A hard question on sum and multiple for P5, with a step-by-step solution, diagrams and the answer. Sum-and-multiple and difference-and-multiple questions appear in the Hua Xia Cup's P3 to P5 first rounds, qualifying rounds and finals.
The question
The school library has twice as many Chinese storybooks as English ones.
Today 4 of each were borrowed, and now there are 3 times as many Chinese storybooks left on the shelves as English ones.
How many English storybooks were there at first?
Show the hints (3)
💡Hint 1: The same number of each were borrowed. Does the difference between them change?
💡Hint 2: The difference is 1 times the English books at first, and also 2 times the English books left.
💡Hint 3: So the English books were 2 parts at first and 1 part after; the 4 borrowed are 1 part, and 1 part is 4.
Worked solution
Have a go first, then read the solution step by step.
1.4 of each were borrowed, so the difference is unchanged (the two orange pieces are the same length).
2.At first the difference is the English books × 1; afterwards it is the English books left × 2.
3.So if the English books were 2 parts, 1 part is left: 1 part went, which is the 4 borrowed. 1 part = 4 ÷ 1 = 4.
4.The English books: 2 × 4 = 8.
5.Check: afterwards 12 Chinese and 4 English, and 4 × 3 = 12 ✓
⚠️Common mistake: thinking that taking 4 from each keeps the ratio. It's the difference that stays the same, not the ratio.
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