The average of 10 numbers is 15 and that of 15 numbers is 20. Find the average of all 25 numbers?

Answer

To find the average of all 25 numbers given the average of two separate groups, we can follow these steps:

### Step 1: Calculate the sum of the first group of numbers
The average of the first group of 10 numbers is given as 15. The formula for average is:

\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

From this, we can rearrange the formula to find the sum:

\[ \text{Sum} = \text{Average} \times \text{Number of observations} \]

For the first group:

\[ \text{Sum of first 10 numbers} = 15 \times 10 = 150 \]

### Step 2: Calculate the sum of the second group of numbers
The average of the second group of 15 numbers is given as 20. Using the same formula:

\[ \text{Sum of second 15 numbers} = 20 \times 15 = 300 \]

### Step 3: Calculate the total sum of all 25 numbers
Now, we can find the total sum of all 25 numbers by adding the sums of the two groups:

\[ \text{Total Sum} = \text{Sum of first 10 numbers} + \text{Sum of second 15 numbers} \]

\[ \text{Total Sum} = 150 + 300 = 450 \]

### Step 4: Calculate the average of all 25 numbers
Finally, we can find the average of all 25 numbers using the total sum and the total number of observations:

\[ \text{Average of all 25 numbers} = \frac{\text{Total Sum}}{\text{Total Number of Observations}} = \frac{450}{25} = 18 \]

### Final Answer
The average of all 25 numbers is **18**.

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