Rakesh has marks in five subjects. The average of his marks in the first three subjects is 79, and the average of his marks in the last three subjects is 86. If his mark in the third subject is 80, what is his average mark across all five subjects?

Introduction / Context:
This is a clever averages question that involves overlapping groups of subjects. You are given the average of the first three subjects, the average of the last three subjects, and the actual mark in the overlapping subject. The task is to determine the overall average in all five subjects. It tests your ability to manipulate sums and averages with overlapping data.

Given Data / Assumptions:

Rakesh has marks in 5 subjects, call them S1, S2, S3, S4 and S5.
Average of first three subjects (S1, S2, S3) = 79.
Average of last three subjects (S3, S4, S5) = 86.
Mark in the third subject S3 = 80.
We need the average of all five subjects combined.

Concept / Approach:
Average = total marks / number of subjects. From the given averages of first three and last three subjects, we can find the sums S1 + S2 + S3 and S3 + S4 + S5. Since S3 is common to both groups and its value is known, we can separate out sums S1 + S2 and S4 + S5. Adding these with S3 gives the total marks in all five subjects, and dividing by 5 yields the required overall average.

Step-by-Step Solution:
Step 1: Convert averages into totals. S1 + S2 + S3 = 79 * 3 = 237. S3 + S4 + S5 = 86 * 3 = 258. Step 2: Use the given value of S3. S3 = 80. So S1 + S2 = 237 - 80 = 157. And S4 + S5 = 258 - 80 = 178. Step 3: Find total marks in all five subjects. Total = (S1 + S2) + S3 + (S4 + S5) = 157 + 80 + 178 = 415. Step 4: Compute overall average. Average over 5 subjects = 415 / 5 = 83.

Verification / Alternative Check:
We can check consistency by reconstructing plausible marks or simply reusing the sums. S1 + S2 + S3 = 237, S3 + S4 + S5 = 258, and S3 = 80 give S1 + S2 = 157 and S4 + S5 = 178 as before. Adding all five subjects again gives 415, so 415 / 5 = 83, confirming the calculation. The value 83 also lies between 79 and 86, which is reasonable because the full set of five subjects includes both groups.

Why Other Options Are Wrong:
81 and 82 are both lower than the correct overall mean and would not yield the correct totals when multiplied by 5.
85 is too high and would require the total to be 425, which contradicts the sums computed from the given averages.

Common Pitfalls:
One common mistake is to simply average 79 and 86, which ignores the overlapping subject and yields 82.5, not an integer and not correct. Another issue is forgetting to subtract the third subject when transitioning between the sum of three subjects and the sum of the other two. Keeping track of the overlapping subject S3 and working systematically with sums avoids these errors.

Final Answer:
Rakesh's average mark across all five subjects is 83.