AMC10 2012 A

AMC10 2012 A · Q8

AMC10 2012 A · Q8. It mainly tests Systems of equations.

The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?

三个整数两两相加的和分别是12、17和19。这三个整数中的中间大小的数是多少？

(A) 4 4

(B) 5 5

(D) 7 7

(E) 8 8

Answer

Correct choice: (D)

正确答案：（D）

Solution

Let the three whole numbers be $a < b < c$. The set of sums of pairs of these numbers is $(a + b, a + c, b + c) = (12, 17, 19)$. Thus $2(a + b + c) = (a + b) + (a + c) + (b + c) = 12 + 17 + 19 = 48$, and $a + b + c = 24$. It follows that $(a, b, c) = (24 - 19, 24 - 17, 24 - 12) = (5, 7, 12)$. Therefore the middle number is 7.

设三个整数为$a < b < c$。两两和为$(a + b, a + c, b + c) = (12, 17, 19)$。因此$2(a + b + c) = (a + b) + (a + c) + (b + c) = 12 + 17 + 19 = 48$，所以$a + b + c = 24$。从而$(a, b, c) = (24 - 19, 24 - 17, 24 - 12) = (5, 7, 12)$。因此中间数是7。

Topics

AL.002 Systems of equations