The true discount on a certain sum of money due after 2 `1/2` years at `6%` per annum is less than the simple interest on the same sum for the same time by Rs. 81. Find the sum-

Options

A. Rs. 4140
B. Rs. 4240
C. Rs. 4150
D. Rs. 4250

Answer

To solve the problem step by step, we need to find the sum of money based on the given conditions about true discount and simple interest. Let's break it down:

### Step 1: Understand the Terms
- **True Discount (TD)**: The difference between the present value (PV) and the future value (FV) of a sum of money due at a certain time.
- **Simple Interest (SI)**: The interest calculated on the principal amount over a period of time.

### Step 2: Set Up the Given Information
- Time (T) = 2.5 years
- Rate (R) = 6% per annum
- The difference between Simple Interest and True Discount = Rs. 81

### Step 3: Define the Variables
Let the sum of money (Principal) be \( P \).

### Step 4: Calculate Simple Interest
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI = \frac{P \times 6 \times 2.5}{100} = \frac{15P}{100} = 0.15P \]

### Step 5: Calculate True Discount
The True Discount can be calculated using the formula:
\[ TD = \frac{SI \times R \times T}{100 + R \times T} \]
Where \( R \times T = 6 \times 2.5 = 15 \).

So,
\[ TD = \frac{SI \times 15}{100 + 15} = \frac{SI \times 15}{115} \]
Substituting \( SI = 0.15P \):
\[ TD = \frac{0.15P \times 15}{115} = \frac{2.25P}{115} \]

### Step 6: Set Up the Equation
According to the problem, the difference between SI and TD is Rs. 81:
\[ SI - TD = 81 \]
Substituting the values:
\[ 0.15P - \frac{2.25P}{115} = 81 \]

### Step 7: Simplify the Equation
To solve for \( P \), we need a common denominator:
\[ 0.15P = \frac{15P}{100} = \frac{15P \times 115}{11500} = \frac{1725P}{11500} \]
Now the equation becomes:
\[ \frac{1725P}{11500} - \frac{2.25P}{115} = 81 \]
Convert \( \frac{2.25P}{115} \) to have the same denominator:
\[ \frac{2.25P \times 100}{11500} = \frac{225P}{11500} \]
Now the equation is:
\[ \frac{1725P - 225P}{11500} = 81 \]
\[ \frac{1500P}{11500} = 81 \]

### Step 8: Solve for \( P \)
Cross-multiplying gives:
\[ 1500P = 81 \times 11500 \]
Calculating the right side:
\[ 1500P = 931500 \]
Now divide both sides by 1500:
\[ P = \frac{931500}{1500} = 621 \]

### Final Answer
The sum of money is Rs. 621.

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