A trader gains 15% after selling an item at 10% discount on the printed price. The ratio of the cost price and printed price of the item is

Options

A. `18 : 23`
B. `17 : 18`
C. `17 : 23`
D. `18 : 25`

Answer

To solve the problem step by step, we need to find the ratio of the cost price (CP) to the printed price (PP) given the conditions in the question.

### Step 1: Define the Variables
Let:
- CP = Cost Price
- PP = Printed Price
- SP = Selling Price

### Step 2: Establish the Selling Price
The trader sells the item at a 10% discount on the printed price. Therefore, the selling price (SP) can be expressed as:
\[ SP = PP - (10\% \text{ of } PP) = PP - 0.1 \times PP = 0.9 \times PP \]

### Step 3: Calculate the Cost Price using Gain Percentage
The trader gains 15% on the cost price. Thus, we can express the selling price in terms of the cost price:
\[ SP = CP + (15\% \text{ of } CP) = CP + 0.15 \times CP = 1.15 \times CP \]

### Step 4: Set the Two Expressions for Selling Price Equal
Since both expressions represent the selling price, we can set them equal to each other:
\[ 0.9 \times PP = 1.15 \times CP \]

### Step 5: Rearranging the Equation
Now, we can rearrange the equation to find the ratio of CP to PP:
\[ \frac{CP}{PP} = \frac{0.9}{1.15} \]

### Step 6: Simplifying the Ratio
To simplify \( \frac{0.9}{1.15} \):
1. Multiply both the numerator and the denominator by 100 to eliminate the decimals:
\[ \frac{90}{115} \]
2. Now, simplify \( \frac{90}{115} \) by dividing both by their greatest common divisor, which is 5:
\[ \frac{90 \div 5}{115 \div 5} = \frac{18}{23} \]

### Conclusion
Thus, the ratio of the cost price to the printed price is:
\[ \frac{CP}{PP} = \frac{18}{23} \]