## Module 2: Calculations and Solving Equations

Algebraic problem solving strategies, learning outcome.

- Use a problem-solving strategy to set up and solve word problems

The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?

Now that we can solve equations, we are ready to apply our new skills to word problems.

Previously, you translated word phrases into algebraic equations using some basic mathematical vocabulary and symbols. Since then you’ve increased your math vocabulary as you learned about more algebraic procedures. You’ve also solved some word problems applying math to everyday situations. This method works as long as the situation is familiar to you and the math is not too complicated.

Now we’ll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We’ll demonstrate the strategy as we solve the following problem.

Pete bought a shirt on sale for $[latex]18[/latex], which is one-half the original price. What was the original price of the shirt?

Solution: Step 1. Read the problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don’t understand, look them up in a dictionary or on the Internet.

- In this problem, do you understand what is being discussed? Do you understand every word?

Step 2. Identify what you are looking for. It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!

- In this problem, the words “what was the original price of the shirt” tell you what you are looking for: the original price of the shirt.

Step 3. Name what you are looking for. Choose a variable to represent that quantity. You can use any letter for the variable, but it may help to choose one that helps you remember what it represents.

- Let [latex]p=[/latex] the original price of the shirt

Step 4. Translate into an equation. It may help to first restate the problem in one sentence, with all the important information. Then translate the sentence into an equation.

Step 6. Check the answer in the problem and make sure it makes sense.

- We found that [latex]p=36[/latex], which means the original price was [latex]\text{\$36}[/latex]. Does [latex]\text{\$36}[/latex] make sense in the problem? Yes, because [latex]18[/latex] is one-half of [latex]36[/latex], and the shirt was on sale at half the original price.

Step 7. Answer the question with a complete sentence.

- The problem asked “What was the original price of the shirt?” The answer to the question is: “The original price of the shirt was [latex]\text{\$36}[/latex].”

If this were a homework exercise, our work might look like this:

We list the steps we took to solve the previous example.

## Problem-Solving Strategy

- Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don’t understand, look them up in a dictionary or on the internet.
- Identify what you are looking for.
- Name what you are looking for. Choose a variable to represent that quantity.
- Translate into an equation. It may be helpful to first restate the problem in one sentence before translating.
- Solve the equation using good algebra techniques.
- Check the answer in the problem. Make sure it makes sense.
- Answer the question with a complete sentence.

Let’s use this approach with another example.

In the next example, we will apply our Problem-Solving Strategy to applications of percent.

The difference of a number and six is [latex]13[/latex]. Find the number.

The sum of twice a number and seven is [latex]15[/latex]. Find the number.

Watch the following video to see another example of how to solve a number problem.

## Contribute!

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- Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/caa57dab-41c7-455e-bd6f-f443cda5[email protected]

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## 2.3: Use a Problem Solving Strategy

## Learning Objectives

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

- Use a problem solving strategy for word problems
- Solve number word problems
- Solve percent applications
- Solve simple interest applications

Before you get started, take this readiness quiz.

- Translate “six less than twice \(x\)” into an algebraic expression. If you missed this problem, review [link] .
- Convert 4.5% to a decimal. If you missed this problem, review [link] .
- Convert 0.6 to a percent. If you missed this problem, review [link] .

- I think I can! I think I can!
- While word problems were hard in the past, I think I can try them now.
- I am better prepared now—I think I will begin to understand word problems.
- I am able to solve equations because I practiced many problems and I got help when I needed it—I can try that with word problems.
- It may take time, but I can begin to solve word problems.
- You are now well prepared and you are ready to succeed. If you take control and believe you can be successful, you will be able to master word problems.

## Use a Problem Solving Strategy for Word Problems

## EXAMPLE \(\PageIndex{1}\)

## Try It! \(\PageIndex{1}\)

## Try It! \(\PageIndex{2}\)

He did seven crossword puzzles

We summarize an effective strategy for problem solving.

## PROBLEM SOLVING STRATEGY FOR WORD PROBLEMS

- Read the problem. Make sure all the words and ideas are understood.
- Identify what you are looking for.
- Name what you are looking for. Choose a variable to represent that quantity.
- 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.
- Solve the equation using proper algebra techniques.
- Check the answer in the problem to make sure it makes sense.
- Answer the question with a complete sentence.

## Solve Number Word Problems

## EXAMPLE \(\PageIndex{2}\)

The sum of seven times a number and eight is thirty-six. Find the number.

