The average of two numbers is 8 and the average of another three numbers is 3. What is the average of all five numbers together?

Introduction / Context:
This question examines how to combine averages from two different groups of numbers to find the overall average of the combined group. It is a common type of problem in aptitude tests involving weighted averages.

Given Data / Assumptions:
- There are two groups of numbers. - First group: 2 numbers with average 8. - Second group: 3 numbers with average 3. - We need the average of all 5 numbers together.

Concept / Approach:
The overall average of combined groups is not simply the average of the two averages. Instead, it is computed using total sums. For each group, total sum = average * number of terms. We compute the total of both groups, add them, and then divide by the total number of terms, which is 5. This is a straightforward application of the definition of average.

Step-by-Step Solution:
Step 1: Let the two numbers in the first group be a and b. Step 2: Average of a and b is 8, so (a + b) / 2 = 8. Step 3: Therefore, a + b = 16. Step 4: Let the three numbers in the second group be c, d and e. Step 5: Average of c, d and e is 3, so (c + d + e) / 3 = 3. Step 6: Therefore, c + d + e = 9. Step 7: Total sum of all five numbers = (a + b) + (c + d + e) = 16 + 9 = 25. Step 8: Total number of numbers = 2 + 3 = 5. Step 9: Average of all five numbers = total sum / total count = 25 / 5. Step 10: Compute the result: 25 / 5 = 5. Step 11: So, the overall average is 5.

Verification / Alternative check:
As a quick check, note that the two averages 8 and 3 are being combined with weights 2 and 3 respectively. The overall average is (2 * 8 + 3 * 3) / (2 + 3) = (16 + 9) / 5 = 25 / 5 = 5. This is a direct application of the weighted average formula and confirms the earlier result.

Why Other Options Are Wrong:
Option A (2) and B (3): Too low and ignore the influence of the larger numbers in the first group. Option D (6): Might come from averaging 8 and 4 incorrectly; it does not match the combined total. Option E (4): Could come from averaging 8 and 3 directly, but that is incorrect because the groups have different sizes.

Common Pitfalls:
A common error is to average the two given averages themselves, such as (8 + 3) / 2 = 5.5, without considering the number of values in each group. While sometimes this may accidentally give a close value, it is not a reliable method. Always convert averages to totals, combine the totals, and then divide by the total number of values.

Final Answer:
The average of all five numbers together is 5.