Word Problems to
Expressions
Turn everyday sentences into algebra by translating words like "more than," "of," and "per" into symbols such as \( + \), \( \times \), and \( = \).
What Is an Expression?
An algebraic expression is a combination of numbers, variables, and operations. Unlike an equation, it has no equals sign on its own. A variable (often \( x \) or \( n \)) is a placeholder for an unknown number.
Why Translation Matters
On the ISEE, many problems describe a situation in plain English. Before you can solve, you must convert the words into math. Getting this first step right is the difference between a correct answer and a careless miss.
Keywords: Add & Subtract
| Phrase (Addition) | Phrase (Subtraction) |
|---|---|
| sum of | difference of |
| more than | less than |
| increased by | decreased by |
| added to | fewer than |
| total | take away |
Keywords: Multiply & Divide
| Phrase (Multiply) | Phrase (Divide) |
|---|---|
| product of | quotient of |
| times | divided by |
| of (with fractions/%) | per |
| twice, double, triple | split equally |
| each, every | ratio of |
Keywords for Equals
An expression has no equals sign, but a word problem often gives you an equation. These phrases signal \( = \):
Expression vs. Equation
| Expression | Equation | |
|---|---|---|
| Has \( = \)? | No | Yes |
| Represents | A quantity | A relationship |
| Can be solved? | No, only simplified | Yes |
| Example | \( 2x + 3 \) | \( 2x + 3 = 11 \) |
A 4-Step Approach
Example 1: Building Up
"Seven more than twice a number." Write an expression.
Example 2: Watch the Order
"Three less than half of a number." Write an expression.
Example 3: A Real Situation
"A taxi charges $3 plus $2 per mile. Write an expression for the cost of \( m \) miles."
Special Case: Consecutive Integers
When a problem mentions consecutive numbers, express them all in terms of one variable.
Common Mistakes
When You Need Parentheses
If an operation applies to a whole group, wrap that group in parentheses first.
| Phrase | Expression |
|---|---|
| twice the sum of \( x \) and 5 | \( 2(x+5) \) |
| 3 times a number, then add 5 | \( 3x + 5 \) |
| half of the difference of \( a \) and \( b \) | \( \frac{1}{2}(a-b) \) |
| the sum of \( x \) and 5, times 2 | \( 2(x+5) \) |
Quick Reference
| Words | Symbol | Example |
|---|---|---|
| more than, sum | \( + \) | \( x + 4 \) |
| less than, difference | \( - \) | \( x - 4 \) |
| of, times, product | \( \times \) | \( 4x \) |
| per, quotient, ratio | \( \div \) | \( \frac{x}{4} \) |
| is, equals, results in | \( = \) | \( x = 4 \) |
Question 1 of 5
Which expression represents "eight less than a number \( n \)"?
Question 2 of 5
Translate: "five more than three times a number \( x \)."
Question 3 of 5
A gym charges a $20 fee plus $15 per month. Which expression gives the cost for \( m \) months?
Question 4 of 5
Which expression represents "twice the sum of \( a \) and 6"?
Question 5 of 5
If \( n \) is the smallest of three consecutive even integers, which expression represents their sum?