If the cost price of 18 pens is equal to the selling price of 12 pens, what is the gain percent?

Options

A. `12.5%`
B. `22.5%`
C. `50%`
D. `36%`

Answer

To solve the problem, we need to find the gain percentage when the cost price of 18 pens is equal to the selling price of 12 pens.

### Step-by-Step Solution:

1. **Define Variables**:
Let the cost price (CP) of one pen be \( x \).
Therefore, the cost price of 18 pens is \( 18x \).

2. **Selling Price Calculation**:
Let the selling price (SP) of one pen be \( y \).
Therefore, the selling price of 12 pens is \( 12y \).

3. **Set Up the Equation**:
According to the problem, the cost price of 18 pens is equal to the selling price of 12 pens:
\[ 18x = 12y \]

4. **Find the Ratio of CP to SP**:
Rearranging the equation gives us:
\[ \frac{CP}{SP} = \frac{12}{18} = \frac{2}{3} \]

5. **Express CP and SP in Terms of a Common Variable**:
If we let the cost price of 18 pens be \( 2k \) (for some \( k \)), then the selling price of 12 pens will be \( 3k \).

6. **Calculate Profit**:
The profit can be calculated as:
\[ \text{Profit} = SP - CP = 3k - 2k = k \]

7. **Calculate Gain Percentage**:
The gain percentage is given by the formula:
\[ \text{Gain Percentage} = \left(\frac{\text{Profit}}{\text{CP}}\right) \times 100 \]
Substituting the values we have:
\[ \text{Gain Percentage} = \left(\frac{k}{2k}\right) \times 100 = \frac{1}{2} \times 100 = 50\% \]

### Final Answer:
The gain percentage is **50%**.