The cash difference between the selling prices of an article at a profit of 4% and 6% is ₹ 3. The ratio of the two selling prices is

Options

A. `51 : 52`
B. `52 : 53`
C. `51 : 53`
D. `51 : 55`

Answer

To solve the problem step by step, let's define the variables and calculate the selling prices based on the given profit percentages.

### Step 1: Define the Cost Price
Let the cost price (CP) of the article be ₹ X.

### Step 2: Calculate Selling Price at 4% Profit
The selling price (SP) when there is a profit of 4% can be calculated as follows:
\[ SP_1 = CP + \text{Profit} = X + \left(\frac{4}{100} \times X\right) = X + \frac{4X}{100} = X + \frac{X}{25} = \frac{25X + 4X}{25} = \frac{29X}{25} \]

### Step 3: Calculate Selling Price at 6% Profit
The selling price (SP) when there is a profit of 6% can be calculated similarly:
\[ SP_2 = CP + \text{Profit} = X + \left(\frac{6}{100} \times X\right) = X + \frac{6X}{100} = X + \frac{3X}{50} = \frac{50X + 6X}{50} = \frac{56X}{50} = \frac{28X}{25} \]

### Step 4: Find the Difference Between Selling Prices
According to the problem, the difference between the two selling prices is ₹ 3:
\[ SP_2 - SP_1 = 3 \]
Substituting the selling prices we calculated:
\[ \frac{28X}{25} - \frac{29X}{25} = 3 \]
This simplifies to:
\[ \frac{28X - 29X}{25} = 3 \]
\[ \frac{-X}{25} = 3 \]
Multiplying both sides by -25 gives:
\[ X = -75 \]
Since cost price cannot be negative, we will consider the absolute value:
\[ X = 75 \]

### Step 5: Calculate the Selling Prices Again
Now, we can calculate the actual selling prices:
\[ SP_1 = \frac{29 \times 75}{25} = 87 \]
\[ SP_2 = \frac{28 \times 75}{25} = 84 \]

### Step 6: Find the Ratio of Selling Prices
Now, we can find the ratio of the two selling prices:
\[ \text{Ratio} = \frac{SP_1}{SP_2} = \frac{87}{84} \]
To simplify this ratio, we can express it as:
\[ \text{Ratio} = \frac{29}{28} \]

### Final Answer
The ratio of the two selling prices is \( \frac{29}{28} \).