The pattern
Working backwards
You know how it ended. Undo each step, last one first.
The story ends with a known number, and every step that led there gets undone in reverse, last step first. Halve where the story doubled, add back what was spent, divide back a percentage profit. Each undoing peels off one layer until the starting amount is showing.
How to spot it
The question gives the final amount after a sequence of changes and asks what the amount was 'at first' or 'to begin with'.
Common mistake
Children undo the steps in the same order they happened rather than in reverse, or forget that a percentage profit is added to the cost price, so the final price is 120% of it, not 100%.
17 worked solutions
Standard
- A tank of oil — morning and afternoon A mechanic had a tank of oil. He used half of it to service cars in the morning. In the afternoon he used a further 3.5 litres and…
- An antique vase — the cost price An antique dealer sold a vase at a 12% profit for $448. What was the cost price of the vase?
- Bicycle shop — sold at a profit A bicycle shop sold a bicycle at a 25% profit for $250. What was the cost price of the bicycle?
- Bus passengers — boarding, alighting, doubling There were some passengers on a bus. At the first stop, 12 passengers boarded and 5 got off. At the second stop, the number of pas…
- Cable car — passengers double at the second station There were some passengers in a cable car. At the first station, 15 passengers boarded and 6 got off. At the second station, the n…
- find a starting time An activity ended at 14:10 after 50 minutes. When did it start?
- find a starting time An activity ended at 10:20 after 75 minutes. When did it start?
- Ivan and Helen — half his money Ivan gave half of his money to Helen. He then spent $18 and was left with $12. How much money did Ivan have at first?
- Jacket — sold at a profit A jacket was sold at a 20% profit for $168. What was the cost price?
- Marbles — a magic trick doubles the jar Ravi had some marbles in a jar. He added 20 marbles and removed 8 marbles. Then a magic trick doubled the number of marbles in the…
- Mrs Lim's withdrawal — half to her son Mrs Lim withdrew some money from the bank. She gave half of it to her son for his tuition fees. She then spent $24 on groceries an…
- Shirt — $85 after a 15% discount A shirt costs $85 after a 15% discount. What was its original price?
- Sticker book — trading and a bonus pack Mei had some stickers in her sticker book. She traded with a friend, receiving 18 stickers and giving away 7 stickers. Then a bonu…
- Stickers — Aiden's packet Aiden had a packet of stickers. He gave half of them to his cousin. He then used 15 stickers to decorate his folder and had 45 sti…
- Wedding cake — sold at a profit A baker sold a wedding cake at a 15% profit for $115. What was the cost price of the cake?
Challenging
- PSLE 2025 — Zach jacket (a) Zach bought a jacket at a 15% discount for $57.80. What was the original price before the discount? (b) Zach's friend later bo…
- Shuttle bus — tourists on and off at two stops A shuttle bus left the airport carrying some tourists. At the first station, 1/4 of the tourists in it alighted and 5 tourists boa…
All the problem patterns → · the revision list
The 33 problem patterns
All 33 patterns, one to a page — the 19 behind Primary 5 and 6 word problems, and the skills the younger years are built on. The bar model, how to spot it in the question, and the mistake children make. A booklet you can print. Free.
A PDF, and a note when new solutions go up.