A word problem is an equation written in sentences. Your job is to translate it, the same way you wrote expressions from words, and then solve it with the balance moves you know. A steady plan helps:
- Name the unknown with a letter, and say what it stands for (with units).
- Translate each phrase into symbols. “Is” becomes $=$.
- Solve the equation.
- Check the answer in the story, not only in your equation, and answer in a sentence.
| Words | Symbols |
|---|---|
| $5$ more than a number | $n + 5$ |
| $5$ less than a number | $n - 5$ |
| the difference of a number and $5$ | $n - 5$ |
| twice the sum of a number and $5$ | $2(n + 5)$ |
| three consecutive integers | $n,\ n + 1,\ n + 2$ |
| three consecutive even integers | $n,\ n + 2,\ n + 4$ |
Why it works
Why is “$5$ less than $n$” written $n - 5$? Try it with a number: $5$ less than $12$ is $7$, and $12 - 5 = 7$, while $5 - 12 = -7$. You start at the number and come down $5$, so the number comes first. Testing a phrase with a small number is a quick way to settle any doubt.
Worked examples
Worked example: Consecutive integers
The sum of three consecutive integers is $87$. Find them.
- Let $n$ be the smallest. The others are $n + 1$ and $n + 2$.
- Translate: $n + (n + 1) + (n + 2) = 87$.
- Combine: $3n + 3 = 87$. Subtract $3$: $3n = 84$. Divide: $n = 28$.
- Check: $28 + 29 + 30 = 87$.
Answer: $28$, $29$ and $30$.
Worked example: A rectangle
A garden's length is $4$ m more than twice its width, and its perimeter is $50$ m. Find the width.
- Let $w$ be the width in meters. The length is $2w + 4$.
- Perimeter is twice the width plus twice the length: $2w + 2(2w + 4) = 50$.
- Distribute and combine: $6w + 8 = 50$. Subtract $8$: $6w = 42$. Divide: $w = 7$.
- Check: the length is $18$ m, and $2(7) + 2(18) = 50$.
Answer: The width is $7$ meters.
Common mistake
Translating “$5$ less than $x$” as $5 - x$. The order matters for subtraction: “less than” flips the order of the words, so it is $x - 5$. Check with a number, as in the box above.
Key idea
Name the unknown, translate phrase by phrase, solve, and check the answer against the story itself.