Rahim lent a sum to Anil at a simple interest of 10% per annum, and at the same time lent to Sunil a sum, which was Rs. 2000 more than the amount lent to Anil at the simple interest of 12 % per annum. At the end of 3 years, both Anil and Sunil returned the principal and interest to Rahim. If Rahim got Rs. 1020 more interest from Sunil than that from Anil, how much did each borrow from Rahim ?

Options

A. Rs. 2500, Rs. 4500
B. Rs. 4000, Rs. 6000
C. Rs. 5000, Rs. 7000
D. Rs. 6000, Rs. 8000

Answer

To solve the problem, we will denote the amount lent to Anil as \( P \). Therefore, the amount lent to Sunil will be \( P + 2000 \).

### Step 1: Calculate the interest for Anil
The formula for simple interest is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

For Anil:
- Principal = \( P \)
- Rate = 10%
- Time = 3 years

So, the interest from Anil will be:
\[ \text{SI}_{\text{Anil}} = \frac{P \times 10 \times 3}{100} = \frac{30P}{100} = 0.3P \]

### Step 2: Calculate the interest for Sunil
For Sunil:
- Principal = \( P + 2000 \)
- Rate = 12%
- Time = 3 years

So, the interest from Sunil will be:
\[ \text{SI}_{\text{Sunil}} = \frac{(P + 2000) \times 12 \times 3}{100} = \frac{36(P + 2000)}{100} = 0.36P + 720 \]

### Step 3: Set up the equation based on the difference in interest
According to the problem, Rahim got Rs. 1020 more interest from Sunil than from Anil:
\[ \text{SI}_{\text{Sunil}} - \text{SI}_{\text{Anil}} = 1020 \]
Substituting the values we calculated:
\[ (0.36P + 720) - (0.3P) = 1020 \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 0.36P + 720 - 0.3P = 1020 \]
\[ 0.06P + 720 = 1020 \]
Subtract 720 from both sides:
\[ 0.06P = 1020 - 720 \]
\[ 0.06P = 300 \]

### Step 5: Solve for \( P \)
Now, divide both sides by 0.06:
\[ P = \frac{300}{0.06} = 5000 \]

### Step 6: Find the amount lent to Sunil
Since the amount lent to Sunil is \( P + 2000 \):
\[ \text{Amount lent to Sunil} = 5000 + 2000 = 7000 \]

### Final Answer
- Amount lent to Anil = Rs. 5000
- Amount lent to Sunil = Rs. 7000