Word Problems to
Algebraic Expressions

Learn to decode everyday language into precise algebraic expressions, turning written clues into symbols you can solve.

SSAT Upper Level: Quantitative

What Is an Algebraic Expression?

An algebraic expression is a combination of numbers, variables, and operations that represents a quantity. Unlike an equation, it has no equals sign.

Variable

A letter (like $ x $ or $ n $) representing an unknown value.

Constant

A fixed number, such as $ 5 $, $ \frac{1}{2} $, or $ -3 $.

Operation

Addition, subtraction, multiplication, or division connecting terms.

Why Does Translation Matter?

On the SSAT, problems are almost never written as pure math. They are wrapped in words. Your job is to unwrap those words into symbols you can manipulate.

The Vocabulary Map

Every operation has a family of signal words. Memorize these and you can decode almost any sentence on the SSAT.

Operation	Signal Words	Example
+ Addition	sum, more than, increased by, plus, total	$ x + 7 $
− Subtraction	difference, less than, decreased by, fewer	$ x - 4 $
× Multiplication	product, times, twice, triple, of	$ 3x $
÷ Division	quotient, divided by, ratio, per, half of	$ \frac{x}{2} $

The Core Translation Rule

Order Matters: Subtraction and Division

Addition and multiplication are commutative: $ a + b = b + a $. But subtraction and division are not. The phrase structure tells you which value comes first.

"Less Than"

"5 less than $ x $" → $ x - 5 $
The value after "less than" comes first.

"Divided Into"

"10 divided into $ x $" → $ \frac{x}{10} $
The value after "into" is the numerator.

Expression vs. Equation

The SSAT may ask you to write an expression or an equation. Knowing the difference prevents wrong answer choices.

Feature	Expression	Equation
Equals Sign	No	Yes
Example	$ 3x + 2 $	$ 3x + 2 = 14 $
Signal Phrase	"Write an expression for..."	"is", "equals", "the result is"
Can You Solve?	Simplify only	Solve for the variable

How to Translate Step by Step

1

Identify the unknown. What does the variable represent? Assign it a letter.

2

Spot the signal words. Underline operation keywords: "more than", "product of", "less than", etc.

3

Determine operand order. For subtraction and division, check which quantity comes first.

4

Assemble the expression. Combine variable, constants, and operations into one expression.

5

Verify with a test value. Plug in a simple number to see if the expression matches the original sentence.

Example 1: Basic Translation

Example 2: Intermediate Translation

1

Unknown: let the number be $ n $.

2

"product of five and a number" → $ 5n $

3

"eight less than" → subtract 8 from the product → $ 5n $ $ - $ $ 8 $

Example 3: Multi-Step Translation

1

Variable: $ d $ is the price per notebook (already given).

2

"buys 4 notebooks" → total before coupon is $ 4d $.

3

"saves her $3" → subtract the coupon → $ 4d $ $ - $ $ 3 $.

Example 4: Consecutive Integers

The Power of Parentheses

When a word problem says "the quantity", "the sum of ... multiplied by", or "all of ... divided by", it signals that you need parentheses to group operations correctly.

Without Parentheses

"Twice a number plus 3" → $ 2x + 3 $

With Parentheses

"Twice the sum of a number and 3" → $ 2(x + 3) $

Common Mistakes to Avoid

✕ Reversing "Less Than"

Writing $ 5 - x $ for "five less than $ x $". Correct: $ x - 5 $.

✕ Forgetting Parentheses

Writing $ 2x + 7 $ for "twice the sum of $ x $ and 7". Correct: $ 2(x + 7) $.

✕ Confusing "Of" with Addition

Writing $ \frac{1}{2} + x $ for "half of $ x $". Correct: $ \frac{x}{2} $.

Quick Reference Table

English Phrase	Expression	Watch Out
"6 more than $ x $"	$ x + 6 $	Order doesn't matter
"6 less than $ x $"	$ x - 6 $	Not $ 6 - x $
"the product of 4 and $ x $"	$ 4x $	Order doesn't matter
"$ x $ divided by 9"	$ \frac{x}{9} $	Not $ \frac{9}{x} $
"twice the sum of $ x $ and 5"	$ 2(x+5) $	Not $ 2x + 5 $
"the square of $ x $, decreased by 1"	$ x^2 - 1 $	Square first, then subtract

Question 1 of 5

Which expression represents "nine less than twice a number"?

"Twice a number" is $ 2n $. "Nine less than" means subtract 9 from that result: $ 2n - 9 $. The phrase "less than" reverses the operand order.

Question 2 of 5

A taxi charges $ d $ dollars per mile plus a $5 flat fee. Which expression gives the cost for $ m $ miles?

The per-mile charge is $ d $ dollars times $ m $ miles: $ dm $. The flat fee of $5 is added: $ dm + 5 $. Each variable has a clear role, and they combine with addition.

Question 3 of 5

Which expression represents "the quotient of the sum of $ x $ and 7, and 3"?

"The sum of $ x $ and 7" is $ x + 7 $, which forms the entire numerator. "Quotient ... and 3" means divide by 3. So the result is $ \frac{x + 7}{3} $. Parentheses (grouping) are critical here.

Question 4 of 5

The sum of three consecutive odd integers can be expressed as:

Consecutive odd integers differ by 2: if the first is $ n $, the next two are $ n+2 $ and $ n+4 $. Their sum is $ n + (n+2) + (n+4) = 3n + 6 $. Choice (a) uses differences of 1, which gives consecutive integers, not odd ones.

Question 5 of 5

A rectangle's length is 3 more than twice its width $ w $. Which expression represents the perimeter?

Length $ = 2w + 3 $. Perimeter $ = 2(\text{length} + \text{width}) = 2((2w+3) + w) = 2(3w+3) = 6w + 6 $. Choice (d), $ 2(2w+3) $, only doubles the length and ignores the two widths.

English Phrase	Expression	Watch Out
"6 more than \( x \)"	\( x + 6 \)	Order doesn't matter
"6 less than \( x \)"	\( x - 6 \)	Not \( 6 - x \)
"the product of 4 and \( x \)"	\( 4x \)	Order doesn't matter
"\( x \) divided by 9"	\( \frac{x}{9} \)	Not \( \frac{9}{x} \)
"twice the sum of \( x \) and 5"	\( 2(x+5) \)	Not \( 2x + 5 \)
"the square of \( x \), decreased by 1"	\( x^2 - 1 \)	Square first, then subtract

Operation	Signal Words	Example
+ Addition	sum, more than, increased by, plus, total	\( x + 7 \)
− Subtraction	difference, less than, decreased by, fewer	\( x - 4 \)
× Multiplication	product, times, twice, triple, of	\( 3x \)
÷ Division	quotient, divided by, ratio, per, half of	\( \frac{x}{2} \)

Feature	Expression	Equation
Equals Sign	No	Yes
Example	\( 3x + 2 \)	\( 3x + 2 = 14 \)
Signal Phrase	"Write an expression for..."	"is", "equals", "the result is"
Can You Solve?	Simplify only	Solve for the variable

Word Problems toAlgebraic Expressions

What Is an Algebraic Expression?

Why Does Translation Matter?

The Vocabulary Map

The Core Translation Rule

Order Matters: Subtraction and Division

Expression vs. Equation

How to Translate Step by Step

Example 1: Basic Translation

Example 2: Intermediate Translation

Example 3: Multi-Step Translation

Example 4: Consecutive Integers

The Power of Parentheses

Common Mistakes to Avoid

Quick Reference Table

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Word Problems to
Algebraic Expressions