A hard question on fraction problems for P5–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
Ho Yin and Hiu Ching have each saved some money.
Ho Yin spends a quarter of his savings on a football, and Hiu Ching spends two thirds of hers on a set of paints.
They spend the same amount.
Now Ho Yin has 330 dollars more left than Hiu Ching.
How many dollars had Ho Yin saved at first?
Show the hints (3)
💡Hint 1: They spend the same: 1/4 of Ho Yin's savings is the same money as 2/3 of Hiu Ching's. Who had more to begin with?
💡Hint 2: Make the two numerators the same: call the money spent 2 parts. How many parts did each have at first?
💡Hint 3: Count the parts each had at first and has left. The difference between the parts left is worth 330 dollars: find one part first.
Worked solution
Have a go first, then read the solution step by step.
1.They spent the same. Make the two tops the same: 1/4 = 2/8 and 2/3 already has 2 on top.
2.Call the money spent 2 parts; then a part is the same size for both: Ho Yin had 8 parts and Hiu Ching 3.
3.Ho Yin has 8 − 2 = 6 parts left; Hiu Ching has 3 − 2 = 1 left.
4.What they have left differs by 6 − 1 = 5 parts, which is 330 dollars: one part = 330 ÷ 5 = 66.
⚠️Common mistake: comparing or subtracting 1/4 and 2/3 as if they were fractions of the same whole. One is of Ho Yin's savings, the other of Hiu Ching's.
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