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

Introduction / Context:
This question is another example of a weight combination puzzle. You are given four box weights and asked which of several candidate totals cannot be formed by taking some or all of the boxes, with the rule that each box can be used at most once. Problems like this appear frequently in aptitude tests because they combine basic arithmetic with systematic reasoning about combinations and totals.

Given Data / Assumptions:
- Box weights: 20 kg, 40 kg, 50 kg and 30 kg. - Each box may be chosen or not chosen, but cannot be repeated. - Combinations can include one, two, three or all four boxes. - Candidate totals (in kg): 140, 130, 90 and 120, plus one more option. - We must find the total weight that cannot be formed from any valid combination of the four weights.

Concept / Approach:
As in other combination questions, the most reliable method is to enumerate all possible distinct sums from the available weights. With only four numbers, this enumeration is quite manageable. After listing all achievable totals, we compare them with the given options. Any option that does not appear in the list must be the impossible total. This method guarantees that no valid sum is missed and makes the answer clear.

Step-by-Step Solution:
Step 1: List single box totals: 20, 30, 40 and 50 kilograms. Step 2: Compute two box sums: - 20 + 40 = 60 - 20 + 50 = 70 - 20 + 30 = 50 - 40 + 50 = 90 - 40 + 30 = 70 - 50 + 30 = 80 Step 3: Compute three box sums: - 20 + 40 + 50 = 110 - 20 + 40 + 30 = 90 - 20 + 50 + 30 = 100 - 40 + 50 + 30 = 120 Step 4: Sum of all four boxes: 20 + 40 + 50 + 30 = 140. Step 5: Collect distinct achievable totals: 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120 and 140. Step 6: Compare with options: 140, 130, 90, 120 and 100. We see that 140, 90, 120 and 100 are present in the achievable list, but 130 does not appear.

Verification / Alternative check:
To double check, we can specifically look for ways to form 130 kilograms. Try combinations involving the larger weights. The sum 50 + 40 + 30 = 120. Adding 20 to any triple would exceed 130 and actually give 140. Pair sums such as 50 + 40 = 90 and 50 + 30 = 80 cannot reach 130 even if a third weight is added because only one remaining weight is left. There is therefore no subset of the four weights that yields exactly 130 kilograms, confirming that it is impossible.

Why Other Options Are Wrong:
- 140 kilograms is achieved by combining all four boxes. - 90 kilograms is obtained from 40 + 50 or from 20 + 40 + 30. - 120 kilograms is obtained from 40 + 50 + 30. - 100 kilograms is obtained from 20 + 50 + 30. Since these totals can all be formed from the given weights, they do not satisfy the requirement of being impossible totals.

Common Pitfalls:
A frequent mistake is to assume that a certain option looks unusual and must be impossible without performing a full combination check. Another pitfall is accidentally reusing a weight, which violates the condition that each box can appear only once in any combination. By listing single, double, triple and full combinations carefully, you ensure that all sums are considered and that the final answer is accurate.

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