2012 AMC 10A Problem 8

Question

The sums of three whole numbers taken in pairs are

12,

17,

and

19.

What is the middle number?

4

5

6

7

8

Add the three pair sums

The total of the three pair sums is twice the sum of the three numbers

Let the three numbers be

a, b, c

where

a \lt b \lt c.

None of them are equal, since all three sums are different.

Then

a + b = 12, a + c = 17,

and

b + c = 19.

Adding all three equations together gives us

2(a + b + c) = 48.

Then

a + b + c = 24,

from which we can subtract

a + c = 17,

to get

b = 7.

Thus, D is the correct answer.

Problem 8 in other years