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

小提示：

如果删去末尾的零得到另一个数，那么较大的数是较小数的十倍。

If deleting the final zero gives the other number, the larger number is ten times the smaller one

大提示：

先用总和求出较小的那个数。

Use the sum to find the smaller number first

视频讲解：

Click to load, then click again to play

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 $\begin{aligned} \dfrac{x}{10} + x &= 17,402 \\ \dfrac{11x}{10} &= 17,402 \\ x &= 15,820 \end{aligned}\text{。}$ 因此两数之差为 $x - \dfrac{x}{10} = 14,238\text{。}$

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