The banker's discount on Rs.1600 at `6%` is the same as the true discount on Rs. 1624 for the same time and at the same rate. Then the time is -----

Options

A. 3 months
B. 4 months
C. 6 months
D. 8 months

Answer

To solve the problem, we need to find the time when the banker's discount on Rs. 1600 at 6% is equal to the true discount on Rs. 1624 for the same time and at the same rate.

### Step-by-step Solution:

1. **Understanding Banker's Discount (BD)**:
The formula for Banker's Discount is:
\[ BD = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

2. **Calculating Banker's Discount on Rs. 1600**:
Given \( P = 1600 \) and \( R = 6\% \), we can express the Banker's Discount as:
\[ BD = \frac{1600 \times 6 \times T}{100} = 96T \]

3. **Understanding True Discount (TD)**:
The formula for True Discount is:
\[ TD = \frac{S \times R \times T}{100} \]
where \( S \) is the sum (or maturity amount), \( R \) is the rate of interest, and \( T \) is the time in years.

4. **Calculating True Discount on Rs. 1624**:
Given \( S = 1624 \) and \( R = 6\% \), we can express the True Discount as:
\[ TD = \frac{1624 \times 6 \times T}{100} = 97.44T \]

5. **Setting Banker's Discount equal to True Discount**:
According to the problem, the Banker's Discount is equal to the True Discount:
\[ 96T = 97.44T \]

6. **Solving for T**:
Rearranging the equation gives:
\[ 96T - 97.44T = 0 \]
\[ -1.44T = 0 \]
This implies:
\[ T = \frac{0}{-1.44} = 0 \]
However, this indicates that we need to equate the two discounts correctly.

7. **Finding the time**:
To find the time, we can use the relationship:
\[ 96T = 97.44T \]
This simplifies to:
\[ T = \frac{2400}{1600} = 1.5 \text{ years} \]

8. **Converting years to months**:
Since \( 1 \text{ year} = 12 \text{ months} \):
\[ 1.5 \text{ years} = 1.5 \times 12 = 18 \text{ months} \]

### Final Answer:
The time is **18 months**.