A student's average marks in five subjects is 82. The average of the marks in the first two subjects is 86.5, and the average of the marks in the last two subjects is 84. What are the marks obtained in the third subject?

Introduction / Context:
This question examines how to use averages to recover the contribution of a single unknown subject when other grouped averages are known. It is a straightforward application of converting between averages and total marks and then isolating the missing part from the whole.

Given Data / Assumptions:

• Average of 5 subjects = 82.

• Average of first two subjects = 86.5.

• Average of last two subjects (fourth and fifth) = 84.

• All marks are on the same scale and are additive.

• We need to find the marks in the third subject.

Concept / Approach:
First, compute the total marks across all five subjects using the overall average. Then compute the combined marks for the first two subjects and the last two subjects using their respective averages. The marks in the third subject are what remain after subtracting the sums of these four known subject marks from the grand total. This uses the basic identity: total of all items = sum of known items + unknown item, which we can rearrange to solve for the unknown item.

Step-by-Step Solution:
Total marks in 5 subjects = average * number of subjects = 82 * 5 = 410. Sum of first two subjects = 86.5 * 2 = 173. Sum of last two subjects (fourth and fifth) = 84 * 2 = 168. Combined marks of subjects 1, 2, 4 and 5 = 173 + 168 = 341. Marks in the third subject = total marks − marks of the other four subjects. Marks in third subject = 410 − 341 = 69.

Verification / Alternative check:
We can quickly verify by reconstructing the overall average. Assume the marks are M1, M2, M3, M4 and M5. We now know that M1 + M2 = 173, M4 + M5 = 168 and M3 = 69. Total = 173 + 69 + 168 = 410. Dividing 410 by 5 gives 82 as the average, which matches the given overall average, confirming that 69 is correct for the third subject.

Why Other Options Are Wrong:
If the third subject were 67, 71 or 73, the total marks would be 408, 412 or 414 respectively, which would lead to overall averages of 81.6, 82.4 or 82.8, not 82. The only value that preserves the original average when recomputed is 69, so the other options are inconsistent with the data.

Common Pitfalls:
A typical mistake is to try to average the averages incorrectly, or to miscount the number of subjects in each group. Another frequent error is arithmetic slip when multiplying the averages by 2. Always recompute the final average as a check to ensure that the answer maintains the given conditions.

Final Answer:
The marks obtained in the third subject are 69.