The difference b/w S.I. % T.D. on a certain sum of money for 6 months at `12(1)/(2)%` per annum is Rs 25 find the sum due??

Options

A. Rs 7500
B. Rs 7000
C. Rs 6800
D. Rs 8500

Answer

To solve the problem step by step, we will find the sum due based on the given information about Simple Interest (SI) and True Discount (TD).

### Step 1: Understand the Problem
We are given that the difference between Simple Interest (SI) and True Discount (TD) on a certain sum of money for 6 months at an interest rate of 12.5% per annum is Rs 25. We need to find the sum due (let's denote it as \( x \)).

**Hint:** Identify the key terms: SI, TD, sum due, and the interest rate.

### Step 2: Convert the Interest Rate and Time
The interest rate is given as \( 12.5\% \) per annum, which can be expressed as \( \frac{25}{2}\% \). The time period of 6 months can be converted to years, which is \( \frac{1}{2} \) years.

**Hint:** Remember to convert the time period into years when dealing with annual interest rates.

### Step 3: Calculate True Discount (TD)
The formula for True Discount (TD) is given by:

\[ TD = \frac{x \cdot r \cdot t}{100 + r \cdot t} \]

Where:
- \( x \) = sum due
- \( r \) = rate of interest = \( \frac{25}{2} \)
- \( t \) = time in years = \( \frac{1}{2} \)

Substituting the values:

\[ TD = \frac{x \cdot \frac{25}{2} \cdot \frac{1}{2}}{100 + \frac{25}{2} \cdot \frac{1}{2}} = \frac{x \cdot \frac{25}{4}}{100 + \frac{25}{4}} = \frac{x \cdot \frac{25}{4}}{\frac{400 + 25}{4}} = \frac{25x}{425} = \frac{x}{17} \]

**Hint:** Make sure to simplify the fraction correctly when substituting values.

### Step 4: Calculate Simple Interest (SI)
The formula for Simple Interest (SI) is given by:

\[ SI = \frac{x \cdot r \cdot t}{100} \]

Substituting the values:

\[ SI = \frac{x \cdot \frac{25}{2} \cdot \frac{1}{2}}{100} = \frac{x \cdot \frac{25}{4}}{100} = \frac{25x}{400} = \frac{x}{16} \]

**Hint:** Ensure to follow the correct formula for SI and simplify it.

### Step 5: Set Up the Equation for the Difference
According to the problem, the difference between SI and TD is Rs 25:

\[ SI - TD = 25 \]

Substituting the expressions for SI and TD:

\[ \frac{x}{16} - \frac{x}{17} = 25 \]

**Hint:** Set up the equation carefully, ensuring you have the correct expressions for SI and TD.

### Step 6: Solve the Equation
To solve the equation, find a common denominator (which is 272):

\[ \frac{17x - 16x}{272} = 25 \]

This simplifies to:

\[ \frac{x}{272} = 25 \]

Multiplying both sides by 272 gives:

\[ x = 25 \times 272 = 6800 \]

**Hint:** When solving for \( x \), ensure you multiply correctly to isolate \( x \).

### Conclusion
The sum due is Rs 6800.

**Final Answer:** The sum due is Rs 6800.