Vandana bought a watch for X 600 and sold it the same day for X 688.50 at a credit of 9 months and this way she gained 2%. Find the rate of interest per annum.

Options

A. `16(2)/(3)%`
B. `15(2)/(3)%`
C. `11(2)/(3)%`
D. `5(2)/(3)%`

Answer

To solve the problem step by step, we will follow the given information and apply the formulas related to true discount and interest rates.

### Step 1: Identify the Cost Price and Selling Price
Vandana bought the watch for ₹600 and sold it for ₹688.50.

### Step 2: Calculate the Gain
The gain can be calculated as:
\[ \text{Gain} = \text{Selling Price} - \text{Cost Price} = 688.50 - 600 = ₹88.50 \]

### Step 3: Calculate the Gain Percentage
The gain percentage is given as 2%. We can express this as:
\[ \text{Gain Percentage} = \frac{\text{Gain}}{\text{Cost Price}} \times 100 \]
From the problem, we know:
\[ 2 = \frac{88.50}{600} \times 100 \]
This confirms that the gain is indeed 2%.

### Step 4: Calculate the Present Worth
Since Vandana sold the watch on credit for 9 months, we need to find the present worth (PW) of the selling price. The present worth can be calculated as:
\[ \text{Present Worth} = \text{Selling Price} - \text{Gain} = 688.50 - 88.50 = ₹600 \]

### Step 5: Calculate the True Discount
The true discount (TD) is the difference between the present worth and the cost price:
\[ \text{True Discount} = \text{Selling Price} - \text{Present Worth} = 688.50 - 600 = ₹88.50 \]

### Step 6: Set Up the True Discount Formula
The formula for true discount is given by:
\[ TD = \frac{\text{Present Worth} \times R \times T}{100} \]
Where:
- TD = True Discount
- Present Worth = ₹600
- R = Rate of interest per annum (unknown)
- T = Time in years = \( \frac{9}{12} = \frac{3}{4} \) years

Substituting the known values into the formula:
\[ 88.50 = \frac{600 \times R \times \frac{3}{4}}{100} \]

### Step 7: Solve for R
Rearranging the equation to solve for R:
\[ 88.50 \times 100 = 600 \times R \times \frac{3}{4} \]
\[ 8850 = 600 \times R \times \frac{3}{4} \]
\[ 8850 = 450R \]
\[ R = \frac{8850}{450} = 19.67\% \]

### Final Answer
The rate of interest per annum is approximately **19.67%**.

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