Question

三个非负整数两两相加得到的和分别为 $12$ ， $17$ 和 $19$ 。这三个数中的中间那个数是多少？

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

$4$

$5$

$6$

$7$

$8$

知识点：方程组

把三个两两和相加。

Add the three pair sums

三个两两和的总和是三个数总和的两倍。

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

设三个数为 $a, b, c$ ，且 $a \lt b \lt c$ 。由于三个和都不同，所以这三个数也互不相等。

题意给出 $a + b = 12, a + c = 17\text{，}$ 以及 $b + c = 19\text{。}$

把三式相加，得到 $2(a + b + c) = 48\text{。}$

因此 $a + b + c = 24\text{，}$ 再减去 $a + c = 17\text{，}$ 得到 $b = 7$ 。

所以正确答案是 D。

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.