Maths olympiad question · P6★★★★ Expert
Ratios: olympiad question 7
An expert question on ratios for P6, with a step-by-step solution and the answer. Ratio questions appear in the Hua Xia Cup's P4 to P6 first rounds, qualifying rounds and finals, such as linked ratios and ratios that change.
The question
Two brothers have savings in the ratio 7 :
1.
The older gives some of his to the younger, and the ratio becomes 1 :
1.
Then the older spends $110 and the younger $80, leaving them in the ratio 1 :
4.
How much did the older brother give the younger?
Show the hints (3)
- Hint 1: When the older gives money to the younger, does the total change? Make the two ratios' total parts the same.
- Hint 2: They spend different amounts, so for the third ratio neither the total nor the difference stays fixed. Let one part after the second ratio be some dollars, and write what each has left.
- Hint 3: Call the total 8 parts: after the gift the older has 4 and the younger 4. After spending, (4 parts − 110) : (4 parts − 80) = 1 : 4.
Worked solution
Have a go first, then read the solution step by step.
- 1.The gift leaves the total alone. The LCM of 7 + 1 = 8 and 1 + 1 = 2 is 8, so call the total 8 parts: the older had 7; after the gift 4 and 4.
- 2.After spending: the older has 4 parts − $110 and the younger 4 parts − $80, in the ratio 1 : 4, so 4 × (4 parts − 110) = 1 × (4 parts − 80).
- 3.Expand: 16 parts − 440 = 4 parts − 80, so 12 parts = 360: 1 part = $30.
- 4.The older gave 7 − 4 = 3 parts: 3 × 30 = $90.
- 5.Check: at first $210 and $30 (7 : 1); after the gift $120 and $120 (1 : 1); after spending $10 and $40 (1 : 4) ✓
- ⚠️Common mistake: assuming the difference or the total stays fixed after spending and lining the third ratio up with the others. They spend different amounts, so set up the parts directly.
Answer: 90 dollars
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