Faster: 15 Techniques to Show Students

Written by Marcus Guido

• Teaching Strategies

“Test time. No calculators.”

You’ll intimidate many students by saying this, but teaching techniques to solve math problems with ease and speed can make it less daunting.

This can also  make math more rewarding . Instead of relying on calculators, students learn strategies that can improve their concentration and estimation skills while building number sense. And, while there are educators who  oppose math “tricks”  for valid reasons, proponents point to benefits such as increased confidence to handle difficult problems.

Here are 15 techniques to show students,  helping them solve math problems faster:

Many students struggle when learning to add integers of three digits or higher together, but changing the process’s steps can make it easier.

The first step is to  add what’s easy.  The second step is to  add the rest.

Let’s say students must find the sum of 393 and 89. They should quickly see that adding 7 onto 393 will equal 400 — an easier number to work with. To balance the equation, they can then subtract 7 from 89.

Broken down, the process is:

• (393 + 7) + (89 – 7)

With this fast technique, big numbers won’t look as scary now.

## 2. Two-Step Subtraction

There’s a similar method for subtraction.

Remove what’s easy. Then remove what’s left.

Suppose students must find the difference of 567 and 153. Most will feel that 500 is a simpler number than 567. So, they just have to take away 67 from the minuend — 567 — and the subtrahend — 153 — before solving the equation.

Here’s the process:

• (567 – 67) – (153 – 67)

Instead of two complex numbers, students will only have to tackle one.

## 3. Subtracting from 1,000

You can  give students confidence  to handle four-digit integers with this fast technique.

To subtract a number from 1,000, subtract that number’s first two digits from 9. Then, subtract the final digit from 10.

Let’s say students must solve 1,000 – 438.  Here are the steps:

This also applies to 10,000, 100,000 and other integers that follow this pattern.

## Multiplication and Division

4. doubling and halving.

When students have to multiply two integers, they can speed up the process when one is an even number. They just need to  halve the even number and double the other number.

Students can stop the process when they can no longer halve the even integer, or when the equation becomes manageable.

Using 33 x 48 as an example,  here’s the process:

The only prerequisite is understanding the 2 times table.

## 5. Multiplying by Powers of 2

This tactic is a speedy variation of doubling and halving.

It simplifies multiplication if a number in the equation is a power of 2, meaning it works for 2, 4, 8, 16 and so on.

Here’s what to do:  For each power of 2 that makes up that number, double the other number.

For example, 9 x 16 is the same thing as 9 x (2 x 2 x 2 x 2) or 9 x 24. Students can therefore double 9 four times to reach the answer:

Unlike doubling and halving, this technique demands an understanding of exponents along with a strong command of the 2 times table.

## 6. Multiplying by 9

For most students, multiplying by 9 — or 99, 999 and any number that follows this pattern — is difficult compared with multiplying by a power of 10.

But there’s an easy tactic to solve this issue, and  it has two parts.

First, students round up the 9  to 10. Second, after solving the new equation, they subtract the number they just multiplied by 10 from the answer.

For example, 67 x 9 will lead to the same answer as 67 x 10 – 67. Following the order of operations will give a result of 603. Similarly, 67 x 99 is the same as 67 x 100 – 67.

Despite more steps, altering the equation this way is usually faster.

## 7. Multiplying by 11

There’s an easier way for multiplying two-digit integers by 11.

Let’s say students must find the product of 11 x 34.

The idea is to put a space between the digits, making it 3_4. Then, add the two digits together and put the sum in the space.

What happens if the sum is two digits? Students would put the second digit in the space and add 1 to the digit to the left of the space.  For example:

It’s multiplication without having to multiply.

## 8. Multiplying Even Numbers by 5

This technique only requires basic division skills.

There are two steps,  and 5 x 6 serves as an example. First, divide the number being multiplied by 5 — which is 6 — in half. Second, add 0 to the right of number.

The result is 30, which is the correct answer.

It’s an ideal, easy technique for students mastering the 5 times table.

## 9. Multiplying Odd Numbers by 5

This is another time-saving tactic that works well when teaching students the 5 times table.

This one has three steps,  which 5 x 7 exemplifies.

First, subtract 1 from the number being multiplied by 5, making it an even number. Second, cut that number in half — from 6 to 3 in this instance. Third, add 5 to the right of the number.

Who needs a calculator?

## 10. Squaring a Two-Digit Number that Ends with 1

Squaring a high two-digit number can be tedious, but there’s a shortcut if 1 is the second digit.

There are four steps to this shortcut,  which 812 exemplifies:

• Subtract 1 from the integer: 81 – 1 = 80
• Square the integer, which is now an easier number: 80 x 80 = 6,400
• Add the integer with the resulting square twice: 6,400 + 80 + 80 = 6,560
• Add 1: 6,560 + 1 = 6,561

This work-around eliminates the difficulty surrounding the second digit, allowing students to work with multiples of 10.

## 11. Squaring a Two-Digit Numbers that Ends with 5

Squaring numbers ending in 5 is easier, as there are  only two parts of the process.

First, students will always make 25 the product’s last digits.

Second, to determine the product’s first digits, students must multiply the number’s first digit — 9, for example — by the integer that’s one higher — 10, in this case.

So, students would solve 952 by designating 25 as the last two digits. They would then multiply 9 x 10 to receive 90. Putting these numbers together, the  result is 9,025.

Just like that, a hard problem becomes easy multiplication for many students.

