Practice questions for this study guide can be found at:

Arithmetic Word Problems Practice Problems for McGraw-Hill's ASVAB

What is a word problem? Basically it is a description of a real-life situation that requires a mathematical solution. To solve a word problem, you need to translate the situation into mathematical terms and then calculate the answer.

All of the questions on the ASVAB Arithmetic Reasoning test are word problems. To do well, you need to learn good word problem–solving skills. The following article will show you many different kinds of arithmetic word problems. For each kind, you'll learn how to translate the facts of the problem into mathematical terms. Then you'll see how to use those terms to set up an equation. Finally, you'll see how to solve that equation for the missing piece of information that you're looking for. To start your study program, look at the following word problem-solving tips.

**Key words** Many word problems contain key words that tell you what mathematical operation you need to use to solve the problem. It pays to know these key words, so be sure you study the following list.

These key words tell you to *add*.

These key words tell you to *subtract*.

These key words tell you to *multiply*.

These key words tell you to *divide*.

These key words tell you to use an *equal sign*.

### Setting Up an Equation and Solving For an Unknown

Before you start tackling word problems, you need to know how to set up an equation and solve for an unknown. For each problem, you will be given certain pieces of information ("*what you know*"). You first need to translate this information into mathematical terms. Then you can use those terms to set up an equation. An *equation* is nothing more than a mathematical expression that indicates that one mathematical term is equal to another. The terms in an equation are shown on opposite sides of an equal sign. The missing piece of information that you are looking for ("*what you need to find*") is called the *unknown*. In your equation, you can represent an unknown by a letter such as x or *a*.

In the pages that follow, you will see how to set up equations for many different kinds of word problems. The process of solving an equation for an unknown is pretty simple. Work through these examples until you get perfectly comfortable with the process.

*Examples*

Solve for *x*.

- 25

*x*= 200

This equation is read, "25 times some unknown number *x* equals 200." Your task is to determine what number *x* is. Here is how to solve this equation.

- Divide both sides of the equation by 25:

Solve for *x*:

To solve this type of problem, cross-multiply (45 times *x* and 3 times 5).

Solve for *x*:

To solve this problem, cross-multiply (10 times 350 and 50 times *x*).

- 50

*x*= 3,500

*x*= 70

To check your answer, merely substitute 70 back into the equation and see if it works. Reduce the fractions to or cross-multiply (10 times 350 and 70 times 50), making 3,500 = 3,500. Either way, you are correct.

### Types of ASVAB Arithmetic Word Problems

In this article, you will learn about many different kinds of word problems that you are likely to see on the ASVAB Arithmetic Reasoning test. For each kind, you will see how to use the information you are given to set up an equation. Then you will see how to solve the equation for the unknown that is the answer to the problem. The following chart shows the different kinds of word problems discussed in this chapter.

### Simple Interest

*Interest* is an amount paid for the use of money. *Interest rate* is the percent paid per year. *Principal* is the amount of money on which interest is paid. *Simple interest* is interest that is computed based only on the principal, the interest rate, and the time. To calculate simple interest, use this formula:

- Interest = principal × rate × time

*I*=

*prt*

*Examples*

Martina has $300 in a savings account that pays simple interest at a rate of 3% per year. How much interest will she earn on that $300 if she keeps it in the account for 5 years?

**Procedure**

*What must you find?*Amount of simple interest*What are the units?*Dollars*What do you know?*Rate = 3% per year, time = 5 years, principal = $300*Create an equation and solve.*

*I*= ($300)(0.03)(5)

*I*= $45

Five years ago, Robin deposited $500 in a savings account that pays simple interest. She made no further deposits, and today the account is worth $750. What is the rate of interest?

*Procedure*

*What must you find?*Rate of simple interest*What are the units?*Percent per year*What do you know?*Interest = $250 ($750 – $500); principal = $500; time = 5 years

*Create an equation and solve.*

### Compound Interest

*Compound interest* is the interest paid on the principal and also on any interest that has already been paid. To calculate compound interest, you can use the formula *I* = *prt*, but you must calculate the interest for each time period and then combine them for a total.

