Let's break down the problem step by step:

1. Charis reads $\frac{3}{5}$ of the book on the first day.
2. The fraction of the book remaining after the first day is: $1 - \frac{3}{5} = \frac{2}{5}$
3. The next day, she reads $\frac{3}{4}$ of the remaining $\frac{2}{5}$ of the book: $\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$
4. In total, the fraction of the book she has read is the sum of what she read on both days: $\frac{3}{5} + \frac{3}{10}$ To add these fractions, we need a common denominator. The least common denominator between 5 and 10 is 10: $\frac{3}{5} = \frac{6}{10}$ So: $\frac{6}{10} + \frac{3}{10} = \frac{9}{10}$ Thus, Charis has read $\frac{9}{10}$ of the book in total.

Would you like further details or clarifications on any part of this solution?

Here are some related questions you can explore:

1. What fraction of the book remains unread after both days?
2. How much of the book would Charis have left if she read $\frac{1}{2}$ of the rest on the second day instead?
3. What if Charis read $\frac{2}{3}$ of the rest instead of $\frac{3}{4}$?
4. How would the total fraction change if Charis read $\frac{4}{5}$ on the first day instead?
5. What if Charis reads the remaining fraction of the book on the third day?

Tip: Always convert fractions to a common denominator when adding or subtracting them.