2010 AMC 8 Problems/Problem 21

Problem

Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read $1/5$ of the pages plus $12$ more, and on the second day she read $1/4$ of the remaining pages plus $15$ pages. On the third day she read $1/3$ 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?

$\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 180\qquad\textbf{(C)}\ 240\qquad\textbf{(D)}\ 300\qquad\textbf{(E)}\ 360$

Solution 1(algebra 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 $62.$

$\begin{align*} \frac{2x}{5}-34&=62\\ 2x - 170 &= 310\\ 2x &= 480\\ x &= \boxed{\textbf{(C)}\ 240} \end{align*}$

Solution 2 (working backwards)

On the last day Hui read $62$ pages. Working backwards, she read $18$ pages more ( $80$ total). $80$ is $\frac{2}{3}$ of the remaining pages, so the third morning Hui had $120$ pages to read. Add back the $15$ and you find $135$ must be $\frac{3}{4}$ of the remaining pages, so there were $180$ pages on the second morning. Add $12$ more, then $192$ is $\frac{4}{5}$ of the total book, which must be $\boxed{\textbf{(C)}\ 240}$ pages long.

Video Solution by OmegaLearn

https://youtu.be/rQUwNC0gqdg?t=1739

~pi_is_3.14

Video by MathTalks

https://youtu.be/mSCQzmfdX-g

Video Solution by WhyMath

https://youtu.be/p2wtDPepQhY

~savannahsolver

2010 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

Page

Toolbox

Search