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Maths olympiad question · P6★★★★ Expert

Averages: olympiad question 11

An expert question on averages for 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

A class of 14 to 37 pupils sat a maths test;

60 or more is a pass.

The passes averaged 86, the fails 54, and the whole class 74.

Then the teacher found a misprinted answer and gave everyone 5 more marks.

After that the passes averaged 84 and the fails 54.

How many pupils went from a fail to a pass?

Show the hints (3)
  1. Hint 1: What the passes have above the class average exactly makes up what the fails are short of it. Use that to find the ratio of passes to fails.
  2. Hint 2: After the bonus the class averages 74 + 5 = 79; find the ratio again the same way. The class size must fit both ratios.
  3. Hint 3: Before, passes : fails = 5 : 3; after, 5 : 1. So the class size is a common multiple of 8 and 6.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.Before: each pass is 86 − 74 = 12 above the class average and each fail 74 − 54 = 20 below, and the two balance: passes × 12 = fails × 20, so passes : fails = 20 : 12 = 5 : 3.
  2. 2.After the bonus the class averages 74 + 5 = 79: passes are 5 above, fails 25 below, so passes : fails = 25 : 5 = 5 : 1.
  3. 3.The class size is the same, so it is a multiple of both 8 and 6 — of 24. Between 14 and 37 the only one is 24, so there are 24 pupils.
  4. 4.Passes before: 24 × 5/8 = 15; after: 24 × 5/6 = 20. Everyone who passed still passes, so 20 − 15 = 5 crossed over.
  5. 5.Check: before, 15 × 86 + 9 × 54 = 1776 = 24 × 74 ✓; after, 20 × 84 + 4 × 54 = 1896 = 24 × 79 ✓
  6. ⚠️Common mistake: taking the new pass average as 86 + 5 = 91. The newcomers were just under 60 and pull it down, so the ratio must be worked out again from the new averages.

Answer: 5 pupils

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