Solving Equations for Unknown Book Pages

Welcome to today’s exploration of a challenging reading problem! We will walk through how to solve these types of logic puzzles and equations to determine the total number of pages in a book, using multiple methods. Let’s dive into the details of each solution.

Method 1: Initial Fraction Method

The problem starts with George reading a book on Monday and Tuesday. On Monday, he read 1/8 of the book. On Tuesday, he read 3/5 of the remaining pages, and he has 42 pages left to read.

Let's denote the total number of pages in the book as x.

On Monday, he read 1/8 of the book, leaving 7/8x remaining.

On Tuesday, he read 3/5 of the remaining pages, which means he read 3/5 * 7/8x 21/4.

So, the pages left after Tuesday are:

$$42 x - frac{7}{8}x - frac{3}{5} cdot frac{7}{8}x 42 frac{40}{40}x - frac{35}{40}x - frac{21}{40}x cdot frac{40}{40}x$$

Putting it together:

$$frac{40}{40}x - frac{35}{40}x - frac{21}{40}x 42 frac{40}{40}x - frac{56}{40}x 42 Rightarrow 42 frac{-16}{40}x 42 Rightarrow 42 frac{-4}{10}x 42 Rightarrow 42 frac{120}{10}x Rightarrow x 120$$

Method 2: Simplified Fraction Method

Another way to approach this problem is to use simpler fractions:

On Monday, George read 1/8 of the book, leaving 7/8 of the book.

On Tuesday, he read 3/5 of the remaining pages, which is 3/5 * 7/8x 21/4.

Let’s solve for x again:

$$42 x - frac{7}{8}x - frac{21}{40}x 42 frac{35}{40}x - frac{21}{40}x Rightarrow 42 frac{14}{40}x Rightarrow 42 frac{7}{20}x Rightarrow x frac{42 cdot 20}{7} 120$$

Method 3: Guess and Check with Fractions

A straightforward method is to guess the total pages and check if the remaining pages fit the conditions:

Let’s start with a simpler fraction, x 75. On Monday, George would read 1/8 of the book (9 pages), leaving 66 pages. On Tuesday, he would read 3/5 of 66 (39.6 pages, rounded to 40 pages), leaving 26 pages. This is not the solution, so let's try again with a larger number.

Let's use 120 as a check:

On Monday, he reads 1/8 * 120 15 pages, leaving 105 pages. On Tuesday, he reads 3/5 * 105 63 pages, leaving 42 pages.

This fits perfectly, so the total number of pages in the book is 120.

Alternative Approach: Direct Calculation

In the alternative approach, we can directly calculate the remaining pages using the fraction method:

Let's denote the pages as x. On Monday, he reads 1/8x pages, leaving 7/8x pages.

On Tuesday, he reads 3/5 * 7/8x 21/4, leaving 42 pages. So:

$$42 frac{40}{40}x - frac{35}{40}x - frac{21}{40}x Rightarrow 42 frac{4}{40}x Rightarrow 42 frac{1}{10}x Rightarrow x 420 / 10 120$$

Conclusion

From all the methods used, we can conclude that the total number of pages in the book is 120.

The book indeed has 120 pages. Let’s verify:

On Monday, he read 1/8 * 120 15 pages, leaving 105 pages.

On Tuesday, he read 3/5 * 105 63 pages, leaving 42 pages.

More Examples

For further practice, you can solve similar problems or create your own scenarios. Here are a couple of additional examples:

Example 1: If someone reads 1/3 of a book on the first day and 3/10 of the remaining pages on the second day, and there are 35 pages left, how many pages are in the book?

Example 2: If someone reads 1/8 of a book on the first day and 1/4 of the remaining pages on the second day, and there are 48 pages left, how many pages are in the book?

These problems can be solved by setting up and solving equations in a similar fashion.