## Try It! \(\PageIndex{3}\)

The sum of four times a number and two is fourteen. Find the number.

## Try It! \(\PageIndex{4}\)

The sum of three times a number and seven is twenty-five. Find the number.

## EXAMPLE \(\PageIndex{3}\)

## Try It! \(\PageIndex{5}\)

## Try It! \(\PageIndex{6}\)

We will use this notation to represent consecutive integers in the next example.

## EXAMPLE \(\PageIndex{4}\)

Find three consecutive integers whose sum is \(−54\).

## Try It! \(\PageIndex{7}\)

Find three consecutive integers whose sum is \(−96\).

## Try It! \(\PageIndex{8}\)

Find three consecutive integers whose sum is \(−36\).

## EXAMPLE \(\PageIndex{5}\)

Find three consecutive even integers whose sum is \(120\).

## Try It! \(\PageIndex{9}\)

Find three consecutive even integers whose sum is 102.

## Try It! \(\PageIndex{10}\)

Find three consecutive even integers whose sum is \(−24\).

## EXAMPLE \(\PageIndex{6}\)

## Try It! \(\PageIndex{11}\)

## Solve Percent Applications

## EXAMPLE \(\PageIndex{7}\)

## Try It! \(\PageIndex{12}\)

## Try It! \(\PageIndex{13}\)

## EXAMPLE \(\PageIndex{8}\)

## Try It! \(\PageIndex{14}\)

## Try It! \(\PageIndex{15}\)

## EXAMPLE \(\PageIndex{9}\)

## Try It! \(\PageIndex{16}\)

## Try It! \(\PageIndex{17}\)

## FIND PERCENT CHANGE

\[\text{change}= \text{new amount}−\text{original amount}\]

change is what percent of the original amount?

## EXAMPLE \(\PageIndex{10}\)

## Try It! \(\PageIndex{18}\)

## Try It! \(\PageIndex{19}\)

Applications of discount and mark-up are very common in retail settings.

The sale price should always be less than the original price.

The list price should always be more than the original cost.

## EXAMPLE \(\PageIndex{11}\)

Liam’s art gallery bought a painting at an original cost of $750. Liam marked the price up 40%. Find

## Try It! \(\PageIndex{20}\)

## Try It! \(\PageIndex{21}\)

## Solve Simple Interest Applications

Interest is calculated as simple interest or compound interest. Here we will use simple interest.

## SIMPLE INTEREST

Interest earned or paid according to this formula is called simple interest .

## EXAMPLE \(\PageIndex{12}\)

## Try It! \(\PageIndex{22}\)

## Try It! \(\PageIndex{23}\)

## EXAMPLE \(\PageIndex{13}\)

Write a complete sentence. The rate of interest was \(4\%.\)

## Try It! \(\PageIndex{24}\)

The rate of simple interest was 6%.

## Try It! \(\PageIndex{25}\)

The rate of simple interest was 5.5%.

In the next example, we are asked to find the principal—the amount borrowed.

## EXAMPLE \(\PageIndex{14}\)

Write a complete sentence. The principal was \($11,450.\)

## Try It! \(\PageIndex{26}\)

## Try It! \(\PageIndex{27}\)

## Key Concepts

\(\text{change}=\text{new amount}−\text{original amount}\)

\(\text{change is what percent of the original amount?}\)

- \( \begin{align*} \text{amount of discount} &= \text{discount rate}· \text{original price} \\ \text{sale price} &= \text{original amount}– \text{discount price} \end{align*}\)
- \(\begin{align*} \text{amount of mark-up} &= \text{mark-up rate}·\text{original price} \\ \text{list price} &= \text{original cost}–\text{mark-up} \end{align*}\)
- If an amount of money, \(P,\) called the principal, is invested or borrowed for a period of t years at an annual interest rate \(r,\) the amount of interest, \(I,\) earned or paid is: \[\begin{aligned} &{} &{} &{I=interest} \nonumber\\ &{I=Prt} &{\text{where} \space} &{P=principal} \nonumber\\ &{} &{\space} &{r=rate} \nonumber\\ &{} &{\space} &{t=time} \nonumber \end{aligned}\]

