A hard question on averages for P5–P6, with a step-by-step solution, diagrams and the answer. Averages appear in the Hua Xia Cup's P4 and P5 first rounds, qualifying rounds and finals.
The question
Five pupils stand in a row.
The three on the left average 129 cm in height, the three on the right average 138 cm, and all five average 135 cm.
How tall is the pupil in the middle?
Show the hints (3)
💡Hint 1: Between the left three and the right three, is anyone counted twice?
💡Hint 2: Left three's total + right three's total = all five's total + the middle pupil.
💡Hint 3: The left three total 387 cm, the right three 414 cm, all five 675 cm.
Worked solution
Have a go first, then read the solution step by step.
1.Left three: 129 × 3 = 387 cm; right three: 138 × 3 = 414 cm.
2.All five: 135 × 5 = 675 cm.
3.The middle pupil is in both groups, so 387 + 414 is all five plus him once more: 387 + 414 − 675 = 126 cm.
⚠️Common mistake: treating the two groups of three as six different pupils, forgetting the middle one is counted twice.
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