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Use a Problem Solving Strategy

Approach word problems with a positive attitude

Approach Word Problems with a Positive Attitude

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. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student in ?

When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.

Start with a fresh slate and begin to think positive thoughts like the student in . Read the positive thoughts and say them out loud.

If we take control and believe we can be successful, we will be able to master word problems.

Think of something that you can do now but couldn't do three years ago. Whether it's driving a car, snowboarding, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!

Use a Problem-solving Strategy for Word Problems

In earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then you've increased your math vocabulary as you learned about more algebraic procedures, and you've had more practice translating from words into algebra.

You have also translated word sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. 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.

Let's use this approach with another example.

Condensed — the full section is in OpenStax Prealgebra 2e.

Solve Number Problems

Now we will translate and solve number problems. In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don't usually arise on an everyday basis, but they provide a good introduction to practicing the Problem Solving Strategy. Remember to look for clue words such as difference, of, and and.

Example

Try it.

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

Solution

Step 1. Read the problem. Do you understand all the words?
Step 2. Identify what you are looking for.	the number
Step 3. Name. Choose a variable to represent the number.	Let $n=\text{the number}$
Step 4. Translate. Restate as one sentence. Translate into an equation.
Step 5. Solve the equation. Add 6 to both sides. Simplify.
Step 6. Check: The difference of 19 and 6 is 13. It checks.
Step 7. Answer the question.	The number is 19.

Example

Try it.

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

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the number
Step 3. Name. Choose a variable to represent the number.	Let $n=\text{the number}$
Step 4. Translate. Restate the problem as one sentence. Translate into an equation.
Step 5. Solve the equation.
Subtract 7 from each side and simplify.
Divide each side by 2 and simplify.
Step 6. Check: is the sum of twice 4 and 7 equal to 15?
Step 7. Answer the question.	The number is 4.

Some number word problems ask you to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. We will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

\[\text{...}-10,-9,-8,-7\text{,...}\]

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Problem Solving Strategy
1. 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.
2. Identify what you are looking for.
3. Name what you are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to first restate the problem in one sentence before translating.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem. Make sure it makes sense.
7. Answer the question with a complete sentence.

Use a Problem Solving Strategy

Use a Problem-solving Strategy for Word Problems

In the following exercises, use the problem-solving strategy for word problems to solve. Answer in complete sentences.

Try it.

Two-thirds of the children in the fourth-grade class are girls. If there are $20$ girls, what is the total number of children in the class?

Solution

There are 30 children in the class.

Try it.

Three-fifths of the members of the school choir are women. If there are $24$ women, what is the total number of choir members?

Try it.

Zachary has $25$ country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

Solution

Zachary has 125 CDs.

Try it.

One-fourth of the candies in a bag of are red. If there are $23$ red candies, how many candies are in the bag?

Try it.

There are $16$ girls in a school club. The number of girls is $4$ more than twice the number of boys. Find the number of boys in the club.

Solution

There are 6 boys in the club.

Try it.

There are $18$ Cub Scouts in Troop $645.$ The number of scouts is $3$ more than five times the number of adult leaders. Find the number of adult leaders.

Try it.

Lee is emptying dishes and glasses from the dishwasher. The number of dishes is $8$ less than the number of glasses. If there are $9$ dishes, what is the number of glasses?

Solution

There are 17 glasses.

Try it.

The number of puppies in the pet store window is twelve less than the number of dogs in the store. If there are $6$ puppies in the window, what is the number of dogs in the store?

Try it.

After $3$ months on a diet, Lisa had lost $\text{12\%}$ of her original weight. She lost $21$ pounds. What was Lisa's original weight?

Solution

Lisa's original weight was 175 pounds.

Try it.

Tricia got a $\text{6\%}$ raise on her weekly salary. The raise was $\text{\$30}$ per week. What was her original weekly salary?

Try it.

Tim left a $\text{\$9}$ tip for a $\text{\$50}$ restaurant bill. What percent tip did he leave?

Solution

18%

Try it.

Rashid left a $\text{\$15}$ tip for a $\text{\$75}$ restaurant bill. What percent tip did he leave?

