A man got Rs. 130 less, as simple interest, when he invested Rs. 2000 for 4 years as compared to investing Rs. 2250 for same duration. What is the rate of interest?

Options

A. a. 12%
B. b. 13%
C. c. 12.5%
D. d. 10.50%

Answer

To find the rate of interest, we can follow these steps:

### Step 1: Define the variables
Let:
- \( P_1 = 2000 \) (the first principal amount)
- \( P_2 = 2250 \) (the second principal amount)
- \( T = 4 \) years (the time period for both investments)
- \( R \) = rate of interest (which we need to find)
- \( S.I_1 \) = simple interest for \( P_1 \)
- \( S.I_2 \) = simple interest for \( P_2 \)

### Step 2: Write the relationship between the simple interests
According to the problem, the simple interest on the first investment is Rs. 130 less than that of the second investment:
\[ S.I_1 = S.I_2 - 130 \]

### Step 3: Write the formula for simple interest
The formula for simple interest is:
\[ S.I = \frac{P \times R \times T}{100} \]
Using this formula, we can express \( S.I_1 \) and \( S.I_2 \):
\[ S.I_1 = \frac{P_1 \times R \times T}{100} = \frac{2000 \times R \times 4}{100} = \frac{8000R}{100} = 80R \]
\[ S.I_2 = \frac{P_2 \times R \times T}{100} = \frac{2250 \times R \times 4}{100} = \frac{9000R}{100} = 90R \]

### Step 4: Substitute the expressions into the relationship
Now, substituting the expressions for \( S.I_1 \) and \( S.I_2 \) into the relationship we wrote in Step 2:
\[ 80R = 90R - 130 \]

### Step 5: Solve for \( R \)
Rearranging the equation:
\[ 80R - 90R = -130 \]
\[ -10R = -130 \]
Dividing both sides by -10:
\[ R = \frac{130}{10} = 13 \]

### Conclusion
The rate of interest \( R \) is \( 13\% \).

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