Question

两个自然数的和为 $17,402$ 。其中一个数能被 $10$ 整除。如果擦去这个数的个位数字，就得到另一个数。这两个数的差是多少？

The sum of two natural numbers is $17,402.$ One of the two numbers is divisible by $10.$ If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?

$10,272$

$11,700$

$13,362$

$14,238$

$15,426$

答案：D

知识点：位值一次方程

难度评级：900

视频讲解：

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文字解答：

设两个数为 $x$ 和 $y$ 。不妨设 $x$ 能被 $10$ 整除，那么 $x$ 的个位数字是 $0$ 。

擦去 $x$ 的个位数字，相当于把它除以 $10$ 。题意还给出 $\dfrac{x}{10} = y$ 。

由两数之和可得因此两数之差为 $\begin{aligned} \dfrac{x}{10} + x &= 17,402 \\ \dfrac{11x}{10} &= 17,402 \\ x &= 15,820. \end{aligned}$ $x - \dfrac{x}{10} = 14,238.$

所以正确答案是 D。

Let $x$ and $y$ be the two numbers. WLOG, let $x$ be divisible by $10.$ Then the units digit of $x$ is $0.$

If we erase the units digit, then we are essentially dividing $x$ by $10.$ The problem statement also gives us that $\dfrac{x}{10} = y.$

Therefore, $\begin{aligned} \dfrac{x}{10} + x &= 17,402 \\ \dfrac{11x}{10} &= 17,402 \\ x &= 15,820. \end{aligned}$ Then $x - \dfrac{x}{10} = 14,238.$

Thus, D is the correct answer.

2021 AMC 10A Spring 第 3 题

答案：D

难度评级：900

视频讲解：

文字解答：

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