The average of 14 numbers is 32. If the average of the last 5 numbers is 26, what is the average of the remaining 9 numbers?

Introduction / Context:
This question focuses on how averages change when a subset of numbers is separated out. You know the overall average for 14 numbers and a separate average for the last 5 numbers. Using these, you must compute the average of the remaining 9 numbers.

Given Data / Assumptions:

• There are 14 numbers in total.
• Their overall average is 32.
• The average of the last 5 numbers is 26.
• We are asked to find the average of the first 9 numbers or equivalently the remaining 9 numbers.

Concept / Approach:
The key idea is that average multiplied by count gives the total sum. We first find the total of all 14 numbers from the overall average. Then we find the total of the last 5 numbers from their average. By subtracting the total of these 5 numbers from the grand total, we obtain the total of the remaining 9 numbers. Finally, we divide that result by 9 to get their average.

Step-by-Step Solution:
Step 1: Total of all 14 numbers = 14 * 32. Step 2: Compute 14 * 32 = 448. Step 3: Total of the last 5 numbers = 5 * 26 = 130. Step 4: Total of the remaining 9 numbers = 448 - 130 = 318. Step 5: Average of the remaining 9 numbers = 318 / 9. Step 6: Compute 318 / 9 = 35.333..., which is written as 35.33 to two decimal places.

Verification / Alternative check:
If the 9 earlier numbers have an average slightly above 35, and the 5 later numbers have an average of 26, it is reasonable that the overall average of all 14 numbers is between these two values at 32. The calculation 448 = 318 + 130 also checks out, so the arithmetic is consistent.

Why Other Options Are Wrong:
An average of 41.33 would be too high, pulling the overall average above 32 when combined with the last 5 numbers at 26. Options 27.5 and 44.5 are even more extreme and do not produce the correct grand total when back calculated. Only 35.33 leads to a total of 318 for the 9 numbers, which correctly combines with 130 for the last 5 to give 448 overall.

Common Pitfalls:
Many learners mistakenly average 32 and 26 directly, which has no mathematical justification here because the two groups have different sizes. Others forget to multiply by the number of terms when converting averages to totals. Always work with totals when combining or separating groups of numbers with known averages.

Final Answer:
The average of the remaining 9 numbers is 35.33.