A hard question on most and least for P4–P5, with a step-by-step solution, diagrams and the answer. Finding the most and the least appears in the Hua Xia Cup from the P1 and P2 first rounds to the P6 papers.
The question
24 pupils in a class sit a maths test marked out of 100, every mark a whole number.
The class average is exactly 78.
What is the fewest pupils who can have scored 70 or more?
Show the hints (3)
💡Hint 1: To keep the high scorers as few as possible, should everyone else score a lot or a little?
💡Hint 2: Everyone else gets at most 69; anyone reaching 70 gets at most 100. First find the total.
💡Hint 3: Say x people reach 70: what is the most everyone can total? It must be at least 1872.
Worked solution
Have a go first, then read the solution step by step.
1.Total: 24 × 78 = 1872 marks.
2.To have as few as possible reach 70: everyone else has at most 69; those who reach it have at most 100.
3.If only 6 reach it: the total is at most 6 × 100 + 18 × 69 = 1842, less than 1872. Impossible.
4.With 7 reaching it, it works: 17 get 69 each (1173 in all), and the other 1872 − 1173 = 699 go to the 7: 6 get 100 and 1 gets 99, all between 70 and 100.
5.So at least 7.
⚠️Common mistake: letting everyone else score very little. That needs more high scorers to make the total; everyone else should get as close to 69 as possible.
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