If the annual rate of simple interest increases from 8% to `12(1)/(2)%` a person's yearly income increased by Rs. 639. The principal amount (in Rs.) is :

Options

A. 14, 200
B. 8200
C. 12500
D. 11250

Answer

To solve the problem, we need to find the principal amount based on the change in the annual rate of simple interest and the corresponding increase in yearly income.

### Step 1: Identify the rates of interest
The initial rate of interest (R1) is 8%, and the new rate of interest (R2) is 12.5% (which is equivalent to \(12 \frac{1}{2}\%\)).

### Step 2: Calculate the difference in interest rates
The difference in interest rates can be calculated as follows:
\[ R2 - R1 = 12.5\% - 8\% = 4.5\% \]

### Step 3: Use the formula for Simple Interest
The formula for simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(P\) = Principal amount
- \(R\) = Rate of interest
- \(T\) = Time (in years)

### Step 4: Calculate the increase in interest
The increase in yearly income due to the increase in interest rate is given as Rs. 639. This increase can be expressed as:
\[ \text{Increase in SI} = \frac{P \times (R2 - R1) \times 1}{100} \]
Substituting the values we have:
\[ 639 = \frac{P \times 4.5}{100} \]

### Step 5: Solve for Principal (P)
To find \(P\), rearrange the equation:
\[ P = \frac{639 \times 100}{4.5} \]
Calculating this gives:
\[ P = \frac{63900}{4.5} = 14200 \]

### Conclusion
The principal amount is Rs. 14,200.

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