Use Variables in
Real-World Problems
Learn how variables represent unknown numbers in everyday situations: from counting coins to splitting pizza. Turn words into math expressions you can solve.
What Is a Variable?
A variable is a letter (like \( x \), \( n \), or \( t \)) that stands in for a number you don't know yet. Think of it as a blank space waiting to be filled.
Why Do Variables Matter?
Variables let you write math for situations you haven't solved yet. Instead of guessing, you set up an expression, then find the answer.
Two Roles of a Variable
A variable can play two different roles depending on the problem:
Example: "Maria's age is \( a \). She is 12." Here \( a = 12 \).
Example: "The perimeter of a square is \( 4s \)." Here \( s \) can be any side length.
From Words to Expressions
The core skill: translate English phrases into math expressions using a variable for the unknown quantity.
Keyword Clues for Operations
| Operation | Keywords | Example Expression |
|---|---|---|
| Addition | more than, sum, increased by, plus | \( x + 7 \) |
| Subtraction | less than, minus, decreased by, fewer | \( x - 3 \) |
| Multiplication | times, product, each, per, of | \( 4x \) |
| Division | divided by, quotient, split, per | \( \frac{x}{2} \) |
Expression vs. Equation
Example: \( 3x + 5 \)
You can evaluate it for different values of \( x \).
Example: \( 3x + 5 = 20 \)
You can solve it to find the value of \( x \).
How to Write an Expression
Example 1: Movie Tickets
Problem: Each movie ticket costs $9. Write an expression for the total cost of \( t \) tickets.
Example 2: Sharing Pizza
Problem: A pizza has 12 slices. You and your friends share it equally. Write an expression for the number of slices each person gets if there are \( p \) people.
Example 3: Savings Account
Problem: Jamal has $35 in savings. Each week he adds $8. Write an expression for his total savings after \( w \) weeks.
Evaluating an Expression
Once you write an expression, you can evaluate it by replacing the variable with a specific number.
Expressions with Multiple Variables
Some real-world problems have more than one unknown. You simply use a different variable for each.
Choosing Good Variable Names
Any letter works, but meaningful letters make your work easier to read and understand.
| Situation | Poor Choice | Better Choice |
|---|---|---|
| Number of hours | \( z \) | \( h \) |
| Distance traveled | \( q \) | \( d \) |
| Number of students | \( y \) | \( s \) or \( n \) |
| Temperature | \( a \) | \( t \) |
Common Mistakes
Summary Cheat Sheet
| English Phrase | Operation | Expression |
|---|---|---|
| 8 more than a number | Add | \( n + 8 \) |
| a number decreased by 4 | Subtract | \( n - 4 \) |
| triple a number | Multiply | \( 3n \) |
| a number divided by 6 | Divide | \( \frac{n}{6} \) |
| 5 less than twice a number | Multiply, then subtract | \( 2n - 5 \) |
| the sum of a number and 10, divided by 3 | Add, then divide | \( \frac{n + 10}{3} \) |
Question 1 of 5
A book costs \( d \) dollars. You buy 4 books. Which expression represents the total cost?
Question 2 of 5
Which expression means "9 less than a number \( n \)"?
Question 3 of 5
Rosa earns $12 per hour. She already has $50 saved. Which expression shows her total money after working \( h \) hours?
Question 4 of 5
A rectangle has a length of 10 cm and a width of \( w \) cm. If you evaluate the perimeter expression \( 2(10 + w) \) when \( w = 6 \), what do you get?
Question 5 of 5
In the expression \( 5t + 3 \), the variable \( t \) can represent "any number in a specified set." What does this mean?