AMC10 2012 B

AMC10 2012 B · Q6

AMC10 2012 B · Q6. It mainly tests Rounding & estimation.

In order to estimate the value of $x - y$ where $x$ and $y$ are real numbers with $x > y > 0$, Xiaoli rounded $x$ up by a small amount, rounded $y$ down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?

为了估算实数 $x - y$ 的值，其中 $x > y > 0$，小丽将 $x$ 向上取整了一小段距离，将 $y$ 向下取整了相同距离，然后减去她取整后的值。以下哪个陈述一定是正确的？

(A) Her estimate is larger than $x - y$. 她的估计值大于 $x - y$。

(B) Her estimate is smaller than $x - y$. 她的估计值小于 $x - y$。

(D) Her estimate equals $y - x$. 她的估计值等于 $y - x$。

(E) Her estimate is 0. 她的估计值为 0。

Answer

Correct choice: (A)

正确答案：（A）

Solution

Consider $x$ and $y$ as points on the real number line, with $x$\nnecessarily to the right of $y$.\nThen $x - y$ is the distance between $x$ and $y$.\nXiaoli’s rounding moved $x$ to the right and moved $y$ to the left. Therefore the\ndistance between them increased, and her estimate is larger than $x - y$.\n\nTo see that the other answer choices are not correct, let $x = 2.9$ and $y = 2.1$,\nand round each by 0.1. Then $x - y = 0.8$ and Xiaoli’s estimated difference is\n$(2.9 + 0.1) - (2.1 - 0.1) = 1.0$.

将 $x$ 和 $y$ 视为实数轴上的点，$x$ 必然在 $y$ 的右侧。那么 $x - y$ 是 $x$ 和 $y$ 之间的距离。小丽的取整将 $x$ 向右移动，将 $y$ 向左移动。因此它们之间的距离增加了，她的估计值大于 $x - y$。为了看出其他选项不正确，取 $x = 2.9$ 和 $y = 2.1$，每个取整 0.1。那么 $x - y = 0.8$，小丽的估计差值为 $(2.9 + 0.1) - (2.1 - 0.1) = 1.0$。

Topics

AR.011 Rounding & estimation