How to find the total number of pages which a book has when the clues given indicate a range?

Question

This problem doesn't seem very complicated but I got stuck at trying to understand what is the meaning of the last clue involving an integer and a range. Can somebody help me?

The problem is as follows:

Marina is reading a novel. The first day she read a third of the book, the second day she read the fourth parts of what it was left, the third day she read half of what it was left to read, the fourth day she read the fifth parts of what it was still left to read, the fifth day she decided to end the novel and found that it was left less than $70$ pages. If she always read an integer number of pages and never read less than $14$ pages. How many pages did the novel had?

The alternatives given in my book were as follows:

$\begin{array}{ll} 1.&\textrm{360 pages}\\ 2.&\textrm{240 pages}\\ 3.&\textrm{180 pages}\\ 4.&\textrm{300 pages}\\ 5.&\textrm{210 pages}\\ \end{array}$

I'm lost with this problem. What would be the correct way to go?. So far what I attempted to do was the following:

I thought that the number of pages that the book has to be $x$.

Since it said that the first day she read a third of the book I defined it as:

$\frac{1}{3}x$

On the second day it is said that she read the fourth parts of what it was left so that would account for:

$\frac{1}{4}\left(x-\frac{1}{3}x\right)=\frac{1}{4}\left(\frac{2x}{3}\right)=\frac{x}{6}$

The third day:

$\frac{1}{2}\left(x-\frac{x}{6}\right)=\frac{1}{2}\left(\frac{5x}{6}\right)=\frac{5x}{12}$

The fourth day:

$\frac{1}{5}\left(x-\frac{5x}{12}\right)=\frac{1}{5}\left(\frac{7x}{12}\right)=\frac{7x}{60}$

The fifth day:

She decides to end reading the novel but, what it was left was less than $70$ pages.

So this would translate as:

$x-\frac{7x}{60}<70$

This would become into:

$\frac{53x}{60}<70$

Therefore:

$53x<4200$

$x<\frac{4200}{53}$

However this fraction is not an integer.

There is also another piece of information which mentioned that she always read no less than $14$ pages.

If during the first day she read a third of the novel then this would be:

$\frac{1}{3}x>14$

So $x>42$

But, on the fourth day she read:

$\frac{7x}{60}>14$

Therefore:

$x>120$

How come x can be greatest than $42$ and at the same time $120$?. Am I understanding this correctly?.

If I were to select the greatest value and put it in the range which I found earlier:

$120<x<\frac{4200}{53}$

and round to the nearest integer:

$120<x<79$

Which doesn't make sense.

If it were $\frac{5x}{12}>14$

$x>33$ (rounded to the nearest integer)

Which would be:

$33<x<79$

But again this doesn't produce an reasonable answer within the specified range in the answers. Did I overlooked something or perhaps didn't understood something right?. Can somebody help me with this inequation problem?.

Answer 1

Compute the fractions of the book read per day:

• On day 1, $\frac13$ of the novel was read, leaving $\frac23$.
• On day 2, $\frac23×\frac14=\frac16$ was read, leaving $\frac23×\frac34=\frac12$.
• On day 3, $\frac12×\frac12=\frac14$ was read, the same fraction being left.
• On day 4, $\frac14×\frac15=\frac1{20}$ was read, leaving $\frac14×\frac45=\frac15$ that was finished off on day 5.

Letting $x$ be the number of pages in the book, because at most 70 pages were left on day 5 we have $\frac15x<70$ or $x<350$. Because at least 14 pages were read per day, including day 4, we have $\frac1{20}x>14$ or $x>280$. Only option 4 satisfies both inequalities, so the novel had 300 pages.

Answer 2

Your method is good, but you made a mistake on the third day. Indeed, she read a third of what was left. So she read :

$$ \frac{1}{2}\left( x - \frac{1}{3}x-\frac{1}{6}x \right) = \frac{1}{2}\frac{1}{2}x = \frac{1}{4}x$$

So on the fourth day, she read $$ \frac{1}{5}\left(1 - \frac{1}{2} - \frac{1}{4}\right)x = \frac{1}{5}\frac{1}{4}x = \frac{1}{20}x$$

So on the fifth days, there's : $$ x\left(1 - \frac{1}{3} - \frac{1}{6} - \frac{1}{4} - \frac{1}{20}\right) = \frac{1}{5}{x}$$ Pages left.

To sum up :

First day : $\frac{1}{3}x$

Second day : $\frac{1}{6}x$

Third day : $\frac{1}{4}x$

Fourth day : $\frac{1}{20}x$

You want $\frac{x}{20} \ge 14$ so $x\ge 280$. Furthermore, you need $\frac{1}{5}x< 70$ so $x<350$.

The only possibility now is answer 4 : 300 pages.