Simple interest on a certain is `(16)/(25)` of the principal. If the number representing the rate of interest in per cent and time in years be equal, then time, for which the principal amount is lent out, is :

Options

A. `5(1)/(2)` years
B. `6(1)/(2)` years
C. 6 years
D. 8 years

Answer

To solve the problem, we need to find the time for which the principal amount is lent out, given that the simple interest is \( \frac{16}{25} \) of the principal and that the rate of interest (R) in percent and the time (T) in years are equal.

### Step-by-Step Solution:

1. **Understanding Simple Interest**:
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal amount
- \( R \) = Rate of interest (in percent)
- \( T \) = Time (in years)

2. **Given Information**:
We know that:
\[ SI = \frac{16}{25} P \]

3. **Equating Simple Interest**:
From the formula, we can substitute the value of SI:
\[ \frac{16}{25} P = \frac{P \times R \times T}{100} \]

4. **Canceling Principal**:
Since \( P \) is common on both sides, we can cancel it (assuming \( P \neq 0 \)):
\[ \frac{16}{25} = \frac{R \times T}{100} \]

5. **Relating Rate and Time**:
We are given that the rate of interest (R) and time (T) are equal. Let's denote this common value as \( x \):
\[ R = T = x \]

6. **Substituting R and T**:
Now, substituting \( R \) and \( T \) in the equation:
\[ \frac{16}{25} = \frac{x \times x}{100} \]
This simplifies to:
\[ \frac{16}{25} = \frac{x^2}{100} \]

7. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 16 \times 100 = 25 \times x^2 \]
Simplifying this:
\[ 1600 = 25x^2 \]

8. **Dividing by 25**:
Dividing both sides by 25:
\[ x^2 = \frac{1600}{25} \]
\[ x^2 = 64 \]

9. **Taking the Square Root**:
Taking the square root of both sides:
\[ x = 8 \]

10. **Conclusion**:
Since \( x \) represents both the rate of interest and the time in years, we conclude that:
\[ \text{Time (T)} = 8 \text{ years} \]

### Final Answer:
The time for which the principal amount is lent out is **8 years**.