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AMC 8 Math Olympiad Number Theory

Page number counting |AMC 8- 2010 -|Problem 21

Try this beautiful problem from Algebra about Page number counting from AMC-8(2010). You may use sequential hints to solve the problem.

Try this beautiful problem from Algebra about Page number counting

Page number counting | AMC-8, 2010 |Problem 21


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 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?

  • 320
  • 240
  • 200

Key Concepts


Algebra

Arithmetic

multiplication

Check the Answer


Answer:$240$

AMC-8, 2010 problem 21

Challenges and Thrills in Pre College Mathematics

Try with Hints


assume that the number of all pages be \(x\)

Can you now finish the problem ……….

count day by day

can you finish the problem……..

Let x be the number of pages in the book

First day ,Hui Read \(\frac{x}{5} + 12\) pages

After first day Remaining pages=\(\{x-(\frac{x}{5}+12)\}\)=\(\frac{4x}{5} -12\)

Second day ,Hui Read \(\frac{1}{4} (\frac{4x}{5} -12) +15=\frac{x}{5} +12\)

After Second day Remaining pages= \((\frac{4x}{5} -12) -(\frac{x}{5} +12)\)=\(\frac{4x}{5} -\frac{x}{5}-24\)=\(\frac{3x}{5} -24\)

Third day,Hui read \(\frac {1}{3} (\frac{3x}{5} -24) +18\) =\((\frac{x}{5} -8+18)\)=\(\frac{x}{5} +10\)

After Third day Remaining pages = \((\frac{3x}{5} -24) -(\frac{x}{5} +10)\) =\(\frac{2x}{5} – 34\)

Now by the condition, \(\frac{2x}{5} – 34 = 62\)

\(\Rightarrow 2x-170=310\)

\(\Rightarrow 2x=480\)

\(\Rightarrow x=240\)

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