Turning Word Problems into Equations

Word problems are not a different kind of maths — they are the same equations with the numbers hidden in sentences. Four steps handle all of them. First, LET: write down clearly what x stands for, including the unit, because 'let x = Amy's age in years' turns every later line into something meaningful. Second, BUILD the equation from the sentences. Third, SOLVE it using the methods you already have. Fourth, ANSWER in a sentence that actually answers the question asked. Step one is the one students skip, and it is the one that makes the rest work.

Most word problems hide two unknowns, and the trick is to use only one letter. If Ben is 4 years older than Amy, call Amy's age x and then Ben's age is x + 4 — described in terms of the same letter. The sentence 'together they are 26' becomes x + (x + 4) = 26, which simplifies to 2x + 4 = 26, giving x = 11. So Amy is 11 and Ben is 15. A useful habit: choose the smaller quantity to be x, so the other becomes a plus rather than a minus. And check both facts, not just one — they add to 26 ✓ and Ben is 4 older ✓.

Two setups appear again and again. Consecutive numbers each go up by one, so they are x, x + 1 and x + 2. 'Three consecutive numbers add to 72' becomes x + (x + 1) + (x + 2) = 72, which is 3x + 3 = 72, giving x = 23 and the numbers 23, 24, 25. The other classic is a rectangle. If the length is 5 more than the width, call the width x and the length x + 5. Perimeter 46 means 2x + 2(x + 5) = 46, so 4x + 10 = 46 and x = 9 — a width of 9 cm and length of 14 cm. Remember a rectangle has four sides, so both doubles are needed.

Money problems follow the same pattern: quantity times price, added up. If 3 pens and 2 books cost $31 and a pen costs $3, let x be the cost of one book in dollars. Then 3 times 3, plus 2x, equals 31 — that is 9 + 2x = 31, so 2x = 22 and x = 11. A book costs $11. Two things to watch. Keep everything in one unit, either dollars or cents, never a mix. And be precise about what x represents: here it is the cost of one book, so if the question asked for the cost of two books the answer would be 22, not 11.

The last step is the one most often dropped. Finding x = 9 is not the answer if the question asked for the length — that is 9 + 5 = 14 cm. Before you stop, ask three things. Did I answer what was actually asked, or did I just find x? Does my answer carry a unit? And is it sensible — an age cannot be negative and a length cannot be zero, so an answer like that means a mistake earlier. A correct value of x with no final sentence loses marks, and it is the easiest mark in the whole question to keep.