AMC12 2015 A

AMC12 2015 A · Q5

AMC12 2015 A · Q5. It mainly tests Manipulating equations, Rounding & estimation.

Amelia needs to estimate the quantity $\frac{a}{b} - c$, 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 $\frac{a}{b} - c$?

阿梅利亚需要估算 $\frac{a}{b} - c$ 的值，其中 $a$、$b$ 和 $c$ 是较大的正整数。她将每个整数四舍五入以便于心算。在以下哪种情况下，她的答案必然大于 $\frac{a}{b} - c$ 的精确值？

(A) She rounds all three numbers up. 她将所有三个数向上取整。

(B) She rounds $a$ and $b$ up, and she rounds $c$ down. 她将$a$和$b$向上取整，将$c$向下取整。

(D) She rounds $a$ up, and she rounds $b$ and $c$ down. 她将$a$向上取整，将$b$和$c$向下取整。

(E) She rounds $c$ up, and she rounds $a$ and $b$ down. 她将$c$向上取整，将$a$和$b$向下取整。

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): As long as $x$ and $y$ and their rounded values are positive, rounding the dividend $x$ up in a division problem $\frac{x}{y}$ makes the answer larger, and rounding $x$ down makes the answer smaller. Similarly, rounding the divisor $y$ up makes the answer smaller, and rounding $y$ down makes the answer larger. In a subtraction problem $x-y$, rounding $x$ up or rounding $y$ down increases the answer, and rounding $x$ down or rounding $y$ up decreases it. Only in choice (D) do all the roundings contribute to increasing the answer. In the other situations, the estimate may be larger or smaller than the exact value, depending on the the amount by which each number is rounded and their values. In particular, the rounding may make the answer smaller. For (A), $\frac{999999}{900}-490>\frac{1000000}{1000}-500.$ For (B), $\frac{999999}{900}-510>\frac{1000000}{1000}-500.$ For (C), $\frac{999999}{1001}-490>\frac{1000000}{1000}-500.$ For (E), $\frac{1000001}{1001}-490>\frac{1000000}{1000}-500.$

答案（D）：只要 $x$ 和 $y$ 及其四舍五入后的值都是正数，在除法问题 $\frac{x}{y}$ 中，把被除数 $x$ 向上取整会使结果变大，把 $x$ 向下取整会使结果变小。同样，把除数 $y$ 向上取整会使结果变小，把 $y$ 向下取整会使结果变大。在减法问题 $x-y$ 中，把 $x$ 向上取整或把 $y$ 向下取整会使结果增大，而把 $x$ 向下取整或把 $y$ 向上取整会使结果减小。只有在选项（D）中，所有的取整都会促使结果增大。在其他情况下，估算值可能大于或小于精确值，这取决于每个数取整的幅度以及它们的数值。特别地，取整可能会使结果变小。对于（A），$\frac{999999}{900}-490>\frac{1000000}{1000}-500.$ 对于（B），$\frac{999999}{900}-510>\frac{1000000}{1000}-500.$ 对于（C），$\frac{999999}{1001}-490>\frac{1000000}{1000}-500.$ 对于（E），$\frac{1000001}{1001}-490>\frac{1000000}{1000}-500.$

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