Six boxes — A, B, C, D, E and F — have different weights. The weight of each box is a whole number of kilograms, and no box weighs less than 4 kg. A warehouse assistant records the following facts after several weighings on a two-pan balance and a platform scale:
1. The total weight of the six boxes is 50 kg.
2. A and B together exactly balance C and E together.
3. F and C together exactly balance E.
4. A and E together exactly balance D and C together.
5. D is the heaviest box.
6. B is heavier than A but lighter than E.
1. The total weight of the six boxes is 50 kg.
2. A and B together exactly balance C and E together.
3. F and C together exactly balance E.
4. A and E together exactly balance D and C together.
5. D is the heaviest box.
6. B is heavier than A but lighter than E.
Question 1
Numerical answer
What is the weight (in kg) of box D?
Show answer and solution
Answer: 13
Deduction. From (3), E = C + F. From (2), B = C + E − A = 2C + F − A. From (4), D = A + E − C = A + F.
Substituting in (1): A + (2C + F − A) + C + (A + F) + (C + F) + F = A + 4C + 4F = 50, so A = 50 − 4(C + F). Since C, F ≥ 4, C + F ≥ 8, and A ≥ 4 gives C + F ≤ 11.
- C + F = 11 → A = 6, E = 11, B = C + 5.
- C + F = 10 → A = 10, E = 10: A = E, not allowed.
- C + F = 9 → A = 14, E = 9, B = C − 5 < 4: not allowed.
- C + F = 8 → A = 18 > E, contradicting (6).
So A = 6, E = 11. B = C + 5 < 11 → C ∈ {4, 5}. C = 5 gives F = 6 = A (not distinct), so C = 4, F = 7, B = 9, D = 6 + 7 = 13 (heaviest ✓).
Weights (kg): A = 6, B = 9, C = 4, D = 13, E = 11, F = 7.
D = A + F = 6 + 7 = 13 kg, which is indeed the heaviest box.
Substituting in (1): A + (2C + F − A) + C + (A + F) + (C + F) + F = A + 4C + 4F = 50, so A = 50 − 4(C + F). Since C, F ≥ 4, C + F ≥ 8, and A ≥ 4 gives C + F ≤ 11.
- C + F = 11 → A = 6, E = 11, B = C + 5.
- C + F = 10 → A = 10, E = 10: A = E, not allowed.
- C + F = 9 → A = 14, E = 9, B = C − 5 < 4: not allowed.
- C + F = 8 → A = 18 > E, contradicting (6).
So A = 6, E = 11. B = C + 5 < 11 → C ∈ {4, 5}. C = 5 gives F = 6 = A (not distinct), so C = 4, F = 7, B = 9, D = 6 + 7 = 13 (heaviest ✓).
Weights (kg): A = 6, B = 9, C = 4, D = 13, E = 11, F = 7.
D = A + F = 6 + 7 = 13 kg, which is indeed the heaviest box.
Question 2
Single correct option
Which box is the second heaviest?
- A. E
- B. B
- C. D
- D. F
Show answer and solution
Answer: (A) E
Weights (kg): A = 6, B = 9, C = 4, D = 13, E = 11, F = 7.
In descending order: D (13), E (11), B (9), F (7), A (6), C (4). The second heaviest is E.
Key step: E = C + F = 11 comes from A + 4(C + F) = 50 with A = 6 forced.
In descending order: D (13), E (11), B (9), F (7), A (6), C (4). The second heaviest is E.
Key step: E = C + F = 11 comes from A + 4(C + F) = 50 with A = 6 forced.
Question 3
Numerical answer
What is the combined weight (in kg) of boxes B and F?
Show answer and solution
Answer: 16
Weights (kg): A = 6, B = 9, C = 4, D = 13, E = 11, F = 7.
B + F = 9 + 7 = 16 kg.
Quick check with the clues: A + B = 15 = C + E ✓; F + C = 11 = E ✓; A + E = 17 = D + C ✓; total = 6 + 9 + 4 + 13 + 11 + 7 = 50 ✓.
B + F = 9 + 7 = 16 kg.
Quick check with the clues: A + B = 15 = C + E ✓; F + C = 11 = E ✓; A + E = 17 = D + C ✓; total = 6 + 9 + 4 + 13 + 11 + 7 = 50 ✓.
Question 4
Single correct option
How many boxes are heavier than box A?
- A. 2
- B. 3
- C. 4
- D. 5
Show answer and solution
Answer: (C) 4
Weights (kg): A = 6, B = 9, C = 4, D = 13, E = 11, F = 7.
A weighs 6 kg. Heavier boxes: B (9), D (13), E (11), F (7) → 4 boxes. Only C (4 kg) is lighter.
A weighs 6 kg. Heavier boxes: B (9), D (13), E (11), F (7) → 4 boxes. Only C (4 kg) is lighter.
Question 5
Single correct option
A seventh box G is now added such that the average weight of all seven boxes is 8 kg. Which of the original six boxes has the same weight as G?
- A. A
- B. C
- C. F
- D. No existing box
Show answer and solution
Answer: (A) A
Weights (kg): A = 6, B = 9, C = 4, D = 13, E = 11, F = 7.
Total of seven boxes = 7 × 8 = 56 kg, so G = 56 − 50 = 6 kg.
The box weighing 6 kg is A. (Note that G's weight follows from fact 1 alone, but identifying the matching box needs the full solution.)
Total of seven boxes = 7 × 8 = 56 kg, so G = 56 − 50 = 6 kg.
The box weighing 6 kg is A. (Note that G's weight follows from fact 1 alone, but identifying the matching box needs the full solution.)
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