Maths olympiad question · P5–P6★★★★ Expert
Chickens and rabbits: olympiad question 7
An expert question on chickens and rabbits for P5–P6, with a step-by-step solution, diagrams and the answer. One of the best-known Olympiad problem types: in the Hua Xia Cup it appears from the P2 first round to the P5 papers.
The question
Tsz-ching took part in an inter-school maths quiz of 40 questions:
8 points for a right answer, 3 off for a wrong one, 0 for a blank.
Tsz-ching scored 68.
She had at least one right, one wrong and one blank, and more wrong than blank.
How many did Tsz-ching get right?
Show the hints (3)
- Hint 1: Look at right and wrong only: 8 × right − 3 × wrong = 68. Is there only one way?
- Hint 2: Each 8 more wrong needs 3 more right to keep the score. Find the fewest wrong answers possible, then list every case — and remember at least one blank.
- Hint 3: The wrong answers can only be 4, 12, 20. Work out the blanks for each, then use the last condition.
Worked solution
Have a go first, then read the solution step by step.
- 1.Let R be right and W wrong: 8 × R − 3 × W = 68, so 8 × R = 68 + 3 × W.
- 2.68 + 3 × W must be a multiple of 8. Trying from W = 1, the first is W = 4; after that each 8 more wrong goes with 3 more right (24 points lost, 24 gained).
- 3.40 questions, at least one blank, so right + wrong is at most 39. Every case: 4 wrong → 10 right, 26 blank; 12 wrong → 13 right, 15 blank; 20 wrong → 16 right, 4 blank ✓.
- 4.Only 20 wrong with 4 blank has more wrong than blank, so 16 right.
- 5.Check: 8 × 16 − 3 × 20 = 128 − 60 = 68 ✓; 16 + 20 + 4 = 40 ✓
- ⚠️Common mistake: supposing all right and treating blanks as wrong, stopping at the first answer found. With three outcomes there are usually several; list them all, then use the conditions.
Answer: 16 questions
Play more like this in Numsprite (unlimited questions, instant marking) →A free maths game: every level free from Easy to Expert (★ to ★★★★), unlimited questions, no download