AMC8 2010

AMC8 2010 · Q21

AMC8 2010 · Q21. It mainly tests Linear equations, Fractions.

Hui is an avid reader. She bought a copy of the best-seller *Math is Beautiful*. On the first day, Hui read $\frac{1}{5}$ of the pages plus 12 more, and on the second day she read $\frac{1}{4}$ of the remaining pages, plus 15 pages. On the third day, she read $\frac{1}{3}$ of the remaining pages, plus 18 pages. She then realized that there were only 62 pages left to read, which she read the next day. How many pages are in this book?

Hui 是一个狂热的读者。她买了一本畅销书《数学很美》。第一天，Hui 阅读了全书页数的 \frac{1}{5} 加上另外 12 页，第二天她阅读了剩余页数的 \frac{1}{4} 加上 15 页。第三天，她阅读了剩余页数的 \frac{1}{3} 加上 18 页。然后她发现只剩下 62 页没读了，她第二天就把这些读完了。这本书总共有多少页？

(A) 120 120

(B) 180 180

(D) 300 300

(E) 360 360

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let $x$ be the number of pages in the book. After the first day, Hui had $\frac{4x}{5}-12$ pages left to read. After the second, she had $\left(\frac{3}{4}\right)\left(\frac{4x}{5}-12\right)-15 = \frac{3x}{5}-24$ left. After the third, she had $\left(\frac{2}{3}\right)\left(\frac{3x}{5}-24\right)-18=\frac{2x}{5}-34$ left. This is equivalent to $62.$ \begin{align*} \frac{2x}{5}-34&=62\\ 2x - 170 &= 310\\ 2x &= 480\\ x &= \boxed{\textbf{(C)}\ 240} \end{align*}

设本书有 $x$ 页。第一天后，剩余页数为 $\frac{4x}{5}-12$。第二天后，剩余页数为 $\left(\frac{3}{4}\right)\left(\frac{4x}{5}-12\right)-15 = \frac{3x}{5}-24$。第三天后，剩余页数为 $\left(\frac{2}{3}\right)\left(\frac{3x}{5}-24\right)-18=\frac{2x}{5}-34$。这等于 62。 \begin{align*} \frac{2x}{5}-34&=62\\ 2x - 170 &= 310\\ 2x &= 480\\ x &= \boxed{\textbf{(C)}\ 240} \end{align*}

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