Question

Both the average (arithmetic mean) and the median of a set of 7 numbers equal 20. If the smallest number in the set is 5 less than half the largest number, what is the largest possible number in the set?

(A) 40

(B) 38

(C) 33

(D) 32

(E) 30

Answer 1

Solution:

(B) 38

we have 7 numbers, out of which 20 is the middle one..

_ _ _ 20 _ _ A
suppose A is the largest number.
A/2 -5 = is the smallest number
to maximize A, suppose that we have all the numbers before 20 equal to the smallest number, and everything between 20 and A is equal to 20.

we know the average, we can find the sum.
20*7 = 140.
we have 3 numbers equal to 20 -> 60total
we have 3 numbers equal to A/2 -5.

now:
3(A/2 - 5) + A = 140-60 = 80.
3A/2 - 15 + A = 80 -> multiply everything by 2, to get rid of the fractions.
3A - 30 +2A = 160
5A=190
A=190/5 = 38.