The average of 9 results is 50. The average of the first four results is 52 and the average of the last four results is 49. What is the value of the fifth result?

Introduction / Context:
This is a classic averages question in which you are given the overall average of several results as well as the averages of subsets of those results. The aim is to find the missing central value. Problems like this test your ability to switch between averages and total sums and to carefully separate different groups within the data.

Given Data / Assumptions:

There are 9 results in total.
The average of all 9 results is 50.
The average of the first 4 results is 52.
The average of the last 4 results is 49.
We are required to find the fifth result, which lies between the first four and the last four results.

Concept / Approach:
The key idea is that average multiplied by the number of items gives the total sum of those items. We can compute the total of all 9 results, the total of the first 4, and the total of the last 4. The middle result is not counted in either of the two groups of four, so we subtract both partial totals from the overall total to obtain the fifth result.

Step-by-Step Solution:
Step 1: Compute the total of all 9 results. Total of 9 results = average * number of results = 50 * 9 = 450. Step 2: Compute the total of the first 4 results. Total of first 4 results = 52 * 4 = 208. Step 3: Compute the total of the last 4 results. Total of last 4 results = 49 * 4 = 196. Step 4: Use these totals to find the fifth result. Total of all 9 results = (first 4 results) + (fifth result) + (last 4 results). So, 450 = 208 + fifth result + 196. Add 208 and 196 to get 404. Therefore, fifth result = 450 - 404 = 46.

Verification / Alternative Check:
To verify, we can imagine example numbers that satisfy the given averages or simply trust the algebra. If the fifth result is 46, then the grand total 450 is exactly split into 208, 46 and 196 for each part. Recomputing the averages from these totals gives back 52 for the first four, 49 for the last four and 50 overall, which confirms our answer.

Why Other Options Are Wrong:
Option 48 would give a total of 208 + 48 + 196 = 452, which would lead to an overall average greater than 50, so it is wrong.
Option 52 would produce 208 + 52 + 196 = 456, again giving a higher overall average than 50.
Option 44 would give a total of 208 + 44 + 196 = 448, which would produce an overall average less than 50, so it is also incorrect.

Common Pitfalls:
A common error is to average 52 and 49 and assume that somehow produces the fifth result. Another mistake is forgetting that the middle result is counted only once in the total, while the groups of four do not include it at all. Students may also mistakenly treat 9 as if it is split into equal parts, which is not the case here because the central value is separate.

Final Answer:
The value of the fifth result is 46.