- 3.1 Use a Problem-Solving Strategy
- Introduction
- 1.1 Introduction to Whole Numbers
- 1.2 Use the Language of Algebra
- 1.3 Add and Subtract Integers
- 1.4 Multiply and Divide Integers
- 1.5 Visualize Fractions
- 1.6 Add and Subtract Fractions
- 1.7 Decimals
- 1.8 The Real Numbers
- 1.9 Properties of Real Numbers
- 1.10 Systems of Measurement
- Key Concepts
- Review Exercises
- Practice Test
- 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
- 2.2 Solve Equations using the Division and Multiplication Properties of Equality
- 2.3 Solve Equations with Variables and Constants on Both Sides
- 2.4 Use a General Strategy to Solve Linear Equations
- 2.5 Solve Equations with Fractions or Decimals
- 2.6 Solve a Formula for a Specific Variable
- 2.7 Solve Linear Inequalities
- 3.2 Solve Percent Applications
- 3.3 Solve Mixture Applications
- 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
- 3.5 Solve Uniform Motion Applications
- 3.6 Solve Applications with Linear Inequalities
- 4.1 Use the Rectangular Coordinate System
- 4.2 Graph Linear Equations in Two Variables
- 4.3 Graph with Intercepts
- 4.4 Understand Slope of a Line
- 4.5 Use the Slope-Intercept Form of an Equation of a Line
- 4.6 Find the Equation of a Line
- 4.7 Graphs of Linear Inequalities
- 5.1 Solve Systems of Equations by Graphing
- 5.2 Solving Systems of Equations by Substitution
- 5.3 Solve Systems of Equations by Elimination
- 5.4 Solve Applications with Systems of Equations
- 5.5 Solve Mixture Applications with Systems of Equations
- 5.6 Graphing Systems of Linear Inequalities
- 6.1 Add and Subtract Polynomials
- 6.2 Use Multiplication Properties of Exponents
- 6.3 Multiply Polynomials
- 6.4 Special Products
- 6.5 Divide Monomials
- 6.6 Divide Polynomials
- 6.7 Integer Exponents and Scientific Notation
- 7.1 Greatest Common Factor and Factor by Grouping
- 7.2 Factor Trinomials of the Form x2+bx+c
- 7.3 Factor Trinomials of the Form ax2+bx+c
- 7.4 Factor Special Products
- 7.5 General Strategy for Factoring Polynomials
- 7.6 Quadratic Equations
- 8.1 Simplify Rational Expressions
- 8.2 Multiply and Divide Rational Expressions
- 8.3 Add and Subtract Rational Expressions with a Common Denominator
- 8.4 Add and Subtract Rational Expressions with Unlike Denominators
- 8.5 Simplify Complex Rational Expressions
- 8.6 Solve Rational Equations
- 8.7 Solve Proportion and Similar Figure Applications
- 8.8 Solve Uniform Motion and Work Applications
- 8.9 Use Direct and Inverse Variation
- 9.1 Simplify and Use Square Roots
- 9.2 Simplify Square Roots
- 9.3 Add and Subtract Square Roots
- 9.4 Multiply Square Roots
- 9.5 Divide Square Roots
- 9.6 Solve Equations with Square Roots
- 9.7 Higher Roots
- 9.8 Rational Exponents
- 10.1 Solve Quadratic Equations Using the Square Root Property
- 10.2 Solve Quadratic Equations by Completing the Square
- 10.3 Solve Quadratic Equations Using the Quadratic Formula
- 10.4 Solve Applications Modeled by Quadratic Equations
- 10.5 Graphing Quadratic Equations in Two Variables

## Learning Objectives

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

- Approach word problems with a positive attitude
- Use a problem-solving strategy for word problems
- Solve number problems

## Be Prepared 3.1

Before you get started, take this readiness quiz.

## Be Prepared 3.2

Solve: 2 3 x = 24 . 2 3 x = 24 . If you missed this problem, review Example 2.16 .

Solve: 3 x + 8 = 14 . 3 x + 8 = 14 . If you missed this problem, review Example 2.27 .

## Approach Word Problems with a Positive Attitude

“If you think you can… or think you can’t… you’re right.”—Henry Ford

Use a Problem-Solving Strategy for Word Problems

## Use a Problem-Solving Strategy to Solve Word Problems.

- 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 algebraic 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.

## Example 3.1

If this were a homework exercise, our work might look like this:

Let’s try this approach with another example.

## Example 3.2

## Example 3.3

The difference of a number and six is 13. Find the number.

The difference of a number and eight is 17. Find the number.

The difference of a number and eleven is −7 . −7 . Find the number.

## Example 3.4

The sum of twice a number and seven is 15. Find the number.

The sum of four times a number and two is 14. Find the number.

The sum of three times a number and seven is 25. Find the number.

