Writing Expressions | Open Textbooks

What Are Algebraic Expressions?

Expressions: Math's Language

An algebraic expression is a combination of numbers, variables, and operations that represents a quantity.

What is a Variable?

x, n, y

A variable is a letter that stands for an unknown number. We use variables when we don't know the exact value yet, or when the value can change.

Key Words and Their Operations

When translating words to math, certain words signal which operation to use:

Operation	Symbol	Key Words
Addition	+	sum, plus, more than, increased by, total, added to, combined
Subtraction	-	difference, minus, less than, decreased by, fewer, take away
Multiplication	x	product, times, of, multiplied by, twice, double, triple
Division	/	quotient, divided by, per, ratio, split, shared equally

Translating Words to Expressions

"five more than a number"

v

n + 5

"a number decreased by 7"

v

n - 7

"twice a number"

v

2n

"a number divided by 4"

v

n / 4

Watch Out for Order! "Less than" and "subtracted from" flip the order.
"5 less than n" means n - 5 (not 5 - n).
"8 subtracted from n" means n - 8.

Writing Multiplication with Variables

In algebra, we often skip the multiplication sign:

Multiplication Notation

3 x n is usually written as 3n
n x n is written as n^2
2 x (n + 5) is written as 2(n + 5)
Numbers always go before variables: write 3n, not n3

Combining Multiple Operations

"three more than twice a number"

v

2n + 3

"half of a number plus 10"

v

n/2 + 10

"5 times the sum of n and 4"

v

5(n + 4)

"the product of 6 and n, minus 2"

v

6n - 2

Strategy: Break phrases into parts. Find the key operation words, identify what's being done to the variable, then build the expression step by step.

Worked Examples

Let's practice translating phrases into algebraic expressions.

Example 1: Simple Expression

Write an expression for: "the sum of a number and 12"

1

Identify the key word:
"sum" means addition.

2

Identify what we're adding:
A number (use a variable like n) and 12.

3

Write the expression:
n + 12

"The sum of a number and 12" = n + 12

Example 2: "Less Than" Phrase

Write an expression for: "9 less than a number"

1

Identify the key phrase:
"less than" means subtraction, but with reversed order.

2

Think about what it means:
"9 less than a number" means start with the number, then subtract 9.

3

Write the expression:
n - 9 (not 9 - n)

"9 less than a number" = n - 9

Example 3: Two Operations

Write an expression for: "four times a number, decreased by 5"

1

Find the first operation:
"four times a number" = 4n

2

Find the second operation:
"decreased by 5" means subtract 5 from the result.

3

Combine:
4n - 5

"Four times a number, decreased by 5" = 4n - 5

Example 4: Using Parentheses

Write an expression for: "double the sum of a number and 7"

1

Find what goes inside:
"the sum of a number and 7" = n + 7

2

Apply the outer operation:
"double" means multiply by 2.

3

Use parentheses to show what's being doubled:
2(n + 7)

"Double the sum of a number and 7" = 2(n + 7)

Example 5: Real-World Context

A movie ticket costs $12. Write an expression for the cost of n tickets.

1

Identify what changes:
The number of tickets (n) is the variable.

2

Identify the operation:
Total cost = price per ticket x number of tickets

3

Write the expression:
12n (or 12 x n)

Cost of n movie tickets = 12n dollars

Practice Problems

Translate each phrase into an algebraic expression.

Problem 1

"a number increased by 15"

Problem 2

"7 less than a number"

Problem 3

"the product of 8 and a number"

Problem 4

"triple a number, plus 4"

Problem 5

"half of the sum of a number and 6"

Check Your Understanding: Expression Translator

Match each phrase to its correct algebraic expression!

Expression Translator Challenge

Score: 0 / 6

Question 1 of 6

Challenge Complete!

0/6

Next Steps

Key Takeaways:

Variables represent unknown or changing numbers
Key words tell you which operation to use
"Less than" and "subtracted from" flip the order
Use parentheses when an operation applies to an entire expression
In multiplication, numbers go before variables (3n, not n3)

Practice identifying key words in word problems
Try writing your own phrases and translating them
Move on to learn how to solve equations!