An expert question on fraction problems for P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P5 and P6 qualifying rounds and finals; harder ones work backwards through fractions, or need clever shortcuts with fractions.
The question
A cha chaan teng bakes a batch of pineapple buns one morning.
In the first hour it sells a sixth of them; in the second hour, two thirds of those left plus 2 more; in the third hour, four sevenths of those left then.
Only whole buns are sold — none is cut up — and some are still left after the three hours.
There were more than 400 buns in the batch.
What is the smallest number of buns the batch could have had?
Show the hints (3)
💡Hint 1: Call the total 6 × k (hour 1 must divide exactly). What is left after hour 1? For hour 2, which amount is split into 3 equal parts?
💡Hint 2: After hour 1, 5 × k is left, which must divide by 3, so k is a multiple of 3. Write k = 3 × j: what is left after hour 2? Don't forget the 2 more.
💡Hint 3: After hour 2, 5 × j − 2 is left, and it must divide by 7. Try j = 1, 2, 3, …
Worked solution
Have a go first, then read the solution step by step.
1.Hour 1 splits the total into 6 equal parts, so the total is a multiple of 6: call it 6 × k; 1 × k go, leaving 5 × k.
2.Hour 2 splits 5 × k into 3 equal parts, so k is a multiple of 3. Write k = 3 × j: the total is 18 × j, and 15 × j is left after hour 1; hour 2 takes 2/3 of it plus 2, leaving 5 × j − 2.
3.Hour 3 splits 5 × j − 2 into 7 equal parts. Trying j: j = 1 gives 3, j = 2 gives 8, j = 3 gives 13, j = 4 gives 18, j = 5 gives 23, j = 6 gives 28; the first that divides by 7 is j = 6, and after that every 7 more.
4.So j = 6, 13, 20, … The total 18 × j must be over 400; the smallest is j = 27: 18 × 27 = 486.
⚠️Common mistake: forgetting that the 2 more changes what is left at the start of hour 3, and answering 504 from the fractions alone; or just taking a multiple of 6, 3, 7. Check hour by hour that what is left then divides exactly.
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