## 12. Calculating Percentages

Cross-multiplication is an  important skill  to develop, but there’s an easier way to calculate percentages.

For example, if students want to know what 65% of 175 is, they can multiply the numbers together and move the decimal place two digits to the left.

The result is 113.75, which is indeed the correct answer.

This shortcut is a useful timesaver on tests and quizzes.

## 13. Balancing Averages

To determine the average among a set of numbers, students can balance them instead of using a complex formula.

Suppose a student wants to volunteer for an average of 10 hours a week over a period of four weeks. In the first three weeks, the student worked for 10, 12 and 14 hours.

To determine the number of hours required in the fourth week, the student must  add how much he or she surpassed or missed the target average  in the other weeks:

• 14 hours – 10 hours = 4 hours
• 12 – 10 = 2
• 10 – 10 = 0
• 4 hours + 2 hours + 0 hours = 6 hours

To learn the number of hours for the final week, the student must  subtract the sum from the target average:

• 10 hours – 6 hours = 4 hours

With practice, this method may not even require pencil and paper. That’s how easy it is.

## Word Problems

14. identifying buzzwords.

Students who struggle to translate  word problems  into equations will benefit from learning how to spot buzzwords — phrases that indicate specific actions.

This isn’t a trick. It’s a tactic.

Teach students to look for these buzzwords,  and what skill they align with in most contexts:

Be sure to include buzzwords that typically appear in their textbooks (or other classroom  math books ), as well as ones you use on tests and assignments.

As a result, they should have an  easier time processing word problems .

## 15. Creating Sub-Questions

For complex word problems, show students how to dissect the question by answering three specific sub-questions.

Each student should ask him or herself:

• What am I looking for?  — Students should read the question over and over, looking for buzzwords and identifying important details.
• What information do I need?  — Students should determine which facts, figures and variables they need to solve the question. For example, if they determine the question is rooted in subtraction, they need the minuend and subtrahend.
• What information do I have?  — Students should be able to create the core equation using the information in the word problem, after determining which details are important.

These sub-questions help students avoid overload.

Instead of writing and analyzing each detail of the question, they’ll be able to identify key information. If you identify students who are struggling with these, you can use  peer learning  as needed.

For more fresh approaches to teaching math in your classroom, consider treating your students to a range of  fun math activities .

## Final Thoughts About these Ways to Solve Math Problems Faster

Showing these 15 techniques to students can give them the  confidence to tackle tough questions .

They’re also  mental math  exercises, helping them build skills related to focus, logic and critical thinking.

A rewarding class equals an  engaging class . That’s an easy equation to remember.

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Unit test. Level up on all the skills in this unit and collect up to 1100 Mastery points! There are lots of strategies we can use to solve equations. Let's explore some different ways to solve equations and inequalities. We'll also see what it takes for an equation to have no solution, or infinite solutions.

12. Mathway Free Math Solver

Mathway's math problem solver is an excellent tool to check your work for free. Just type in the math problem into the interactive calculator using its prescribed syntax, and it will give you the correct answer. It's like having a private tutor 24/7/365 with you at all times! Note: If you want more help, and therefore, would want to see how ...

13. Equation Solver

Step 1: Enter the Equation you want to solve into the editor. The equation calculator allows you to take a simple or complex equation and solve by best method possible. Step 2: Click the blue arrow to submit and see the result! The equation solver allows you to enter your problem and solve the equation to see the result.

14. Mathway

Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Mathway. Visit Mathway on the web. Start 7-day free trial on the app. Start 7-day free trial on the app.

15. Step-by-Step Math Problem Solver

Help Tutorial Solve an equation, inequality or a system. Example: 2x-1=y,2y+3=x

16. Math Problem Solver

Our math problem solver that lets you input a wide variety of math math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects. Basic Math Math Word Problems Pre-Algebra Algebra Geometry Graphing Trigonometry Precalculus Calculus Statistics Finite Math

17. Equation Solver: Wolfram|Alpha

For equation solving, Wolfram|Alpha calls the Wolfram Language's Solve and Reduce functions, which contain a broad range of methods for all kinds of algebra, from basic linear and quadratic equations to multivariate nonlinear systems. In some cases, linear algebra methods such as Gaussian elimination are used, with optimizations to increase ...

18. How to Solve Math Problems Faster: 15 Techniques to Show Students

Here are 15 techniques to show students, helping them solve math problems faster: Addition and Subtraction. 1. Two-Step Addition ... — Students should determine which facts, figures and variables they need to solve the question. For example, if they determine the question is rooted in subtraction, they need the minuend and subtrahend. ...

19. Wolfram Problem Generator: Online Practice Questions & Answers

Only Wolfram Problem Generator directly integrates the popular and powerful Step-by-step Solutions from Wolfram|Alpha. You can use a single hint to get unstuck, or explore the entire math problem from beginning to end. Online practice problems for math, including arithmetic, algebra, calculus, linear algebra, number theory, and statistics.

20. MathGPT

MathGPT is an AI-powered math problem solver, integral calculator, derivative cacluator, polynomial calculator, and more! Try it out now and solve your math homework!

21. Free Math Worksheets

Khan Academy's 100,000+ free practice questions give instant feedback, don't need to be graded, and don't require a printer. Math Worksheets. Khan Academy. Math worksheets take forever to hunt down across the internet. Khan Academy is your one-stop-shop for practice from arithmetic to calculus. Math worksheets can vary in quality from ...

22. Question.AI-AI Math Calculator 12+