When 2 is subtracted from each of the given n numbers, then the sum of the numbers so obtained is 102. When 5 is subtracted from each of them, then the sum of the numbers so obtained is 12. What is the average of the given n numbers?

Let for 'n' numbers the average be 'x'.

So, the total sum of 'n' numbers would be 'nx'.

If 2 is subtracted from each 'n' numbers, then the resulted value to be subtracted becomes = 2n

Thus, value of the total sum now = (nx - 2n)

Given that, this value equals to 102.

So, nx - 2n = 102 ...(1)

Again when 5 is subtracted from each 'n' numbers, then the resulted value to be subtracted becomes = 5n

Thus, value of the total sum now = (nx - 5n)

Given that, this value equals to 12.

So, nx - 5n = 12 ...(2)

Subtracting (2) from (1), we get:

nx - 2n - (nx - 5n) = 102 - 12

⇒ -2n + 5n = 90

⇒ 3n = 90

⇒ n = 90/3 = 30

There are 30 numbers.

Putting n = 30, in eqn.(1), we get:

(30)x - 2(30) = 102

⇒ 30x - 60 = 102

⇒ 30x = 162

⇒ x = 162/30 = 5.4

$$\therefore$$ The average of 30 numbers is 5.4.