Book Lines Calculation

The problem describes a book with a certain number of pages and lines per page. When the number of lines per page changes, the number of pages changes, but the total amount of text (which means the total number of lines in the book) remains the same.

Let's represent the given information:

• Original number of pages = 40
• Original number of lines per page = $x$

The total number of lines in the original book is the product of the number of pages and the number of lines per page.

Total lines (Original) = Original pages $\times$ Original lines per page

Total lines (Original) = $40 \times x = 40x$

Now consider the modified scenario:

• Number of lines per page is reduced by 2, so new lines per page = $x - 2$
• Number of pages increases by 10, so new number of pages = $40 + 10 = 50$

The total number of lines in the modified book is the product of the new number of pages and the new number of lines per page.

Total lines (Modified) = New pages $\times$ New lines per page

Total lines (Modified) = $50 \times (x - 2)$

Identical Text Implies Equal Total Lines

The problem states that the text is identical in both scenarios. This means the total number of lines must be the same in the original book and the modified book.

Total lines (Original) = Total lines (Modified)

$40x = 50(x - 2)$

Solving for the Value of x

Now, we need to solve the equation $40x = 50(x - 2)$ for the value of $x$.

First, distribute the 50 on the right side of the equation:

$40x = 50 \times x - 50 \times 2$

$40x = 50x - 100$

Next, collect the terms with $x$ on one side and the constant term on the other side. Subtract $40x$ from both sides:

$0 = 50x - 40x - 100$

$0 = 10x - 100$

Add 100 to both sides:

$100 = 10x$

Finally, divide both sides by 10 to find the value of $x$:

$x = \frac{100}{10}$

$x = 10$

The value of $x$, which is the original number of lines per page, is 10.

Verifying the Solution

Let's check if $x=10$ works with the problem description:

• Original book: 40 pages, 10 lines/page. Total lines = $40 \times 10 = 400$
• Modified book: (10-2) = 8 lines/page, 50 pages. Total lines = $50 \times 8 = 400$

Since the total lines are equal (400 = 400), our value for $x$ is correct.