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:
Here we are given the average of a total set of numbers and the average of a subset of those numbers. Specifically, 14 numbers have a certain average, and the last 5 numbers within this group have a lower average. The aim is to find the average of the remaining 9 numbers. This is a typical mixture problem where we need to make use of sums derived from averages and then separate the contributions of subgroups.

Given Data / Assumptions:

• Total number of numbers = 14.
• Average of all 14 numbers = 32.
• Average of the last 5 numbers = 26.
• We must find the average of the first 9 numbers (the remaining numbers).

Concept / Approach:
The sum of all 14 numbers can be obtained by multiplying the overall average by 14. Similarly, the sum of the last 5 numbers is the average of those 5 multiplied by 5. The sum of the first 9 numbers is simply the total sum minus the sum of the last 5. Once we know that sum, dividing by 9 gives the required average. This approach uses the idea that total = subgroup one + subgroup two, and is very common in average problems.

Step-by-Step Solution:
Step 1: Compute the total sum of all 14 numbers. Total sum = 14 * 32. 14 * 32 = 14 * (30 + 2) = 420 + 28 = 448. Step 2: Compute the sum of the last 5 numbers. Average of last 5 numbers = 26. Sum of last 5 numbers = 5 * 26 = 130. Step 3: Find the sum of the first 9 numbers. Sum of first 9 numbers = total sum − sum of last 5 numbers. Sum of first 9 = 448 − 130 = 318. Step 4: Compute the average of the first 9 numbers. Average of first 9 numbers = 318 / 9. 9 * 35 = 315, remainder 3, so 318 / 9 = 35 + 3/9 = 35.333..., which we write as 35.33 to two decimal places.
Verification / Alternative check:
We can verify that mixing averages works out. Weighted contribution of first 9 numbers = 9 * 35.33 ≈ 318. Weighted contribution of last 5 numbers = 5 * 26 = 130. Total ≈ 318 + 130 = 448, and 448 / 14 = 32, which matches the original average.
Why Other Options Are Wrong:
An average of 41.33 or 44.5 would give a total for the first 9 numbers that is too large and would raise the overall average above 32. An average of 27.5 or 32 for the first 9 numbers would produce totals that do not combine with the last 5 numbers to give an overall average of 32.
Common Pitfalls:
Some students mistakenly average 32 and 26 directly, without taking into account the fact that they apply to different group sizes. Another mistake is to mix up the number of terms, using 14 instead of 9 when computing the average of the remaining numbers.
Final Answer:
The average of the remaining 9 numbers is 35.33 (approximately).