*Example*

Ricardo bought a $1,000 savings bond that earns 5% interest compounded annually. How much interest will he earn in two years?

*Procedure*

*What must you find?*Amount of compound interest*What are the units?*Dollars*What do you know?*Principal = $1,000; rate = 5%; time = 2 years

*Create an equation and solve.*

To find the compound interest, calculate the amount earned in the first year. Add that amount to the principal, then calculate the interest earned in the second year. Total the amount of interest earned in the two years.

Year 1:

*I*=

*prt*

*I*= ($1,000)(0.05)(1)

*I*= $50

New principal = $1,050

Year 2:

*I*=

*prt*

*I*= ($1,050)(0.05)(1)

*I*= $52.50

So the total compound interest paid in two years is $50 + $52.50 = $102.50.

### Ratio and Proportion

On the ASVAB, you will almost certainly encounter word problems that will require you to work with ratios and proportions.

*Example 1 *

Kim reads an average of 150 pages per week. At that rate, how many weeks will it take him to read 1,800 pages?

*Procedure:*

*What must you find?*How long it will take to read 1,800 pages.*What are the units?*Weeks*What do you know?*150 pages read each week; 1,800 pages to be read

*Set up a proportion and solve.*

Substitute values into the equation.

Cross-multiply:

150*x* = 1,800

*x* = 12 weeks

**Example 2**

It takes 8 hours to fill a swimming pool that holds 3,500 gallons of water. At that rate, how many hours will it take to fill a pool that holds 8,750 gallons?

*Procedure:*

*What must you find?*How long it will take to fill the 8,750-gallon pool*What are the units?*Hours*What do you know?*Number of hours for 3,500 gallons*Set up a proportion and solve.*

Cross-multiply:

3500*x* = (8)(8,750)

*x* = 20 hours

**Example 3**

An airplane travels the 1,700 miles from Phoenix to Nashville in 2.5 hours. Flying at the same speed, the plane could travel the 2,550 miles from Phoenix to Boston in how many hours?

*Procedure*

*What must you find?*Time it would take to fly 2,550 miles*What are the units?*Hours*What do you know?*The plane traveled 1,700 miles in 2.5 hours

*Set up a proportion and solve.*

Substitute values.

Cross-multiply:

1,700*x* = (2.5)(2,550)

1,700*x* = 6,375

*x* = 3.75 hours

### Motion

Motion problems deal with how long it will take to get from point *a* to point *b* if you are traveling at a certain steady rate. To solve them, use this formula:

- Distance = rate × time

*d*=

*rt*

*Example*

If a racing boat travels at a steady rate of 80 miles per hour, how many miles could it travel in 3.5 hours?

*Procedure*

*What must you find?*Distance traveled in 3.5 hours*What are the units?*Miles*What do you know?*Rate = 80 miles per hour; time = 3.5 hours*Create an equation and solve.*

*d* = *rt*

Substitute values into the formula:

*d* = (80)(3.5)

*d* = 280 miles

### Percent

There are likely to be word problems involving percent on both the Arithmetic Reasoning and Math Knowledge tests of the ASVAB.

*Example 1 *

Lilly's bill at a restaurant is $22.00, and she wants to leave a 15% tip. How much money should her tip be?

*Procedure*

*What must you find?*Amount of tip*What are the units?*Dollars and cents*What do you know?*Total bill = $22.00; Percent of tip = 15*Create an equation and solve.*

Tip = 15% × 22.00

Substitute and solve.

*t* = (0.15)(22.00)

*t* = $3.30

**Example 2**

Frederick earns $1,500 per month at his job, but 28% of that amount is deducted for taxes. What is his monthly take-home pay?

*Procedure*

*What must you find?*Monthly take-home pay*What are the units?*Dollars and cents*What do you know?*Monthly pay before taxes = $1,500; percent deducted = 28%*Create an equation and solve.*

Take-home pay is 1,500 minus 28% × 1,500.

