A sum was lent at simple interest at a certain rate for 2 years. Had it been lent at 3% higher rate, it would have fetched ₹ 300 more. The original sum of money was—

Options

A. 4000
B. 5000
C. 6000
D. 7000

Answer

To solve the problem step by step, we will use the formula for simple interest and set up equations based on the information given.

### Step-by-Step Solution:

1. **Understand the Problem**:
We need to find the original sum of money (let's denote it as \( P \)) that was lent at a certain rate for 2 years. If it had been lent at a rate 3% higher, it would have earned ₹300 more in interest.

2. **Write the Formula for Simple Interest**:
The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

3. **Calculate Simple Interest for the Original Rate**:
For the original rate \( R \) over 2 years, the simple interest is:
\[ SI_1 = \frac{P \times R \times 2}{100} \]

4. **Calculate Simple Interest for the Higher Rate**:
If the rate were 3% higher, the new rate would be \( R + 3 \). The simple interest for this case over 2 years would be:
\[ SI_2 = \frac{P \times (R + 3) \times 2}{100} \]

5. **Set Up the Equation**:
According to the problem, the difference in interest earned between the two scenarios is ₹300:
\[ SI_2 - SI_1 = 300 \]
Substituting the expressions for \( SI_1 \) and \( SI_2 \):
\[ \frac{P \times (R + 3) \times 2}{100} - \frac{P \times R \times 2}{100} = 300 \]

6. **Simplify the Equation**:
Combining the terms:
\[ \frac{2P(R + 3) - 2PR}{100} = 300 \]
This simplifies to:
\[ \frac{2P \times 3}{100} = 300 \]

7. **Solve for \( P \)**:
Multiply both sides by 100:
\[ 2P \times 3 = 30000 \]
Divide both sides by 6:
\[ P = \frac{30000}{6} = 5000 \]

8. **Conclusion**:
The original sum of money lent was ₹5000.