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Word problems

The hardest part of a word problem isn't the algebra. It's translating English into an equation. Here's the recipe.

From sentence to equation

Five steps: read, name, write, solve, answer.

1. Read it
2. Name the unknown
3. Write the equation
4. Solve
5. Answer in words

Sam buys 3 notebooks and a £2 pen. He spends £14 in total. How much is one notebook?

Step 1 of 5

The five-step recipe

  1. Read it — twice, slowly. Underline the question.
  2. Name the unknown — give it a letter and a unit.
  3. Write the equation — turn each clue into a piece of math.
  4. Solve — use the balance laws.
  5. Answer in words — and check it makes sense.

English-to-algebra dictionary

  • Sum, total, increased by → +
  • Difference, less than, decreased by → −
  • Product, of, times → ×
  • Quotient, per, ratio → ÷
  • Is, equals, gives, results in → =
  • A number, an unknown → x (or any letter)

Watch the order on 'less than'

"5 less than x" is x − 5, not 5 − x. The phrase reads right-to-left into the equation. Same for "more than".

Worked example

A rope is cut into two pieces. One piece is 3 m longer than the other. The whole rope is 21 m. How long is the shorter piece?

  1. Let s be the length of the shorter piece (m).
  2. The longer piece is s + 3.
  3. Together: s + (s + 3) = 21.
  4. Solve: 2s + 3 = 21 → 2s = 18 → s = 9.
  5. The shorter piece is 9 m. Check: 9 + 12 = 21 ✓.

Quick check

  1. Three consecutive integers add to 72. What are they?
  2. I think of a number, multiply it by 4, add 3, and get 23. What was the number?

Answers: 23, 24, 25 (the middle one is 72 ÷ 3); and 5.

Quick check

Translate '5 less than x' into algebra.

Quick check

Three consecutive integers add to 72. What is the middle one?