Question

Amelia 需要估算 $\dfrac{a}{b}-c$ ，其中 $a$ ， $b$ ，和 $c$ 都是很大的正整数。她把每个整数都取整，使计算更容易心算。在哪种情况下，她的答案一定会大于精确值 $\dfrac{a}{b}-c$ ？

Amelia needs to estimate the quantity $\dfrac{a}{b}-c,$ where $a,$ $b,$ and $c$ are large positive integers. She rounds each of the integers so that the calculation will be easier to do mentally. In which of these situations will her answer necessarily be greater than the exact value of $\dfrac{a}{b}-c?$

她把三个数都向上取整。

She rounds all three numbers up.

她把 $a$ 和 $b$ 向上取整，把 $c$ 向下取整。

She rounds $a$ and $b$ up, and she rounds $c$ down.

她把 $a$ 和 $c$ 向上取整，把 $b$ 向下取整。

She rounds $a$ and $c$ up, and she rounds $b$ down.

她把 $a$ 向上取整，把 $b$ 和 $c$ 向下取整。

She rounds $a$ up, and she rounds $b$ and $c$ down.

她把 $c$ 向上取整，把 $a$ 和 $b$ 向下取整。

She rounds $c$ up, and she rounds $a$ and $b$ down.

答案：D

知识点：估算不等式

难度评级：1270

解答：

要使 $\dfrac{a}{b}$ 变大，应把分子 $a$ 向上取整，把分母 $b$ 向下取整。要使 $-c$ 变大，应把 $c$ 向下取整。

只有选项 $\text{(D)}$ 同时做到这三点：把 $a$ 向上取整，同时把 $b$ 和 $c$ 向下取整，因此每个变化都会使估算值高于精确值。其他选项中至少有一个变化方向相反，所以不能保证估算值更大。

因此，正确答案是 D。

To make $\dfrac{a}{b}$ larger, round the numerator $a$ up and the denominator $b$ down. To make $-c$ larger, round $c$ down.

Only choice $\text{(D)}$ does all three: it rounds $a$ up while rounding $b$ and $c$ down, so every change pushes the estimate above the exact value. In the other choices at least one change works the wrong way, so the estimate is not guaranteed to be larger.

Thus, the correct answer is D.

2015 AMC 12A 第 5 题

答案：D

难度评级：1270

解答：

其他年份的第 5 题