The simple interest at `R%` p.a on a certain principal P is `((9)/(16))^(th)` of it .
R is equal to the number of years (N) ,find the value of N.

Options

A. `8.5`
B. 7
C. `7.5`
D. 6

Answer

To solve the problem, we need to find the value of \( N \) given that the simple interest at \( R\% \) per annum on a principal \( P \) is \( \frac{9}{16} \) of \( P \), and that \( R \) is equal to the number of years \( N \).

### Step-by-Step Solution:

1. **Understand the Simple Interest Formula**:
The formula for simple interest (SI) is given by:
\[ \text{SI} = \frac{P \times R \times N}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest per annum, and \( N \) is the number of years.

2. **Set Up the Equation**:
According to the problem, the simple interest is \( \frac{9}{16} P \). Therefore, we can set up the equation:
\[ \frac{P \times R \times N}{100} = \frac{9}{16} P \]

3. **Cancel \( P \)**:
Since \( P \) is common on both sides of the equation and \( P \neq 0 \), we can cancel \( P \):
\[ \frac{R \times N}{100} = \frac{9}{16} \]

4. **Substitute \( R \) with \( N \)**:
We know from the problem that \( R = N \). Substituting \( R \) with \( N \) gives us:
\[ \frac{N \times N}{100} = \frac{9}{16} \]
This simplifies to:
\[ \frac{N^2}{100} = \frac{9}{16} \]

5. **Cross-Multiply to Solve for \( N^2 \)**:
Cross-multiplying gives:
\[ 16N^2 = 900 \]

6. **Divide by 16**:
Now, divide both sides by 16:
\[ N^2 = \frac{900}{16} \]

7. **Simplify \( N^2 \)**:
Simplifying \( \frac{900}{16} \) gives:
\[ N^2 = 56.25 \]

8. **Take the Square Root**:
To find \( N \), take the square root of both sides:
\[ N = \sqrt{56.25} = 7.5 \]

### Final Answer:
The value of \( N \) is \( 7.5 \).