2012 AMC 12A Problem 6

Attempt Problem 6 of the 2012 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2012 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

The sums of three whole numbers taken in pairs are

12,

17,

and

19.

What is the middle number?

4

5

6

7

8

Adding all three pairwise sums counts each number twice

The total

a+b+c

is half of

12+17+19

; subtract each pair sum to get the third number

Let the numbers be

a \lt b \lt c.

Adding the three pairwise sums gives

2(a+b+c) = 12+17+19

= 48,

a+b+c = 24.

Then

a = 24-19 = 5,

b = 24-17 = 7,

and

c = 24-12 = 12.

The middle number is

7.

Thus, the correct answer is D.

Problem 6 in other years