The price of a book varies directly as the no. of pages in it and inversely as the time periods in years that have elapsed since the date of purchasing. Two books cost the same, however, the no. of pages in the first book is triple of the second book. If the first book is sold on 18 years ago, how long ago was the second book sold?

Options

A. 54 years
B. 9 years
C. 6 years
D. 3 years

Answer

To solve the problem, we need to understand the relationship between the price of the books, the number of pages, and the time elapsed since their purchase.

1. **Define Variables**:
- Let \( P_1 \) be the price of the first book.
- Let \( P_2 \) be the price of the second book.
- Let \( n_1 \) be the number of pages in the first book.
- Let \( n_2 \) be the number of pages in the second book.
- Let \( t_1 \) be the time since the first book was sold (18 years).
- Let \( t_2 \) be the time since the second book was sold (unknown).

2. **Establish Relationships**:
- According to the problem, the price of a book varies directly as the number of pages and inversely as the time since it was purchased. This can be expressed as:
\[ P \propto \frac{n}{t} \]
- Therefore, we can write:
\[ P_1 = k \cdot \frac{n_1}{t_1} \quad \text{and} \quad P_2 = k \cdot \frac{n_2}{t_2} \]
where \( k \) is a constant of proportionality.

3. **Given Information**:
- The first book has three times the number of pages as the second book:
\[ n_1 = 3n_2 \]
- The first book was sold 18 years ago:
\[ t_1 = 18 \]

4. **Set Up the Equation**:
- Since both books cost the same, we can set the prices equal to each other:
\[ P_1 = P_2 \]
- Substituting the expressions for \( P_1 \) and \( P_2 \):
\[ k \cdot \frac{n_1}{t_1} = k \cdot \frac{n_2}{t_2} \]
- The \( k \) cancels out (assuming \( k \neq 0 \)):
\[ \frac{n_1}{t_1} = \frac{n_2}{t_2} \]

5. **Substituting Known Values**:
- Substitute \( n_1 = 3n_2 \) and \( t_1 = 18 \):
\[ \frac{3n_2}{18} = \frac{n_2}{t_2} \]

6. **Cross Multiply**:
- Cross multiplying gives:
\[ 3n_2 \cdot t_2 = 18n_2 \]
- Dividing both sides by \( n_2 \) (assuming \( n_2 \neq 0 \)):
\[ 3t_2 = 18 \]

7. **Solve for \( t_2 \)**:
- Dividing both sides by 3:
\[ t_2 = 6 \]

8. **Conclusion**:
- The second book was sold 6 years ago.

### Summary of Steps:
1. Define variables for prices, pages, and time.
2. Establish the relationship between price, pages, and time.
3. Use the given information to set up equations.
4. Substitute known values into the equations.
5. Cross multiply to eliminate fractions.
6. Solve for the unknown time.