## Example 3.5

One number is five more than another. The sum of the numbers is 21. Find the numbers.

One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.

## Try It 3.10

The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.

## Example 3.6

## Try It 3.11

## Try It 3.12

The sum of two numbers is −18 . −18 . One number is 40 more than the other. Find the numbers.

## Example 3.7

One number is ten more than twice another. Their sum is one. Find the numbers.

## Try It 3.13

One number is eight more than twice another. Their sum is negative four. Find the numbers.

## Try It 3.14

One number is three more than three times another. Their sum is −5 . −5 . Find the numbers.

## Example 3.8

The sum of two consecutive integers is 47. Find the numbers.

## Try It 3.15

The sum of two consecutive integers is 95 . 95 . Find the numbers.

## Try It 3.16

The sum of two consecutive integers is −31 . −31 . Find the numbers.

## Example 3.9

Find three consecutive integers whose sum is −42 . −42 .

## Try It 3.17

Find three consecutive integers whose sum is −96 . −96 .

## Try It 3.18

Find three consecutive integers whose sum is −36 . −36 .

## Example 3.10

Find three consecutive even integers whose sum is 84.

## Try It 3.19

Find three consecutive even integers whose sum is 102.

## Try It 3.20

Find three consecutive even integers whose sum is −24 . −24 .

## Example 3.11

## Try It 3.21

## Try It 3.22

## Section 3.1 Exercises

Use the Approach Word Problems with a Positive Attitude

In the following exercises, prepare the lists described.

In the following exercises, solve each number word problem.

The sum of a number and eight is 12. Find the number.

The sum of a number and nine is 17. Find the number.

The difference of a number and 12 is three. Find the number.

The difference of a number and eight is four. Find the number.

The sum of three times a number and eight is 23. Find the number.

The sum of twice a number and six is 14. Find the number.

The difference of twice a number and seven is 17. Find the number.

The difference of four times a number and seven is 21. Find the number.

Three times the sum of a number and nine is 12. Find the number.

Six times the sum of a number and eight is 30. Find the number.

One number is six more than the other. Their sum is 42. Find the numbers.

One number is five more than the other. Their sum is 33. Find the numbers.

The sum of two numbers is 20. One number is four less than the other. Find the numbers.

The sum of two numbers is 27. One number is seven less than the other. Find the numbers.

The sum of two numbers is −45 . −45 . One number is nine more than the other. Find the numbers.

The sum of two numbers is −61 . −61 . One number is 35 more than the other. Find the numbers.

The sum of two numbers is −316 . −316 . One number is 94 less than the other. Find the numbers.

The sum of two numbers is −284 . −284 . One number is 62 less than the other. Find the numbers.

One number is one more than twice another. Their sum is −5 . −5 . Find the numbers.

One number is six more than five times another. Their sum is six. Find the numbers.

The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.

The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.

The sum of two consecutive integers is 77. Find the integers.

The sum of two consecutive integers is 89. Find the integers.

The sum of two consecutive integers is −23 . −23 . Find the integers.

The sum of two consecutive integers is −37 . −37 . Find the integers.

The sum of three consecutive integers is 78. Find the integers.

The sum of three consecutive integers is 60. Find the integers.

Find three consecutive integers whose sum is −3 . −3 .

Find three consecutive even integers whose sum is 258.

Find three consecutive even integers whose sum is 222.

Find three consecutive odd integers whose sum is 171.

Find three consecutive odd integers whose sum is 291.

Find three consecutive even integers whose sum is −36 . −36 .

Find three consecutive even integers whose sum is −84 . −84 .

Find three consecutive odd integers whose sum is −213 . −213 .

Find three consecutive odd integers whose sum is −267 . −267 .

## Everyday Math

Buying in Bulk Alicia bought a package of eight peaches for $3.20. Find the cost of each peach.

## Writing Exercises

What has been your past experience solving word problems?

When you start to solve a word problem, how do you decide what to let the variable represent?

What are consecutive odd integers? Name three consecutive odd integers between 50 and 60.

ⓑ If most of your checks were:

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Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction

- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
- Publisher/website: OpenStax
- Book title: Elementary Algebra 2e
- Publication date: Apr 22, 2020
- Location: Houston, Texas
- Book URL: https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
- Section URL: https://openstax.org/books/elementary-algebra-2e/pages/3-1-use-a-problem-solving-strategy

## Math Problem Solving Strategies

## Problem Solving Strategies

The strategies used in solving word problems:

## Solving Word Problems

## Problem Solving Strategy: Guess And Check

Using the guess and check problem solving strategy to help solve math word problems.

