The difference between the simple interest and discount on a certain sum of money due 1 year 9 months at `4%` is Rs. `7cdot35`. What is the sum?

Options

A. Rs. 1605
B. Rs. 1805
C. Rs. 1525
D. Rs. 1625

Answer

To solve the problem step by step, we need to find the sum of money (let's denote it as \( P \)) given the difference between Simple Interest (SI) and True Discount (TD) over a period of 1 year and 9 months at a rate of 4% is Rs. 7.35.

### Step 1: Understand the relationship between SI and TD
The difference between Simple Interest and True Discount can be represented as:
\[ \text{SI} - \text{TD} = 7.35 \]

### Step 2: Convert the time period into years
1 year and 9 months can be converted into years:
\[ 1 \text{ year} + \frac{9 \text{ months}}{12} = 1 + \frac{3}{4} = \frac{7}{4} \text{ years} \]

### Step 3: Calculate the Simple Interest (SI)
The formula for Simple Interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal (the sum we want to find)
- \( R \) = Rate of interest (4%)
- \( T \) = Time in years (\(\frac{7}{4}\))

Substituting the values:
\[ \text{SI} = \frac{P \times 4 \times \frac{7}{4}}{100} = \frac{P \times 7}{100} \]

### Step 4: Calculate the True Discount (TD)
The True Discount can be calculated as:
\[ \text{TD} = \frac{\text{SI}}{1 + \frac{R \times T}{100}} = \frac{\text{SI}}{1 + \frac{4 \times \frac{7}{4}}{100}} = \frac{\text{SI}}{1 + \frac{7}{100}} = \frac{\text{SI}}{\frac{107}{100}} = \text{SI} \times \frac{100}{107} \]

### Step 5: Set up the equation
Now we can express the difference in terms of \( P \):
\[ \text{SI} - \text{TD} = 7.35 \]
Substituting the expressions for SI and TD:
\[ \frac{P \times 7}{100} - \left(\frac{P \times 7}{100} \times \frac{100}{107}\right) = 7.35 \]

### Step 6: Simplify the equation
\[ \frac{P \times 7}{100} \left(1 - \frac{100}{107}\right) = 7.35 \]
Calculating \( 1 - \frac{100}{107} \):
\[ 1 - \frac{100}{107} = \frac{107 - 100}{107} = \frac{7}{107} \]
So the equation becomes:
\[ \frac{P \times 7}{100} \times \frac{7}{107} = 7.35 \]

### Step 7: Solve for \( P \)
Multiplying both sides by \( \frac{100 \times 107}{7} \):
\[ P \times 7 = 7.35 \times \frac{100 \times 107}{7} \]
\[ P = 7.35 \times \frac{100 \times 107}{7 \times 7} \]
Calculating the right side:
\[ P = 7.35 \times \frac{10700}{49} \]
Calculating \( 7.35 \times 218.3673469387755 \approx 1605 \)

Thus, the sum \( P \) is Rs. 1605.

### Final Answer
The sum is Rs. 1605.