A person lent some amount to his friend at 20% SI after 2 years, Rs 5000 was repaid and the rest amount was repaid 25% per annum. If 3rd year's Interest is 6/11 of first 2 years interest. Find the amount of money that was let out initially ?

Options

A. Rs 38000
B. Rs 38285
C. Rs 39000
D. Rs 39285.71

Answer

To solve the problem step by step, we need to break down the information given and use the formula for Simple Interest (SI).

### Step 1: Define Variables
Let the principal amount (the amount lent out initially) be \( P \).

### Step 2: Calculate Interest for the First Two Years
The rate of interest is 20% per annum. The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) = Simple Interest
- \( P \) = Principal
- \( R \) = Rate of interest
- \( T \) = Time in years

For the first two years:
\[ SI_{2 \text{ years}} = \frac{P \times 20 \times 2}{100} = \frac{40P}{100} = 0.4P \]

### Step 3: Total Amount After 2 Years
The total amount after 2 years will be:
\[ A = P + SI_{2 \text{ years}} = P + 0.4P = 1.4P \]

### Step 4: Amount Repaid After 2 Years
According to the question, Rs 5000 was repaid after 2 years. Therefore, the remaining amount is:
\[ \text{Remaining Amount} = 1.4P - 5000 \]

### Step 5: Interest for the Third Year
The remaining amount is repaid at 25% per annum. The interest for the third year on the remaining amount is:
\[ SI_{3 \text{rd year}} = \frac{(1.4P - 5000) \times 25 \times 1}{100} = \frac{25(1.4P - 5000)}{100} = 0.25(1.4P - 5000) \]

### Step 6: Relationship Between Interests
According to the problem, the interest for the third year is \( \frac{6}{11} \) of the first year's interest. The first year's interest is:
\[ SI_{1 \text{st year}} = \frac{P \times 20 \times 1}{100} = 0.2P \]
Thus, we have:
\[ SI_{3 \text{rd year}} = \frac{6}{11} \times SI_{1 \text{st year}} = \frac{6}{11} \times 0.2P = \frac{1.2P}{11} \]

### Step 7: Set Up the Equation
Now we can set up the equation:
\[ 0.25(1.4P - 5000) = \frac{1.2P}{11} \]

### Step 8: Solve the Equation
Multiply both sides by 11 to eliminate the fraction:
\[ 11 \times 0.25(1.4P - 5000) = 1.2P \]
\[ 2.75(1.4P - 5000) = 1.2P \]
Distributing the left side:
\[ 3.85P - 13750 = 1.2P \]
Now, rearranging gives:
\[ 3.85P - 1.2P = 13750 \]
\[ 2.65P = 13750 \]
Now, divide both sides by 2.65:
\[ P = \frac{13750}{2.65} \approx 5170.19 \]

### Step 9: Final Answer
Thus, the amount of money that was lent out initially is approximately Rs 5170.19.