Find the total number of pages in the book:
Marie has finished \( \frac{3}{4} \) of a book. She found that she has read 106 more pages than the number of pages left to read. What is the total number of pages in the book?
Solution:
Let \(x\) be the total number of pages in the book.
Step 1: Define what we know
Marie finished \( \frac{3}{4} \)of the book, which means she has read \( \frac{3}{4}x \)pages.
She has \(106\) more pages left to read, which means she has \(x – \frac{3}{4}x = \frac{1}{4}x \) pages remaining.
According to the problem, the number of pages she has read is 106 more than the number of pages left to read. Therefore, we can write the equation:
\[
\frac{3}{4}x = \frac{1}{4}x + 106
\]
Step 2: Solve the equation
First, subtract \( \frac{1}{4}x \)from both sides:
\[
\frac{3}{4}x – \frac{1}{4}x = 106
\]
\[
\Rightarrow \frac{2}{4}x = 106
\]
\[
\Rightarrow \frac{1}{2}x = 106
\]
Now, multiply both sides by \(2\):
\Rightarrow x = 212 \text{ pages}
\]
Step 3: Verify the result
Total number of pages is \(212\).
Pages read: \( \frac{3}{4} * 212 = 159 \)
Pages left: \( \frac{1}{4} * 212 = 53 \)
The difference: \( 159 – 53 =106 \), which matches the given information.
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