Maths olympiad question · P6★★★★ Expert
Percentages and mixtures: olympiad question 11
An expert question on percentages and mixtures for P6, with a step-by-step solution and the answer. Common in the Hua Xia Cup's P6 first round and finals: discounts, a rise followed by a fall, and mixtures.
The question
Cup A holds 400 g of salt water at 31%, and cup B 300 g of salt water.
First half of A is poured into B and stirred; then half of B is poured back into A and stirred.
Cup A is now 24%.
What percentage was cup B at first?
Show the hints (3)
- Hint 1: Track both cups' weights and salt step by step. How many grams does cup A end with, and how much salt is in it?
- Hint 2: Cup A's final salt = what stayed in A + what came back from B. Work back from that to the salt in B before the pour back.
- Hint 3: Cup A ends with 450 g holding 108 g of salt, 62 g of it from A's own half.
Worked solution
Have a go first, then read the solution step by step.
- 1.Half of A, 200 g, goes into B: A keeps 200 g with 200 × 31% = 62 g of salt; B becomes 300 + 200 = 500 g.
- 2.Half of B, 250 g, goes back: A holds 200 + 250 = 450 g at 24%, so 450 × 24% = 108 g of salt.
- 3.The salt that came back = 108 − 62 = 46 g: half of B's salt before the pour back, so B then held 46 × 2 = 92 g.
- 4.Of that, 62 g came in with A's half, so B began with 92 − 62 = 30 g of salt: 30 ÷ 300 = 10%.
- 5.Check: B has 300 × 10% = 30 g; after A's half, 92 g in 500 g; half of it, 46 g, returns; cup A: (62 + 46) ÷ 450 = 24% ✓
- ⚠️Common mistake: treating the pour back as B's original salt water at its original strength. What comes back is B already mixed with A's half — find the salt after mixing first.
Answer: 10 %
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