A boy read $\frac{3}{8}$th of a book on one day and $\frac{4}{5}$th of the remainder on another day. If there were 30 pages unread, how many pages did the book contain?

### Concept & Logic This problem utilizes sequential fraction reduction. Unlike standard fraction addition, the second fraction applies only to the "remainder" of the first operation. You must calculate the remaining fraction after the first day, apply the second fraction to that specific remainder, and then equate the final unread fraction to the absolute number of pages left. ### Step-by-Step Solution * **Given:** First day reading = $\frac{3}{8}$ of total. Second day reading = $\frac{4}{5}$ of the remainder. Unread pages = 30. Let total pages be $x$. * **Calculation:** Determine the fraction of the book remaining after the first day: $$ \text{Remainder after Day 1} = 1 - \frac{3}{8} = \frac{5}{8} $$ * Calculate the fraction of the *total* book read on the second day: $$ \text{Read on Day 2} = \frac{4}{5} \times \frac{5}{8} = \frac{4}{8} = \frac{1}{2} $$ * Determine the total fraction of the book read over both days: $$ \text{Total Read} = \frac{3}{8} + \frac{1}{2} = \frac{3}{8} + \frac{4}{8} = \frac{7}{8} $$ * Find the fraction representing the unread portion: $$ \text{Unread Fraction} = 1 - \frac{7}{8} = \frac{1}{8} $$ * Equate the unread fraction to the given number of unread pages: $$ \frac{1}{8}x = 30 $$ $$ x = 30 \times 8 = 240 $$ ### Exam Strategy & Shortcut Use the **Successive Remaining Fraction Multiplier**. If he reads $\frac{3}{8}$, the unread portion is $\frac{5}{8}$. If he then reads $\frac{4}{5}$ of *that*, the new unread portion is $\frac{1}{5}$ of it. So, the final unread fraction of the whole book is simply the product of the unread ratios: $\frac{5}{8} \times \frac{1}{5} = \frac{1}{8}$. Since $\frac{1}{8}$ of the book is 30 pages, the whole book is $30 \times 8 = 240$ pages. This single-line multiplication takes only seconds. ### Common Pitfall A very common mistake is simply adding $\frac{3}{8}$ and $\frac{4}{5}$ together, completely ignoring the crucial phrase "of the remainder". This leads to an impossible fraction larger than 1. ### Final Answer Therefore, the correct answer is **240**.