The average of 31 numbers is 19. The average of the first 8 of these numbers is 24. What is the average of the remaining 23 numbers?

Introduction / Context:
This question involves splitting a data set into two parts and using averages to find the mean of the remaining subset. We are given the overall average of all 31 numbers and the average of the first 8 numbers. Using these, we must determine the average of the remaining 23 numbers.

Given Data / Assumptions:

• Total number of numbers = 31.
• Average of all 31 numbers = 19.
• Average of first 8 numbers = 24.
• We need the average of the remaining 23 numbers.

Concept / Approach:
The basic idea is to convert averages into sums, because total sum = average * number of terms. We first compute the total sum of all 31 numbers using the overall average. Then we compute the sum of the first 8 numbers using their average. Subtracting this from the total gives the sum of the remaining 23 numbers. Dividing that sum by 23 gives the required average.

Step-by-Step Solution:
Total number of numbers = 31. Average of all 31 numbers = 19. Total sum of all numbers = 31 * 19. Total sum = 589. Average of first 8 numbers = 24. Sum of first 8 numbers = 8 * 24 = 192. Sum of remaining 23 numbers = total sum - sum of first 8 numbers. Sum of remaining 23 numbers = 589 - 192 = 397. Average of remaining 23 numbers = 397 / 23. Average = 397 / 23 ≈ 17.26.
Verification / Alternative check:
We can check that the combined effect of averages is consistent. If the remaining 23 numbers have average about 17.26, their total is about 397. Adding back the first group: 192 + 397 = 589, which divided by 31 gives 19, matching the given overall average.
Why Other Options Are Wrong:
Option B (23.25) and option C (26.2) are too high. If those were correct, the overall average would be much greater than 19. Option D (25.45) is even larger and clearly inconsistent with the overall average of 19, given that the first 8 numbers already have an average of 24. Option A (17.26): Correct approximate value of 397 divided by 23.
Common Pitfalls:
Some students mistakenly average 19 and 24 directly instead of converting to sums. Others may incorrectly divide the sum of the remaining numbers by 31 instead of 23. Calculation errors while multiplying or subtracting can also lead to wrong totals and thus wrong averages.
Final Answer:
The average of the remaining 23 numbers is approximately 17.26.