The mean value of a set of 12 numbers is 65. If each of the first 4 numbers is increased by 9 and each of the remaining 8 numbers is decreased by 3, what will be the new mean?

The correct answer is 66.

Step 1: Calculate the initial sum of the 12 numbers.
The mean of a set of numbers is given by:

$\text{Mean}=\frac{\text{Sum of numbers}}{\text{Total count of numbers}}$

Given that the mean of 12 numbers is 65, the original sum (${\text{Sum}}_{\text{old}}$) is:

${\text{Sum}}_{\text{old}}=12\times 65=780$

Step 2: Determine the net change in the total sum.
The first 4 numbers are each increased by 9, so their total increase is:

$4\times 9=36$

The remaining 8 numbers are each decreased by 3, so their total decrease is:

$8\times 3=24$

Therefore, the net change in the sum is:

$\text{Net Change}=36-24=12$

Step 3: Calculate the new total sum.
The new sum (${\text{Sum}}_{\text{new}}$) of the numbers is:

${\text{Sum}}_{\text{new}}=780+12=792$

Step 4: Calculate the new mean.
Divide the new sum by the total count of numbers (12):

$\text{New Mean}=\frac{792}{12}=66$

Thus, the new mean of the 12 numbers is 66.