The cash difference between the selling price of an article at a profit of 8% and 10% is Rs.3. The ratio of two selling prices is:

Options

A. `54:56 `
B. `54:55`
C. `54:57`
D. `54:59`

Answer

To solve the problem, we need to find the ratio of the two selling prices of an article sold at different profit percentages. Here’s a step-by-step solution:

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

### Step 2: Calculate Selling Prices (SP)
1. **Selling Price at 8% Profit:**
\[ SP_1 = CP + 8\% \text{ of } CP = X + \frac{8}{100}X = \frac{108}{100}X \]

2. **Selling Price at 10% Profit:**
\[ SP_2 = CP + 10\% \text{ of } CP = X + \frac{10}{100}X = \frac{110}{100}X \]

### Step 3: Set Up the Equation for the Difference in Selling Prices
According to the problem, the difference between the two selling prices is Rs. 3:
\[ SP_2 - SP_1 = 3 \]
Substituting the expressions for SP:
\[ \frac{110}{100}X - \frac{108}{100}X = 3 \]

### Step 4: Simplify the Equation
Combine the terms:
\[ \frac{110X - 108X}{100} = 3 \]
\[ \frac{2X}{100} = 3 \]

### Step 5: Solve for X
Multiply both sides by 100:
\[ 2X = 300 \]
Now divide by 2:
\[ X = 150 \]

### Step 6: Calculate the Selling Prices
1. **Selling Price at 8% Profit:**
\[ SP_1 = \frac{108}{100} \times 150 = 162 \]

2. **Selling Price at 10% Profit:**
\[ SP_2 = \frac{110}{100} \times 150 = 165 \]

### Step 7: Find the Ratio of the Selling Prices
Now we find the ratio of the two selling prices:
\[ \text{Ratio} = \frac{SP_1}{SP_2} = \frac{162}{165} \]

### Step 8: Simplify the Ratio
To simplify:
\[ \frac{162}{165} = \frac{54}{55} \]

### Final Answer
The ratio of the two selling prices is \( 54 : 55 \).

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