The weights of four boxes are 30 kg, 20 kg, 50 kg and 90 kg. Each box can be used at most once in any combination. Which of the following values cannot be the total weight, in kilograms, of any possible combination of these boxes?

Introduction / Context:

This question is another variant of the subset-sum style problems using small integer weights. We must determine which of the given totals cannot be made by adding together some or all of the four given box weights, with each box used at most once. Such problems test careful enumeration and number sense.

Given Data / Assumptions:

• Box weights are 30 kg, 20 kg, 50 kg and 90 kg.
• Each box can appear at most once in a combination.
• We may select one, two, three or all four boxes.
• We have to find which option cannot be expressed as a valid sum.

Concept / Approach:

Since the number of boxes is small, we can compute all possible sums explicitly. Listing all single-box, two-box, three-box and four-box totals guarantees that we do not miss any combination. Finally, we compare this list to the options to see which total is missing and therefore impossible.

Step-by-Step Solution:

Verification / Alternative check:

We can also reason directly about 200 kg. The largest possible sum using all four boxes is 190 kg, which is already less than 200 kg. Therefore, there is no way to reach 200 kg without exceeding the available weights. Because the total of all four boxes is below 200, no subset of these boxes can possibly reach or exceed 200 kg, confirming that 200 kg is impossible.

Why Other Options Are Wrong:

Option 190 kg is equal to the sum of all four boxes: 20 + 30 + 50 + 90.

Option 140 kg can be formed either as 50 + 90 or as 20 + 30 + 90.

Option 160 kg is formed as 20 + 50 + 90.

Common Pitfalls:

Some candidates may incorrectly assume that if all weights are less than 100 then any large total is impossible, overlooking that sums of multiple boxes can exceed 100. Others may forget to test the total of all boxes. Carefully checking both logic and arithmetic helps avoid these mistakes.

Final Answer:

The total weight that cannot be formed is 200 kilograms.