The average of 9 test 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 (middle) result?

Introduction / Context:
This question involves averages over overlapping subsets of a sequence of nine results. With the overall average and the averages of the first four and last four results given, we must determine the middle result. It uses the basic principle that total sum is the product of average and number of items, and that the total is the sum of parts.

Given Data / Assumptions:

• There are 9 results in sequence.

• Average of all 9 results = 50.

• Average of the first 4 results = 52.

• Average of the last 4 results = 49.

• The fifth result is the one that is not included in either group of 4.

Concept / Approach:
First compute the total sum of all 9 results using the overall average. Then compute the sum of the first four results and separately the sum of the last four results using their averages. The fifth result plus the sums of the first four and last four must equal the total sum of all nine results. Therefore the fifth result is the difference between the grand total and the combined sums of the first and last four results.

Step-by-Step Solution:
Total of 9 results = 9 * 50 = 450. Total of first 4 results = 4 * 52 = 208. Total of last 4 results = 4 * 49 = 196. Let the fifth result be x. Then total of all 9 results = sum of first 4 + x + sum of last 4. So, 208 + x + 196 = 450. Combine known sums: 208 + 196 = 404. Therefore, 404 + x = 450, so x = 450 − 404 = 46.

Verification / Alternative check:
We can verify by reconstructing the averages. With the fifth result as 46, the total is 208 (first four) + 46 + 196 (last four) = 450. Average over 9 results = 450 / 9 = 50, which matches the original overall average. The first four and last four sums still give averages of 52 and 49 respectively, so all given conditions remain satisfied and the answer is consistent.

Why Other Options Are Wrong:
If the fifth result were 48, the total would be 208 + 48 + 196 = 452 and the overall average would be 452 / 9, not 50. Values 52 or 44 would similarly lead to totals of 456 or 448, which also give incorrect overall averages. Only 46 gives a total of 450 and preserves all the given averages.

Common Pitfalls:
The most common error is to forget that the fifth result is not counted in either of the four result averages and to incorrectly include it in those totals. Another mistake is miscomputing the partial sums 208 and 196. Carefully writing out the equation total = first four + fifth + last four, and checking each multiplication, helps to avoid such errors.

Final Answer:
The fifth result is 46.