Translating word problems into equations

A word problem is just a story whose answer is hidden behind words.

Your job is to translate the story into an equation, and then solve it

with the standard techniques.

Three-step recipe

1) Name the unknown. Pick a letter — usually `x` — for the value

you do not know.

2) Build the equation. Rewrite the story as a mathematical

sentence using `x`.

3) Solve. Apply the usual linear-equation techniques and check

the answer makes sense in the story.

Example 1 — sum of ages

Sarah is some age `x`. Matt is 3 times older. Together they are

48 years old. How old is Sarah?

1. Name: Sarah's age = `x`. Matt's age = `3x`.
2. Equation: `x + 3x = 48`.
3. Solve: `4x = 48` → `x = 12`. So Sarah is 12 and Matt is 36.
4. Check: `12 + 36 = 48`. ✓

Example 2 — rectangle perimeter

A rectangle's length is 5 cm more than its width `x`. Its

perimeter is 42 cm. What is the width?

1. Name: width = `x`, length = `x + 5`.
2. Equation: `2(x + (x + 5)) = 42` → `2(2x + 5) = 42`.
3. Solve: `4x + 10 = 42` → `4x = 32` → `x = 8`. Width = 8 cm,

length = 13 cm.

1. Check: `2 · (8 + 13) = 2 · 21 = 42`. ✓

Example 3 — budget

You start with $50</strong>andaftershoppingyouhave<strong>$8 left. How

much did you spend (`x`)?

1. Name: spending = `x`.
2. Equation: `x + 8 = 50` → `x = 42`.
3. Check: `42 + 8 = 50`. ✓

Common phrases → math

Try it yourself

• Word problem — equation
• Equation px + q = r
• Equation p · (x + q) = r