*T* = 1,500 – (1,500 × 0.28)

*T* = 1,500 – (420)

*T* = $1080

**Example 3**

40 is 80% of what number?

*Procedure:*

*What must you find?*Number of which 40 is 80%*What are the units?*Numbers*What do you know?*40 is 80% of some larger number

*Create an equation and solve.*

### Percent Change

Some ASVAB word problems ask you to calculate the percent change from one number or amount to another.

*Example 1 *

Samantha now earns $300 per month working at a cosmetics store, but starting next month her monthly salary will be $375. Her new salary will be what percent increase over her current salary?

*Procedure*

*What must you find?*Percent change from current salary*What are the units?*Percent*What do you know?*Current pay = $300/month; pay after the raise = $375/month*Create an equation and solve.*

Substitute values and solve:

**Example 2**

On his sixteenth birthday, Brad was 60 inches tall.

On his seventeenth birthday, he was 65 inches tall. What was the percent increase in Brad's height during the year?

*Procedure*

*What must you find?*Percent change in height*What are the units?*Percent*What do you know?*Starting height = 60 inches; height after a year = 65 inches*Create an equation and solve.*

Substitute values and solve.

**Example 3**

At a certain store, every item is discounted by 15% off the original price. If Kevin buys a CD originally priced at $15.00 and a baseball cap originally priced at $11.50, how much money will he save?

*Procedure*

*What must you find?*Total amount saved*What are the units?*Dollars and cents*What do you know?*Percent change = 15%; original price for two items = $15.00 + $11.50 = $26.50*Create an equation and solve.*

Substitute values and solve.

### Numbers and Number Relationships

Pay attention to the key words in this type of word problem.

*Example 1 *

If the sum of two numbers is 45 and one number is 5 more than the other, what are the two numbers?

*Procedure*

*What must you find?*Value of each number*What are the units?*Numbers*What do you know?*Sum of two numbers is 45; one number is 5 more than the other*Create an equation and solve.*

Let *x* be the smaller number.

*x* + (*x* + 5) = 45

Solve for *x*:

*x* + *x* + 5 = 45

2*x* = 40

*x* = 20

So the two numbers are 20 and 20 + 5 = 25.

**Example 2**

One number is twice the size of another, and the two numbers together total 150. What are the two numbers?

*Procedure*

*What must you find?*Value of each number*What are the units?*Numbers*What do you know?*One number is twice the size of the other, and the sum of the numbers is 150.*Create an equation and solve.*

Let *x* be the smaller number.

*x* + 2*x* = 150

3*x* = 150

*x* = 50

So the smaller number is 50 and the larger number is 100.

### Age

Some word problems ask you to calculate a person's age given certain facts.

*Examples*

Jessica is 26 years old. Two years ago she was twice as old as her brother Ned. How old is Ned now?

*Procedure*

*What must you find?*Ned's age now*What are the units?*Years*What do you know?*When Jessica was 24, she was 2 times as old as Ned*Create an equation and solve.*

Prepare an equation that shows the relationship.

Let *x* = Ned's age two years ago.

2*x* = 24

*x* = 12

Ned's age two years later = 12 + 2 = 14 years

### Measurement

Some word problems will ask you to use what you know about units of measure to solve problems.

*Example*

How many cups of milk are in 5 pints of milk?

*Procedure*

*What must you find?*The number of cups of milk in 5 pints*What are the units?*Cups*What do you know?*According to the chart in Chapter 8, there are 2 cups in 1 pint.*Create an equation and solve.*

Let *x* = the number of cups in 5 pints.

1 pint = 2 cups

5 × 1 pint = 5 × 2 cups (Multiply both sides of the equation by 5)

5 pints = 10 cups = *x* cups

*x* = 10

So there are 10 cups in 5 pints.

Practice questions for this study guide can be found at:

Arithmetic Word Problems Practice Problems for McGraw-Hill's ASVAB

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