Try it.

Yuki bought a dress on sale for $\text{\$72}.$ The sale price was $\text{60\%}$ of the original price. What was the original price of the dress?

Solution

The original price was $120.

Try it.

Kim bought a pair of shoes on sale for $\text{\$40.50}.$ The sale price was $\text{45\%}$ of the original price. What was the original price of the shoes?

Solve Number Problems

In the following exercises, solve each number word problem.

Try it.

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

Solution

Try it.

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

Try it.

The difference of a number and twelve is $3.$ Find the number.

Solution

Try it.

The difference of a number and eight is $4.$ Find the number.

Try it.

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

Solution

Try it.

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

Try it.

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

Solution

Try it.

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

Try it.

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

Solution

−5

Try it.

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

Try it.

One number is six more than the other. Their sum is forty-two. Find the numbers.

Solution

18, 24

Try it.

One number is five more than the other. Their sum is thirty-three. Find the numbers.

Try it.

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

Solution

8, 12

Try it.

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

Try it.

A number is one more than twice another number. Their sum is negative five. Find the numbers.

Solution

−2, −3

Try it.

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

Try it.

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

Solution

4, 10

Try it.

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

Try it.

One number is fourteen less than another. If their sum is increased by seven, the result is $85.$ Find the numbers.

Solution

32, 46

Try it.

One number is eleven less than another. If their sum is increased by eight, the result is $71.$ Find the numbers.

Try it.

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

Solution

38, 39

Try it.

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

Try it.

The sum of two consecutive integers is $-23.$ Find the integers.

Solution

−11, −12

Try it.

The sum of two consecutive integers is $-37.$ Find the integers.

Try it.

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

Solution

25, 26, 27

Try it.

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

Try it.

Find three consecutive integers whose sum is $-36.$

Solution

−11, −12, −13

Try it.

Find three consecutive integers whose sum is $-3.$

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Translate $\text{“6}$ less than twice $\text{x}\text{”}$ into an algebraic expression.
If you missed this problem, review .

Cevabı açıkla.

$2x-6$
Solve: $\frac{2}{3}x=24.$
If you missed this problem, review .

Cevabı açıkla.

$36$
Solve: $3x+8=14.$
If you missed this problem, review .

Cevabı açıkla.

$2$
Pete bought a shirt on sale for $\text{\$18},$ which is one-half the original price. What was the original price of the shirt?
Cevabı açıkla.
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 that 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 $p=$ 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 5. Solve the equation using good algebra techniques. Even if you know the answer right away, using algebra will better prepare you to solve problems that do not have obvious answers.
Write the equation.
Multiply both sides by 2.
Simplify.
Step 6. Check the answer in the problem and make sure it makes sense.
- We found that $p=36,$ which means the original price was $\text{\$36}.$ Does $\text{\$36}$ make sense in the problem? Yes, because $18$ is one-half of $36,$ 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 $\text{\$36}.”$
If this were a homework exercise, our work might look like this:
Joaquin bought a bookcase on sale for $\text{\$120},$ which was two-thirds the original price. What was the original price of the bookcase?

Cevabı açıkla.

$180
Two-fifths of the people in the senior center dining room are men. If there are $16$ men, what is the total number of people in the dining room?

Cevabı açıkla.

40

Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought $11$ apples to the picnic. How many bananas did he bring?

Cevabı açıkla.

Step 1. Read the problem.
Step 2. Identify what you are looking for.	How many bananas did he bring?
Step 3. Name what you are looking for. Choose a variable to represent the number of bananas.	Let $b=\text{number of bananas}$
Step 4. Translate. Restate the problem in one sentence with all the important information. Translate into an equation.
Step 5. Solve the equation.
Subtract 3 from each side.
Simplify.
Divide each side by 2.
Simplify.
Step 6. Check: First, is our answer reasonable? Yes, bringing four bananas to a picnic seems reasonable. The problem says the number of apples was three more than twice the number of bananas. If there are four bananas, does that make eleven apples? Twice 4 bananas is 8. Three more than 8 is 11.
Step 7. Answer the question.	Yash brought 4 bananas to the picnic.

Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was $3$ more than the number of notebooks. He bought $5$ textbooks. How many notebooks did he buy?

Cevabı açıkla.

2
Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is seven more than the number of crossword puzzles. He completed $14$ Sudoku puzzles. How many crossword puzzles did he complete?

Cevabı açıkla.

7

Nga's car insurance premium increased by $\text{\$60},$ which was $\text{8\%}$ of the original cost. What was the original cost of the premium?

Cevabı açıkla.

Step 1. Read the problem. Remember, if there are words you don't understand, look them up.
Step 2. Identify what you are looking for.	the original cost of the premium
Step 3. Name. Choose a variable to represent the original cost of premium.	Let $c=\text{the original cost}$
Step 4. Translate. Restate as one sentence. Translate into an equation.
Step 5. Solve the equation.
Divide both sides by 0.08.
Simplify.	$c=750$
Step 6. Check: Is our answer reasonable? Yes, a $750 premium on auto insurance is reasonable. Now let's check our algebra. Is 8% of 750 equal to 60?
Step 7. Answer the question.	The original cost of Nga's premium was $750.

Pilar's rent increased by $\text{4\%}.$ The increase was $\text{\$38}.$ What was the original amount of Pilar's rent?

Cevabı açıkla.

$950
Steve saves $\text{12\%}$ of his paycheck each month. If he saved $\text{\$504}$ last month, how much was his paycheck?

Cevabı açıkla.

$4,200

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

Cevabı açıkla.

Step 1. Read the problem. Do you understand all the words?
Step 2. Identify what you are looking for.	the number
Step 3. Name. Choose a variable to represent the number.	Let $n=\text{the number}$
Step 4. Translate. Restate as one sentence. Translate into an equation.
Step 5. Solve the equation. Add 6 to both sides. Simplify.
Step 6. Check: The difference of 19 and 6 is 13. It checks.
Step 7. Answer the question.	The number is 19.

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

Cevabı açıkla.

25
The difference of a number and eleven is $-7.$ Find the number.

Cevabı açıkla.

4

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

Cevabı açıkla.

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the number
Step 3. Name. Choose a variable to represent the number.	Let $n=\text{the number}$
Step 4. Translate. Restate the problem as one sentence. Translate into an equation.
Step 5. Solve the equation.
Subtract 7 from each side and simplify.
Divide each side by 2 and simplify.
Step 6. Check: is the sum of twice 4 and 7 equal to 15?
Step 7. Answer the question.	The number is 4.

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

Cevabı açıkla.

3
The sum of three times a number and seven is $25.$ Find the number.

Cevabı açıkla.

6

One number is five more than another. The sum of the numbers is twenty-one. Find the numbers.

Cevabı açıkla.

Step 1. Read the problem.
Step 2. Identify what you are looking for.		You are looking for two numbers.
Step 3. Name. Choose a variable to represent the first number. What do you know about the second number? Translate.		Let $n=\text{1st number}$ One number is five more than another. $n+5={2}^{\text{nd}}\text{ number}$
Step 4. Translate. Restate the problem as one sentence with all the important information. Translate into an equation. Substitute the variable expressions.		The sum of the numbers is 21. The sum of the 1st number and the 2nd number is 21.
Step 5. Solve the equation.
Combine like terms.
Subtract five from both sides and simplify.
Divide by two and simplify.
Find the second number too.
Substitute n = 8

Step 6. Check:
Do these numbers check in the problem? Is one number 5 more than the other? Is thirteen, 5 more than 8? Yes. Is the sum of the two numbers 21?
Step 7. Answer the question.		The numbers are 8 and 13.

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

Cevabı açıkla.

9, 15
The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.

Cevabı açıkla.

27, 31

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

Cevabı açıkla.

Step 1. Read the problem.
Step 2. Identify what you are looking for.		two numbers
Step 3. Name. Choose a variable. What do you know about the second number? Translate.		Let n = 1^st number One number is 4 less than the other. n - 4 = 2^nd number
Step 4. Translate. Write as one sentence. Translate into an equation. Substitute the variable expressions.		The sum of two numbers is negative fourteen.
Step 5. Solve the equation.
Combine like terms.
Add 4 to each side and simplify.
Divide by 2.
Substitute $n=-5$ to find the 2^nd number.


