The weights of four boxes are 100 kg, 70 kg, 50 kg and 90 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 arithmetic reasoning question deals with combinations of given weights. You are provided with four box weights and several candidate total weights. The task is to identify which total cannot be obtained by selecting some or all of the boxes, with the important restriction that each box can be used at most once in any combination. This type of problem checks your ability to think systematically and to handle simple but careful arithmetic operations.

Given Data / Assumptions:
- Box weights: 100 kg, 70 kg, 50 kg and 90 kg. - Each box can either be included or excluded in a combination, but cannot be repeated. - We can form totals using one, two, three or all four boxes. - Candidate totals (in kg): 310, 260, 230, 210 and 240. - We must find the total weight that is impossible to obtain from any valid combination.

Concept / Approach:
The safest strategy is to list all possible distinct sums that can be obtained from the four weights. Since only four boxes are involved, the number of subsets is small enough to enumerate directly. After obtaining all achievable totals, we simply compare them with the options. Any option that does not appear in this list of sums is the required answer. This exhaustive but short approach avoids guesswork and ensures that no valid combination is overlooked.

Step-by-Step Solution:
Step 1: Single box totals are: 50, 70, 90 and 100 kilograms. Step 2: Two box combinations and their sums: - 100 + 70 = 170 - 100 + 50 = 150 - 100 + 90 = 190 - 70 + 50 = 120 - 70 + 90 = 160 - 50 + 90 = 140 Step 3: Three box sums: - 100 + 70 + 50 = 220 - 100 + 70 + 90 = 260 - 100 + 50 + 90 = 240 - 70 + 50 + 90 = 210 Step 4: Four box sum: 100 + 70 + 50 + 90 = 310. Step 5: Collect all distinct achievable totals: 50, 70, 90, 100, 120, 140, 150, 160, 170, 190, 210, 220, 240, 260 and 310. Step 6: Compare these with the options. We see that 310, 260, 210 and 240 are achievable, but 230 does not appear in the list.

Verification / Alternative check:
An additional check is to try to construct 230 kilograms directly. To obtain 230, one might attempt sums like 100 + 70 + 50 (which is 220) or 100 + 90 + 50 (which is 240). Any pair such as 100 + 70 gives 170, and adding 50 or 90 overshoots 230. Similarly, 90 + 70 = 160, and adding 50 gives 210, still not equal to 230. There is no way to select these four weights without repetition to sum exactly to 230 kilograms, which confirms our earlier conclusion.

Why Other Options Are Wrong:
- 310 kilograms is achieved by using all four boxes together. - 260 kilograms is obtained from 100 + 70 + 90. - 210 kilograms is obtained from 70 + 50 + 90. - 240 kilograms is obtained from 100 + 50 + 90. Because these totals can be formed by appropriate selections, they are not correct answers to the question that asks for an impossible total.

Common Pitfalls:
Many candidates attempt to rely on quick mental arithmetic without listing all combinations, and in the process they may miss a valid sum or incorrectly conclude that a total is impossible. Another common error is to accidentally count a box more than once, violating the condition that a box can be used only once. Systematic listing of single, double, triple and full combinations helps avoid these issues and leads to an accurate answer.

Final Answer:
The total weight that cannot be obtained from any combination of the four boxes is 230 kilograms.