An expert question on sum and difference for P5–P6, with a step-by-step solution, diagrams and the answer. A regular in primary maths competitions such as the Hua Xia Cup, from the P1 and P2 first rounds to the P5 finals.
The question
A post-office clerk has four parcels.
Each weighs a whole number of kilograms, and no two weigh the same.
He weighs them two at a time, every pair once — six weighings in all.
The smallest reading was 14 kg and the largest 37 kg.
Of the other four he wrote down only three — 18, 25, 33 kg — and forgot the fourth.
How heavy is the heaviest of the parcels, in kilograms?
Show the hints (3)
💡Hint 1: Add all six readings: how many times is each item counted? And which two items make the smallest reading, and the largest?
💡Hint 2: Find the forgotten reading first. With all six: the smallest is the two lightest, the next lightest + third lightest, the largest the two heaviest. Which middle reading is 2nd + 3rd lightest? Try both.
💡Hint 3: All four weigh 14 + 37 = 51 kg, so the six readings add to 153; the forgotten one is 153 − 127 = 26 kg.
Worked solution
Have a go first, then read the solution step by step.
1.Each item is weighed with each of the other three, 3 times in all (look at A), so the six readings add up to 3 × the total weight.
2.The smallest reading must be the two lightest and the largest the two heaviest, so all four weigh 14 + 37 = 51 kg.
3.The six readings add to 3 × 51 = 153; the five known add to 127, so the forgotten one is 153 − 127 = 26 kg. All six, smallest first: 14, 18, 25, 26, 33, 37.
4.Call them A, B, C, D from lightest. A + B = 14, A + C = 18, C + D = 37, and the second largest is B + D = 33. Of the middle readings 25 and 26, one is A + D and the other B + C.
5.If B + C = 25: 2A = 14 + 18 − 25 = 7, which is odd — A wouldn't be whole. Impossible.
6.So B + C = 26. (A + B) + (A + C) is two A's more than B + C: 2A = 14 + 18 − 26 = 6, A = 3 kg, B = 14 − 3 = 11 kg.
7.C = 18 − 3 = 15 kg, D = 37 − 15 = 22 kg.
8.Check: 3 + 11 + 15 + 22 = 51 ✓, B + D = 11 + 22 = 33 ✓, A + D = 25, B + C = 26 ✓
⚠️Common mistake: taking the three written readings as the 2nd to 4th smallest without finding the forgotten one first; or assuming the third smallest must be B + C. The two middle readings need checking case by case.
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