The weights of four boxes are 20 kg, 40 kg, 60 kg and 90 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 again asks you to reason about which totals can be formed by combining a set of four box weights, each used at most once. It checks your ability to systematically list or infer all possible sums and then identify a total that cannot occur. Such questions build both numerical fluency and logical organisation of cases.

Given Data / Assumptions:

• Weights of the four boxes: 20 kg, 40 kg, 60 kg, 90 kg.
• Each box can be used either once or not used in a combination.
• Candidate totals are 150 kg, 170 kg, 120 kg and 160 kg.
• No fractional use of a box is allowed.

Concept / Approach:
As before, we look at all possible sums formed by the four given numbers. There are sums from single box selections, two box selections, three box selections and all four boxes. Once we know this set of possible totals, we simply check which of the options is not present in that set. It is very important to be systematic so that no combination is missed or double counted.

Step-by-Step Solution:
Step 1: Write the weights: 20, 40, 60, 90.Step 2: Sums using one box: 20, 40, 60, 90.Step 3: Sums using two boxes: 20 + 40 = 60, 20 + 60 = 80, 20 + 90 = 110, 40 + 60 = 100, 40 + 90 = 130, 60 + 90 = 150.Step 4: Sums using three boxes: 20 + 40 + 60 = 120, 20 + 40 + 90 = 150, 20 + 60 + 90 = 170, 40 + 60 + 90 = 190.Step 5: Sum using all four boxes: 20 + 40 + 60 + 90 = 210.Step 6: Collect all distinct sums: 20, 40, 60, 80, 90, 100, 110, 120, 130, 150, 170, 190, 210.Step 7: Compare each candidate total with this list.

Verification / Alternative check:
Check each option: 150 is present in two ways, 60 + 90 and 20 + 40 + 90. 170 appears as 20 + 60 + 90. 120 is present as 20 + 40 + 60. However, 160 does not appear in the list of possible sums. You can further attempt to build 160 directly: the largest three weights sum to 40 + 60 + 90 = 190, which is too large, and 90 + 60 = 150. Adding 20 to that gives 170, not 160. Other combinations such as 90 + 40 = 130 or 60 + 40 = 100 cannot be extended to 160 without reusing a number. This reconfirms that 160 is impossible.

Why Other Options Are Wrong:
150 kg is possible by 60 + 90 and also by 20 + 40 + 90.170 kg is possible by 20 + 60 + 90.120 kg is possible by 20 + 40 + 60.Therefore, these totals cannot be answers to the impossibility question.

Common Pitfalls:
Learners may wrongly assume that if a number lies between two achievable sums it must also be achievable, which is not true. Others may forget to consider three box combinations and only check pairs, which could lead to incorrect elimination of some options. Writing the combinations in an organised way, as shown above, is the safest technique and prevents oversight.

Final Answer:
The total weight that cannot be obtained from any combination of the four boxes is 160 kilograms.