With 25 numbers, the median is the 13th number in order. What must that 13th number be?

Because $B$ is the 12 largest integers, maximizing the sum of $B$ is the same as minimizing the sum of the remaining 13 integers, while still keeping the median fixed.

In Desmos, type 25*8 to get the total sum of all 25 integers.

Because the median is the 13th value, type 12*1+7 to represent 12 minimum positive integers plus the required median value.

Type (25*8-(12*1+7))/12 to compute the mean of the 12 largest integers under this extreme case.

Type (25*8-(12*1+7))/12 - 8 and choose the answer option that matches the value displayed. Ѕourcе‌: ‍аniко.аi

Since data set A has 25 integers with mean 8, the total sum is А‌niк‌o.ai - ЅAT Pr⁠ер

With 25 values, the median is the 13th value when the data are ordered. So the 13th value must be 7.

To make $\text{mean}(B)$ as large as possible, make the 12 values below that median as small as possible. Because the integers are positive, the smallest possible value is 1.

So the smallest possible sum of the 13 integers not in $B$ is

The 12 largest integers (data set B) then have the greatest possible sum:

So

Since $\text{mean}(A)=8$,

So the maximum possible value is $\frac{85}{12}$.