If the annual rate of simple interest increased from 8% to 12 1/2%, a person's yearly income increased by Rs 459. The principal amount (in Rs) is:

Options

A. 8200
B. 12500
C. 10200
D. 11.25

Answer

To find the principal amount given the increase in yearly income due to a change in the interest rate, we can follow these steps:

### Step 1: Understand the problem
We know that the annual rate of simple interest increased from 8% to 12.5%. The increase in income due to this change is Rs 459. We need to find the principal amount (P).

### Step 2: Set up the equation for the increase in interest
The increase in interest can be calculated using the formula for simple interest, which is:
\[ \text{Interest} = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time (in years)

The increase in interest due to the change in rate is:
\[ \text{Increase in Interest} = \text{Interest at 12.5%} - \text{Interest at 8%} \]

### Step 3: Write the equations for both rates
1. Interest at 12.5%:
\[ I_1 = \frac{P \times 12.5 \times 1}{100} = \frac{12.5P}{100} \]

2. Interest at 8%:
\[ I_2 = \frac{P \times 8 \times 1}{100} = \frac{8P}{100} \]

### Step 4: Find the increase in interest
The increase in interest is:
\[ \text{Increase} = I_1 - I_2 \]
Substituting the values we found:
\[ \text{Increase} = \frac{12.5P}{100} - \frac{8P}{100} \]
\[ \text{Increase} = \frac{(12.5 - 8)P}{100} = \frac{4.5P}{100} \]

### Step 5: Set the increase equal to Rs 459
Now, we set the increase equal to Rs 459:
\[ \frac{4.5P}{100} = 459 \]

### Step 6: Solve for P
To solve for P, multiply both sides by 100:
\[ 4.5P = 45900 \]
Now, divide both sides by 4.5:
\[ P = \frac{45900}{4.5} \]
Calculating this gives:
\[ P = 10200 \]

### Conclusion
The principal amount is Rs 10,200.