Find the mean of 75 numbers, if the mean of 45 of them is 18 and the mean of the remaining ones is 13.

Answer

To find the mean of 75 numbers given the mean of 45 of them is 18 and the mean of the remaining 30 is 13, we can follow these steps:

### Step 1: Calculate the sum of the first 45 numbers
The mean of the first 45 numbers is given as 18. The sum of these numbers can be calculated using the formula:

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

So, for the first 45 numbers:

\[ \text{Sum of first 45 numbers} = 18 \times 45 = 810 \]

### Step 2: Calculate the sum of the remaining 30 numbers
The mean of the remaining 30 numbers is given as 13. Similarly, we can calculate the sum of these numbers:

\[ \text{Sum of remaining 30 numbers} = 13 \times 30 = 390 \]

### Step 3: Calculate the total sum of all 75 numbers
Now, we can find the total sum of all 75 numbers by adding the sums calculated in Steps 1 and 2:

\[ \text{Total Sum} = \text{Sum of first 45 numbers} + \text{Sum of remaining 30 numbers} = 810 + 390 = 1200 \]

### Step 4: Calculate the mean of all 75 numbers
Finally, we can find the mean of all 75 numbers using the formula:

\[ \text{Mean} = \frac{\text{Total Sum}}{\text{Total Number of Observations}} = \frac{1200}{75} \]

Calculating this gives:

\[ \text{Mean} = 16 \]

### Final Answer:
The mean of all 75 numbers is **16**.

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