Step 6. Check:
Is −9 four less than −5? Is their sum −14?
Step 7. Answer the question.		The numbers are −5 and −9.

The sum of two numbers is negative twenty-three. One number is $7$ less than the other. Find the numbers.

Cevabı açıkla.

−8, −15
The sum of two numbers is negative eighteen. One number is $40$ more than the other. Find the numbers.

Cevabı açıkla.

−29, 11

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

Cevabı açıkla.

Step 1. Read the problem.
Step 2. Identify what you are looking for.		two numbers
Step 3. Name. Choose a variable. One number is ten more than twice another.		Let x = 1^st number 2x + 10 = 2^nd number
Step 4. Translate. Restate as one sentence.		Their sum is one.
Translate into an equation
Step 5. Solve the equation.
Combine like terms.
Subtract 10 from each side.
Divide each side by 3 to get the first number.
Substitute to get the second number.


Step 6. Check.
Is 4 ten more than twice −3? Is their sum 1?
Step 7. Answer the question.		The numbers are −3 and 4.

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

Cevabı açıkla.

−4, 0
One number is three more than three times another. Their sum is negative five. Find the numbers.

Cevabı açıkla.

−2, −3

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

Cevabı açıkla.

Step 1. Read the problem.
Step 2. Identify what you are looking for.		two consecutive integers
Step 3. Name.		Let n = 1^st integer n + 1 = next consecutive integer
Step 4. Translate. Restate as one sentence. Translate into an equation.
Step 5. Solve the equation.
Combine like terms.
Subtract 1 from each side.
Divide each side by 2.
Substitute to get the second number.


Step 6. Check:
Step 7. Answer the question.		The two consecutive integers are 23 and 24.

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

Cevabı açıkla.

47, 48
The sum of two consecutive integers is $-31.$ Find the numbers.

Cevabı açıkla.

−15, −16

Find three consecutive integers whose sum is $42.$

Cevabı açıkla.

Step 1. Read the problem.
Step 2. Identify what you are looking for.		three consecutive integers
Step 3. Name.		Let n = 1^st integer n + 1 = 2^nd consecutive integer n + 2 = 3^rd consecutive integer
Step 4. Translate. Restate as one sentence. Translate into an equation.
Step 5. Solve the equation.
Combine like terms.
Subtract 3 from each side.
Divide each side by 3.
Substitute to get the second number.


Substitute to get the third number.


Step 6. Check:
Step 7. Answer the question.		The three consecutive integers are 13, 14, and 15.

Find three consecutive integers whose sum is $96.$

Cevabı açıkla.

31, 32, 33
Find three consecutive integers whose sum is $-36.$

Cevabı açıkla.

−11, −12, −13
Two-thirds of the children in the fourth-grade class are girls. If there are $20$ girls, what is the total number of children in the class?

Cevabı açıkla.

There are 30 children in the class.
Three-fifths of the members of the school choir are women. If there are $24$ women, what is the total number of choir members?
Zachary has $25$ country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?

Cevabı açıkla.

Zachary has 125 CDs.
One-fourth of the candies in a bag of are red. If there are $23$ red candies, how many candies are in the bag?
There are $16$ girls in a school club. The number of girls is $4$ more than twice the number of boys. Find the number of boys in the club.

Cevabı açıkla.

There are 6 boys in the club.
There are $18$ Cub Scouts in Troop $645.$ The number of scouts is $3$ more than five times the number of adult leaders. Find the number of adult leaders.
Lee is emptying dishes and glasses from the dishwasher. The number of dishes is $8$ less than the number of glasses. If there are $9$ dishes, what is the number of glasses?

Cevabı açıkla.

There are 17 glasses.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Use a Problem Solving Strategy

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

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Kendini dene.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Daha fazlası Arithmetic

Order of operations Fractions Percentages Powers and roots