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1.5.4: Words that Describe Mathematical Operations

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  • Expressions for Real-Life Situations

    Kate needs to read 2 books over summer vacation. She doesn't have the books yet so she's not sure how many pages are in each book. Kate doesn't want to wait until the last minute to read the books so she is planning to read the same number of pages each day of her summer vacation. If Kate's summer vacation is 72 days long, how can she write a variable expression to represent the number of pages she will need to read each day?

    Here you will learn how to write variable expressions to represent real world situations.

    Writing Variable Expressions for Real-Life Situations

    A variable expression is a mathematical phrase that contains numbers, operations, and variables.

    Here are some examples of variable expressions:

    • \(3x+y\)
    • \(10r−x\)
    • \(b^3+2\)
    • \(mx−4\)

    You can use a variable expression to describe a real world situation where one or more quantities have an unknown value or can change in value.

    To write a variable expression for a real world situation:

    1. Figure out which quantity in the situation is unknown and define a variable to represent the unknown quantity.
    2. Write an expression using the variable to represent the situation. Look for key words to help you figure out what mathematical operations you should use.

    Here is an example.

    Ralph is a baker who makes the same number of loaves of bread each day. He uses 5 cups of flour in each loaf of bread. Write an expression to represent the number of cups of flour he uses each day making bread.

    First, decide on your variable. You don't know how many loaves of bread Ralph makes each day, but you know every day it is the same.

    Let i equal the number of loaves of bread Ralph makes each day.

    Next, write a variable expression using i. For each of the i loaves of bread Ralph uses 5 cups of flour. This means he uses 5i cups of flour to make i loaves of bread.

    5i

    The answer is Ralph uses 5i cups of flour each day making bread where i is the number of loaves of bread he bakes each day.

    Here is another example.

    Consider again Ralph who makes the same number of loaves of bread each day. Last Friday he only sold half of the loaves that he made. Write an expression to represent the number of loaves he sold.

    First, decide on your variable. You can use the same variable as last time since you still don't know the number of loaves of bread Ralph makes each day.

    Let i equal the number of loaves of bread Ralph makes each day.

    Next, write a variable expression using i. If last Friday he had i loaves of bread but only sold half of them, he sold 12i or i/2 loaves of bread.

    i/2

    The answer is Ralph sold i/2 loaves of bread last Friday where i is the number of loaves of bread he bakes each day.

    Examples

    Example \(\PageIndex{1}\)

    Earlier, you were given a problem about Kate and her summer reading.

    She has to read two books over her 72-day summer vacation. Her plan is to read the same number of pages each day.

    Solution

    First, Kate should decide on her variables. She doesn't know the number of pages in either book, so she will need two variables.

    Let x equal the number of pages in the first book.

    Let y equal the number of pages in the second book.

    Next, write a variable expression using x and y. In total Kate will have to read x+y pages. She is going to read the same number of pages each of the 72 days of vacation. This means she should divide the total number of pages by 72 to find out how many pages she needs to read each day. Her expression is

    (x+y)/72

    The answer is Kate needs to read (x+y)/72 pages each day if x is the number of pages in the first book and y is the number of pages in the second book.

    Example \(\PageIndex{1}\)

    John runs the same number of miles each day. Write an expression to represent the number of miles John ran in June last year.

    Solution

    First, decide on your variable. You don't know how many miles John runs each day, but you know every day it is the same.

    Let m equal the number of miles John runs each day.

    Next, write a variable expression using m. In one day John runs m miles. In 2 days John runs 2m miles. The question asks for an expression that represents how many miles John ran in June. June is a month with 30 days. So your expression is

    30 m

    The answer is John ran 30 m miles in June last year where m is the number of miles he runs each day.

    Example \(\PageIndex{1}\)

    Karen bakes the same number of cookies each day in her bakery. Write an expression to represent the total number of cookies Karen bakes in a week.

    Solution

    First, decide on your variable. You don't know how many cookies Karen bakes each day, but you know every day it is the same.

    Let c equal the number of cookies Karen bakes each day.

    Next, write a variable expression using c. In one day Karen bakes c cookies. In 2 days Karen bakes 2c cookies. The question asks for an expression that represents the total number of cookies Karen bakes in a week. A week has 7 days. So your expression is

    7c

    The answer is Karen bakes 7c cookies each week where c is the number of cookies she bakes each day.

    Example \(\PageIndex{1}\)

    Eden caught 14 fish and then ate some. Write an expression to represent the number of fish Eden has left.

    Solution

    First, decide on your variable. You don't know how many fish Eden ate.

    Let x equal the number of fish Eden ate.

    Next, write a variable expression using x. Eden started with 14 fish, but then ate some. This means she will end up with less fish than she started with. You need to subtract the number of fish she ate from the number she started with. Your expression is

    14−x

    The answer is Eden now has 14−x fish where x is the number of fish Eden ate.

    Example \(\PageIndex{1}\)

    Jessica reads the same number of books each month. Write an expression to represent the number of books Jessica reads in 2 years.

    Solution

    First, decide on your variable. You don't know how many books Jessica reads each month, but you know every month it is the same.

    Let b equal the number of books Jessica reads each month.

    Next, write a variable expression using b. In one month Jessica reads b books. In 2 months Jessica reads 2b books. The question asks for an expression that represents the total number of books Jessica reads in 2 years. There are 12 months in a year so there are 24 months in 2 years. So your expression is

    24b

    The answer is Jessica reads 24b books in 2 years where b is the number of books she reads each month.

    Review

    Choose a variable to represent the number and write a variable expression for the following phrases.

    1. 19 decreased by a number
    2. 4 less than the product of 4 and a number
    3. 30 more than a number
    4. 12 more than 3 times a number
    5. A number divided by seven
    6. A number and seven divided by two
    7. 16 times an unknown number
    8. A number divided by eight
    9. The quantity of a number and seven times two
    10. The quantity of a number and five divided by five
    11. 22 decreased by a number
    12. Seventeen less than three times a number

    Solve each problem.

    1. A librarian has 4 times as many mystery books as romances. She lends out 12 mysteries. How many mysteries does she have now if she started with 15 romances?
    2. In Saturday’s basketball game, Roman scored a fourth of his teams points. If the team scored 48 points total, how many points did Roman score? Write an expression and solve.
    3. At the garden show daffodil bulbs cost $3 and tulip bulbs cost $4. Latoya buys 7 tulip bulbs and twice as many daffodil bulbs as tulips bulbs. How much does she spend total? Write an expression and solve.

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 1.6.

    Additional Resources

    PLIX: Play, Learn, Interact, eXplore: Shopping for Sisters

    Video: Evaluating an Expression with Multiple Variables

    Activity: Variable Expressions Discussion Questions

    Practice: Words that Describe Mathematical Operations

    Real World Application: Cheez Its

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