Question

When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1. How many integers were in the original list?

Answer 1

Answer: 4.

Step-by-step explanation:

Let the number of integers initially be n, and their mean be x.

So, given:

$\frac{nx + 15}{n + 1} = x + 2$

$nx + 15 = (n + 1)(x + 2) = nx + 2 + x + 2n$

$x + 2n = 13$

Also, on appending 1:

$\frac{nx + 15 + 1}{n + 2} = x + 2 - 1$

$\frac{nx + 16}{n + 2} = x + 1$

$nx + 16 = (n + 2)(x + 1) = nx + 2 + n + 2x$

$n + 2x = 14$

$2n + 4x = 28$

$2n + x = 13$ (look above)

$3x = 15$ => $x = 5$

And, $n = 14 - 2x = 14 - 10 = 4$

Number of integers in original list = n = 4.