Goods were bought for Rs. 600 and sold the same day for Rs. `650cdot25` at a credit of 9 months and still there was a gain of `2%`. The rate per cent is-

Options

A. `6 1/3 %`
B. `8 1/3 %`
C. `8%`
D. `7 43/61 %`

Answer

To solve the problem step by step, we need to find the rate percent given the cost price, selling price, and the gain percentage.

### Step 1: Calculate the Selling Price (SP) considering the gain
The goods were bought for Rs. 600, and there was a gain of 2%.

To find the Selling Price (SP) when there is a gain of 2%, we can use the formula:
\[ \text{SP} = \text{CP} + \text{Gain} \]
Where Gain can be calculated as:
\[ \text{Gain} = \frac{\text{Gain Percentage}}{100} \times \text{CP} \]

So,
\[ \text{Gain} = \frac{2}{100} \times 600 = 12 \]
Thus,
\[ \text{SP} = 600 + 12 = 612 \]

### Step 2: Identify the Present Worth (PW)
The goods were sold for Rs. 650.25 at a credit of 9 months. The Present Worth (PW) is the amount that would need to be invested today to yield the future value (FV) of Rs. 650.25 after 9 months.

Given that the SP is Rs. 612, we can equate this to the Present Worth (PW) for the future value of Rs. 650.25.

### Step 3: Calculate the Simple Interest (SI)
The difference between the Selling Price and Present Worth gives us the Simple Interest (SI):
\[ \text{SI} = \text{FV} - \text{PW} \]
\[ \text{SI} = 650.25 - 612 = 38.25 \]

### Step 4: Calculate the Rate of Interest
The formula for Simple Interest is given by:
\[ \text{SI} = \frac{PW \times R \times T}{100} \]
Where:
- SI = Simple Interest
- PW = Present Worth
- R = Rate of interest
- T = Time in years

Since the time is 9 months, we convert it to years:
\[ T = \frac{9}{12} = \frac{3}{4} \text{ years} \]

Now substituting the values we have:
\[ 38.25 = \frac{612 \times R \times \frac{3}{4}}{100} \]

### Step 5: Solve for R
Rearranging the equation to find R:
\[ R = \frac{38.25 \times 100}{612 \times \frac{3}{4}} \]
\[ R = \frac{3825}{612 \times 0.75} \]
\[ R = \frac{3825}{459} \]
\[ R = 8.3333 \]

Thus, the rate percent is:
\[ R = 8 \frac{1}{3} \% \]

### Final Answer
The rate percent is \( 8 \frac{1}{3} \% \).

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