The weights of four boxes are 90 kg, 30 kg, 20 kg and 50 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 question continues the theme of reasoning with combinations of fixed weights. You must identify which suggested total weight cannot be achieved using any subset of the four given box weights, where each box can appear at most once in a combination. It is a useful test of systematic enumeration and logical checking.

Given Data / Assumptions:

• Box weights: 90 kg, 30 kg, 20 kg, 50 kg.
• Each box may be used at most once in forming a total.
• Candidate totals: 190 kg, 170 kg, 100 kg, 150 kg.
• We work with whole kilograms only, no fractions.

Concept / Approach:
The approach is to list all possible sums from 1 box, 2 boxes, 3 boxes and all 4 boxes. Once we have the set of all distinct sums, we simply check which option is missing from this set. The presence or absence of a total is then an objective matter, not a guess.

Step-by-Step Solution:
Step 1: Write the weights: 90, 30, 20, 50.Step 2: Sums using one box: 20, 30, 50, 90.Step 3: Sums using two boxes: 20 + 30 = 50, 20 + 50 = 70, 20 + 90 = 110, 30 + 50 = 80, 30 + 90 = 120, 50 + 90 = 140.Step 4: Sums using three boxes: 20 + 30 + 50 = 100, 20 + 30 + 90 = 140, 20 + 50 + 90 = 160, 30 + 50 + 90 = 170.Step 5: Sum using all four boxes: 20 + 30 + 50 + 90 = 190.Step 6: Collect all distinct sums: 20, 30, 50, 70, 80, 90, 100, 110, 120, 140, 160, 170, 190.Step 7: Compare 190, 170, 100 and 150 against this list.

Verification / Alternative check:
We can see that 190 is achievable by using all four boxes: 20 + 30 + 50 + 90 = 190. The total 170 is achievable with 30 + 50 + 90. The total 100 is achievable with 20 + 30 + 50. The remaining candidate is 150. Try to form 150 directly: 90 + 50 = 140, and adding 20 gives 160, adding 30 gives 170, but we are not allowed to use more than two extra boxes beyond 90 and 50 at a time if we want 150. Combinations like 90 + 30 = 120 and 90 + 20 = 110 also do not lead to 150 without using a third weight that overshoots or misfits. This confirms that 150 cannot be reached.

Why Other Options Are Wrong:
190 kg is possible using all the boxes together.170 kg is possible using 30 + 50 + 90.100 kg is possible using 20 + 30 + 50.These totals therefore are not impossible and cannot be the correct answer.

Common Pitfalls:
Sometimes students overlook the three box combinations and check only pairs, leading them to wrongly classify some achievable totals as impossible. Others may misadd one of the combinations and incorrectly think they have found a way to reach 150, which is not actually the case. Writing all combinations cleanly and double checking the arithmetic avoids these issues.

Final Answer:
The total weight that cannot be obtained from any combination of the four boxes is 150 kilograms.