2022 AMC 8 Problem 16

Question

Four numbers are written in a row. The average of the first two is

21,

the average of the middle two is

26,

and the average of the last two is

30.

What is the average of the first and last of the numbers?

24

25

26

27

28

Convert each average into a sum

Find

a+b+c+d

, then subtract

b+c

Let the numbers be

a,b,c,d

in order.

Since the average of

a

and

b

is

21,

we know their sum is

21\cdot2 = 42.

Since the average of

c

and

d

is

30,

we know their sum is

30\cdot2 = 60.

Since

a+b = 42

and

c+d = 60 ,

we know

\begin{align*} a+b+c+d &= 42 + 60\\ &= 102 \end{align*}

Now, with the average of

b

and

c

being

26,

we know their sum is

26\cdot2=52.

This means

b+c = 52.

Subtracting this result from the sum of all the terms yields

a+d =50.

Since

\dfrac{a+d}{2}=25,

our answer is

25.

Thus, the correct answer is B.

Problem 16 in other years