## Problem Solving : Make A Table And Look For A Pattern

- Identify - What is the question?
- Plan - What strategy will I use to solve the problem?
- Solve - Carry out your plan.
- Verify - Does my answer make sense?

## Find A Pattern Model (Intermediate)

a) The number of dots required for 7 rectangles is 52.

b) If the figure has 73 dots, there would be 10 rectangles.

The number of dots for 7 layers of triangles is 36.

Example: The following figures were formed using matchsticks.

a) Based on the above series of figures, complete the table below.

b) How many triangles are there if the figure in the series has 9 squares?

c) How many matchsticks would be used in the figure in the series with 11 squares?

Total = 6 + 5 + 4 + 3 + 2 + 1 = 21 handshakes.

## The following are some other examples of problem solving strategies.

Finding A Pattern (Basic) Finding A Pattern (Intermediate) Finding A Pattern (Advanced)

EVERYTHING YOU NEED FOR THE YEAR >>> ALL ACCESS

## Math Problem Solving Strategies

Grab your Problem Solving Strategy Freebie Here !

## 1. C.U.B.E.S.

- Why I like it: Gives students a very specific ‘what to do.’
- Why I don’t like it: With all of the annotating of the problem, I’m not sure that students are actually reading the problem. None of the steps emphasize reading the problem but maybe that is a given.

## 2. R.U.N.S.

- Why I like it: Students are forced to think about what type of problem it is (factoring, division, etc) and then come up with a plan to solve it using a strategy sentence. This is a great strategy to teach when you are tackling various types of problems.
- Why I don’t like it: Though I love the opportunity for students to write in math, writing a strategy statement for every problem can eat up a lot of time.

## 3. U.P.S. CHECK

U.P.S. Check stands for understand, plan, solve, and check.

- Why I like it: I love that there is a check step in this problem solving strategy. Students having to defend the reasonableness of their answer is essential for students’ number sense.
- Why I don’t like it: It can be a little vague and doesn’t give concrete ‘what to dos.’ Checking that students completed the ‘understand’ step can be hard to see.

## 4. Maneuvering the Middle Strategy AKA K.N.O.W.S.

UPDATE: IT DOES HAVE A NAME! Thanks to our lovely readers, Wendi and Natalie!

- Know: This will help students find the important information.
- Need to Know: This will force students to reread the question and write down what they are trying to solve for.
- Organize: I think this would be a great place for teachers to emphasize drawing a model or picture.
- Work: Students show their calculations here.
- Solution: This is where students will ask themselves if the answer is reasonable and whether it answered the question.

## 5. Digital Learning Struggle

## Printable and Digital Math Performance Tasks

Check out these related products from my shop.

## Reader Interactions

Great idea! Thanks so much for sharing with our readers!

LOVE this idea! Will definitely use it this year! Thank you!

I would love an anchor chart for RUBY

That’s brilliant! Thank you for sharing!

Going off of your idea, Natalie, how about the following?

I’m doing this one. Love it. Thank you!!

## IMAGES

## VIDEO

## COMMENTS

Problem-Solving Strategy · Read the word problem. Make sure you understand all the words and ideas. · Identify what you are looking for. · Name what you are

PROBLEM SOLVING STRATEGY FOR WORD PROBLEMS. Read the problem. Make sure all the words and ideas are understood. Identify what you are looking

https://www.youtube.com/@MathematicsTutor Register for Summer Math Classes with Anil

Use solved problems to engage students in analyzing algebraic reasoning and strategies. Compared to elementary mathematics work like arithmetic, solving algebra

Now that we can solve equations, we are ready to apply our new skills to word problems. We will develop a strategy we can use to solve any

Use a Problem-Solving Strategy for Word Problems. We have reviewed translating English phrases into algebraic expressions, using some basic

Understand the problem; Devise a plan – Translate; Carry out the plan – Solve; Look – Check and Interpret. Review Questions. Use the information

Plan and Compare Alternative Approaches to Solving Problems · Step 1: Understand · Step 2: Strategy · Step 3: Apply strategy/Solve · Step 4: Check.

Problem Solving : Make A Table And Look For A Pattern · Identify - What is the question? · Plan - What strategy will I use to solve the problem? · Solve - Carry

Here is where I typically struggle with problem solving strategies: 1) modeling the strategy in my own teaching weeks after I have taught