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 th 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 establish the relationship between the price of the books, the number of pages, and the time elapsed since their purchase.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The price \( P \) of a book varies directly as the number of pages \( N \) and inversely as the time period \( T \) in years since the book was purchased. This can be expressed mathematically as:
\[ P = k \frac{N}{T} \]
where \( k \) is a constant of proportionality.

2. **Setting Up the Equations**:
For the first book:
- Let \( N_1 \) be the number of pages in the first book.
- Let \( T_1 \) be the time in years since the first book was sold (which is 18 years).
- The price equation for the first book is:
\[ P = k \frac{N_1}{T_1} \]

For the second book:
- Let \( N_2 \) be the number of pages in the second book.
- Let \( T_2 \) be the time in years since the second book was sold (which we need to find).
- The price equation for the second book is:
\[ P = k \frac{N_2}{T_2} \]

3. **Given Information**:
- The number of pages in the first book is triple that of the second book:
\[ N_1 = 3N_2 \]
- The first book was sold 18 years ago:
\[ T_1 = 18 \]

4. **Equating the Prices**:
Since both books cost the same, we can set the two price equations equal to each other:
\[ k \frac{N_1}{T_1} = k \frac{N_2}{T_2} \]
We can cancel \( k \) from both sides (assuming \( k \neq 0 \)):
\[ \frac{N_1}{T_1} = \frac{N_2}{T_2} \]

5. **Substituting Known Values**:
Substitute \( N_1 = 3N_2 \) and \( T_1 = 18 \) into the equation:
\[ \frac{3N_2}{18} = \frac{N_2}{T_2} \]

6. **Cross Multiplying**:
Cross-multiply to solve for \( T_2 \):
\[ 3N_2 \cdot T_2 = 18 \cdot N_2 \]
Assuming \( N_2 \neq 0 \), we can divide both sides by \( N_2 \):
\[ 3T_2 = 18 \]

7. **Solving for \( T_2 \)**:
Divide both sides by 3:
\[ T_2 = \frac{18}{3} = 6 \]

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