If the rate of interest rises from `6 1/2` to `8%` a man's annual income increases by Rs. 4050. Find the capital.

Options

A. Rs. 270000
B. Rs. 370000
C. Rs. 300000
D. None of these

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Convert the mixed fraction to an improper fraction
The rate of interest rises from \(6 \frac{1}{2}\%\) to \(8\%\). First, we need to convert \(6 \frac{1}{2}\%\) into a decimal or improper fraction.

\[ 6 \frac{1}{2} = 6 + \frac{1}{2} = \frac{12}{2} + \frac{1}{2} = \frac{13}{2}\% \]

### Step 2: Calculate the increase in interest rate
Now, we find the increase in the interest rate:

\[ \text{Increase in interest rate} = 8\% - 6 \frac{1}{2}\% = 8\% - \frac{13}{2}\% = \frac{16}{2}\% - \frac{13}{2}\% = \frac{3}{2}\% \]

### Step 3: Convert the percentage increase to a decimal
Next, we convert \(\frac{3}{2}\%\) into a decimal for easier calculations:

\[ \frac{3}{2}\% = \frac{3}{2} \times \frac{1}{100} = \frac{3}{200} = 0.015 \]

### Step 4: Relate the increase in income to the capital
The increase in annual income due to the rise in interest rate is given as Rs. 4050. We can express this relationship using the formula for simple interest:

\[ \text{Increase in income} = \text{Capital} \times \text{Increase in interest rate} \]

Let \(C\) be the capital. Then, we have:

\[ 4050 = C \times 0.015 \]

### Step 5: Solve for capital
Now, we can solve for \(C\):

\[ C = \frac{4050}{0.015} \]

Calculating this gives:

\[ C = 4050 \div 0.015 = 270000 \]

### Final Answer
Thus, the capital is Rs. 270,000.

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