The weights of four boxes are 40 kg, 30 kg, 50 kg and 20 kg. Which of the following cannot be obtained as the total weight, in kilograms, of any combination of these boxes if each box is used at most once?

Introduction / Context:
This is another combination of weights puzzle. You must determine which proposed total weight cannot be formed using any subset of the four given box weights, where each box may be used at most once. As with the earlier problems of this type, the key is to be systematic and avoid guessing.

Given Data / Assumptions:

• Box weights: 40 kg, 30 kg, 50 kg, 20 kg.
• Each box can be selected at most once in any combination.
• Candidate totals: 140 kg, 130 kg, 90 kg, 100 kg.
• All weights are in whole kilograms.

Concept / Approach:
We list the sums of one box, two box, three box and four box combinations and then compare the set of possible sums with the answer options. The option that does not appear among the achievable sums is the total that cannot be formed. This method ensures an accurate and complete analysis.

Step-by-Step Solution:
Step 1: Record the weights: 40, 30, 50, 20.Step 2: Sums using one box: 20, 30, 40, 50.Step 3: Sums using two boxes: 20 + 30 = 50, 20 + 40 = 60, 20 + 50 = 70, 30 + 40 = 70, 30 + 50 = 80, 40 + 50 = 90.Step 4: Sums using three boxes: 20 + 30 + 40 = 90, 20 + 30 + 50 = 100, 20 + 40 + 50 = 110, 30 + 40 + 50 = 120.Step 5: Sum using all four boxes: 20 + 30 + 40 + 50 = 140.Step 6: Collect distinct sums: 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 140.Step 7: Compare the candidate totals 140, 130, 90 and 100 with this list.

Verification / Alternative check:
The total 140 appears as the sum of all four boxes. The total 90 appears in two ways: 40 + 50 and 20 + 30 + 40. The total 100 appears as 20 + 30 + 50. However, 130 is not present in the list of sums. To double check, attempt to build 130: 50 + 40 + 30 = 120, and adding 20 gives 140. There is no way to pick a subset of the four weights that adds to 130 without exceeding or falling short of it. This confirms that 130 is impossible to obtain.

Why Other Options Are Wrong:
140 kg is possible using all four boxes.90 kg is possible using 40 + 50 or 20 + 30 + 40.100 kg is possible using 20 + 30 + 50.Therefore, these totals are valid and cannot be the correct answer.

Common Pitfalls:
Some learners may try to mentally add combinations and accidentally skip a three box combination such as 20 + 30 + 50, leading to the wrong conclusion about 100. Others might think that since 130 lies between 120 and 140 it should be achievable, which is not guaranteed with discrete weights. A thorough and written enumeration is the best method to avoid these errors.

Final Answer:
The total weight that cannot be obtained from any combination of these four boxes is 130 kilograms.