The average of n numbers is m and the average of m numbers is n. Taking all the numbers together the average is :

Question

Accepted Answer

To solve this problem, we need to determine the average of all the numbers when we combine the two sets of numbers.

1. **Explanation for Option 4:**

Let's denote the first set of numbers as having a total of $ n $ numbers, each with an average of $ m $. Therefore, the sum of these $ n $ numbers is $ n \times m $.

Similarly, the second set of numbers has $ m $ numbers, each with an average of $ n $. Thus, the sum of these $ m $ numbers is $ m \times n $.

When we combine both sets, the total number of numbers is $ n + m $, and the total sum of all numbers is $ n \times m + m \times n = 2mn $.

Therefore, the average of all the numbers combined is given by the total sum divided by the total number of numbers:

$$
   \text{Average} = \frac{2mn}{n + m}
   $$

This matches Option 4, which is $\frac{2mn}{m+n}$.

2. **Explanation for why other options are incorrect:**

- **Option 1: $ m + n $**

This option suggests that the average of all numbers is simply the sum of the two averages. However, this does not take into account the number of elements in each set and their respective contributions to the total sum. Therefore, it does not correctly represent the average of the combined set.

- **Option 2: $ mn $**

This option implies that the average of all numbers is the product of the two averages. This is incorrect because the average should be a weighted mean based on the number of elements in each set, not a simple product of the averages.

- **Option 3: $\frac{mn}{m+n}$**

This option suggests a harmonic mean of the two averages, which is not applicable in this context. The correct approach is to consider the total sum and total number of elements, leading to the formula in Option 4.

In conclusion, Option 4 is the correct answer because it accurately calculates the average of all numbers by considering both the total sum and the total number of elements in the combined set.