AMC12 2014 A

AMC12 2014 A · Q6

AMC12 2014 A · Q6. It mainly tests Linear equations, Money / coins.

The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?

一个两位数与其数字反转后的数的差，是该数两位数字之和的5倍。两位数与其反转数之和是多少？

(A) 44 44

(B) 55 55

(D) 99 99

(E) 110 110

Answer

Correct choice: (D)

正确答案：（D）

Solution

Let the two digits be $a$ and $b$. Then, $5a + 5b = 10a + b - 10b - a = 9a - 9b$, or $2a = 7b$. This yields $a = 7$ and $b = 2$ because $a, b < 10$. Then, $72 + 27 = \boxed{\textbf{(D) }99}.$

设两位数为$a$和$b$。则$5a + 5b = 10a + b - 10b - a = 9a - 9b$，即$2a = 7b$。因为$a, b < 10$，得$a = 7$，$b = 2$。则$72 + 27 = \boxed{\textbf{(D) }99}$。

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