How to solve arithmetic word problems
Word problems: model first, compute second
Insight problems punish equation-free arithmetic. Write the tiny model — two quantities and their relation — and the computation is two lines.
Updated
Sum-and-difference, ages, work rates, remainders: each is a two-line model wearing a story. The classic error is grabbing numbers and operating on them ('two numbers, sum 30' → 30÷2) before the relation is written down.
Sum & difference
Two numbers with sum S and difference D: the larger is (S+D)/2, the smaller (S−D)/2. Derive it once — add the equations x+y=S and x−y=D — and never re-derive under pressure again.
Work rates
Machines finishing in a and b hours together need ab/(a+b) hours — rates add, times don't. The trap answer is always the average (a+b)/2. Sanity check: together must be faster than the faster machine.
Remainders
'Packing N apples, b per box, leaves r over': full boxes = (N−r)/b. Remove the remainder first; dividing before subtracting is the engineered mistake.
The strategy
- Name the unknown you're actually asked for — larger number? change? boxes? Misreading the question wastes correct arithmetic.
- Write the one-line relation before computing anything.
- Compute, then check against the story: do your two numbers really add to S and differ by D?
- Under time pressure, test-fit the options into the story — often faster than solving forward.
Common traps
- Averaging when rates should be combined.
- Answering the intermediate quantity (the total cost) instead of the asked quantity (the change).
- Off-by-one on 'how many full boxes' style questions.
Speed · ~60s/item: 45–90 seconds. The model is the speed trick: stories are slow to re-read, equations are instant to re-check.
Try it yourself: 4 practice questions
Pick an answer, then reveal it. Nothing is recorded, there is no clock, and none of these questions is ever used in a test.
Two numbers add up to 54. Their difference is 22. What is the larger number?
Reveal the answer
The answer is C — 38.
Show why — the worked reasoning
Two numbers add up to 50. Their difference is 16. What is the larger number?
Reveal the answer
The answer is D — 33.
Show why — the worked reasoning
A notebook costs 7 units more than a pen. A pen costs 5 units. What do 5 pens and one notebook cost together?
Reveal the answer
The answer is B — 37.
Show why — the worked reasoning
A notebook costs 3 units more than a pen. A pen costs 5 units. What do 8 pens and one notebook cost together?
Reveal the answer
The answer is C — 48.
Show why — the worked reasoning
Try questions like this
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Questions people ask
Will word problems questions be on the free test?
Maybe. The free test gives numerical reasoning 5 of its 20 questions, spread across 13 different question types, so any one test may or may not include this kind.