Translating word problems into algebra
Worded algebra is half English comprehension, half algebra. Examiners give marks for setting up the equation as well as solving it, so always show your "let x = …" line.
The four-step pattern
- Read the question twice. Identify what's being asked for.
- Define the variable(s). "Let x = the number of …".
- Translate phrases into algebraic equations.
- Solve, then answer the question in words with units.
Common phrase translations
| Phrase | Algebra |
|---|---|
| "5 more than x" | x + 5 |
| "the sum of x and y" | x + y |
| "x less than 10" | 10 − x |
| "twice x" | 2x |
| "x more than three times y" | 3y + x |
| "the product of x and y" | xy |
| "x divided by y" | x/y |
| "the next consecutive integer" | x + 1 |
| "consecutive even integers" | 2n, 2n + 2 |
| "double a number, then add 7" | 2x + 7 |
| "a number plus its reciprocal" | x + 1/x |
✦Worked example— Worked example 1 — age problem
Tom is twice as old as his sister. The sum of their ages is 36. Find their ages.
Let s = sister's age. Then Tom's age is 2s. Equation: s + 2s = 36 ⇒ 3s = 36 ⇒ s = 12.
Sister 12, Tom 24.
✦Worked example— Worked example 2 — perimeter
A rectangle is 5 cm longer than it is wide. Its perimeter is 38 cm. Find the dimensions.
Let w = width. Length = w + 5. Perimeter = 2w + 2(w + 5) = 4w + 10 = 38 ⇒ w = 7.
Width 7 cm, length 12 cm. Check: 2 × 7 + 2 × 12 = 14 + 24 = 38 ✓.
✦Worked example— Worked example 3 — money/tickets
3 adult tickets and 2 child tickets cost £29. The cost of an adult ticket is £4 more than a child's. Find both prices.
Let c = child price. Adult = c + 4. Total: 3(c + 4) + 2c = 29 ⇒ 5c + 12 = 29 ⇒ c = 3.4. Adult = £7.40, Child = £3.40.
✦Worked example— Worked example 4 — quadratic context
A rectangular garden has length 3 m more than its width. Its area is 70 m². Find the dimensions.
Let w = width. Length = w + 3. w(w + 3) = 70 ⇒ w² + 3w - 70 = 0 ⇒ (w + 10)(w - 7) = 0. w = -10 rejected; w = 7 m. Length = 10 m.
⚠Common mistakes— Common mistakes (examiner traps)
- Not defining variables. Write "Let x = …" — examiners reward the precision.
- Wrong phrase translation. "x less than 10" is 10 − x, not x − 10.
- Missing units in the final answer. £, cm, kg, etc.
- Failing to reject impossible solutions (negative ages, lengths).
- Not answering the actual question. "Find their ages" — give both ages, not just x.
➜Try this— Quick check
I think of a number, multiply by 3, then subtract 5; the result is 13. What's the number?
Let n = the number. 3n − 5 = 13 ⇒ 3n = 18 ⇒ n = 6.
AI-generated · claude-opus-4-7 · v3-deep-algebra