What rate of interest is obtained from investing in (12)3/4 percent stock when the price is at a premium of 2 percent?

Options

A. `25%`
B. `(8)1//3%`
C. `(12)1/2%`
D. `(11)2/3%`

Answer

To find the rate of interest obtained from investing in a \( 12 \frac{3}{4} \% \) stock when the price is at a premium of 2%, we can follow these steps:

### Step 1: Convert the mixed fraction to an improper fraction
The stock is \( 12 \frac{3}{4} \% \). We first convert this to an improper fraction:
\[ 12 \frac{3}{4} = 12 + \frac{3}{4} = \frac{48}{4} + \frac{3}{4} = \frac{51}{4} \]

### Step 2: Determine the market value of the stock
The stock is at a premium of 2%. The face value of the stock is typically considered to be 100. Therefore, the market value is:
\[ \text{Market Value} = 100 + 2\% \text{ of } 100 = 100 + 2 = 102 \]

### Step 3: Calculate the rate of interest
The rate of interest can be calculated using the formula:
\[ \text{Rate of Interest} = \left( \frac{\text{Dividend}}{\text{Market Value}} \right) \times 100 \]
Substituting the values we have:
\[ \text{Rate of Interest} = \left( \frac{\frac{51}{4}}{102} \right) \times 100 \]

### Step 4: Simplify the expression
Now, we simplify the expression:
\[ \text{Rate of Interest} = \frac{51}{4} \times \frac{100}{102} \]
This can be simplified further:
\[ = \frac{51 \times 100}{4 \times 102} \]
Calculating \( 4 \times 102 = 408 \):
\[ = \frac{5100}{408} \]

### Step 5: Reduce the fraction
Now we can simplify \( \frac{5100}{408} \):
\[ = \frac{5100 \div 102}{408 \div 102} = \frac{50}{4} = 12.5 \]

### Step 6: Convert to percentage
Thus, the rate of interest is:
\[ 12.5\% \]

### Final Answer
The rate of interest obtained from investing in \( 12 \frac{3}{4} \% \) stock when the price is at a premium of 2% is \( 12 \frac{1}{2} \% \).

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