If the cost price f 12 pens is equal to the selling price of 8 pens, the gain percent is ?

Options

A. 0.12
B. 305
C. 0.5
D. 0.6

Answer

To solve the problem, we need to find the gain percentage when the cost price of 12 pens is equal to the selling price of 8 pens.

### Step-by-Step Solution:

1. **Define Variables**:
Let the cost price of one pen be \( CP \) and the selling price of one pen be \( SP \).

2. **Set Up the Equation**:
According to the problem, the cost price of 12 pens is equal to the selling price of 8 pens. Therefore, we can write the equation:
\[ 12 \times CP = 8 \times SP \]

3. **Rearranging the Equation**:
We can rearrange the equation to express the relationship between \( CP \) and \( SP \):
\[ \frac{CP}{SP} = \frac{8}{12} \]
Simplifying this gives:
\[ \frac{CP}{SP} = \frac{2}{3} \]

4. **Finding the Profit**:
Since \( CP < SP \), we can find the profit. The profit can be calculated as:
\[ Profit = SP - CP \]
To express \( SP \) in terms of \( CP \), we can rearrange the ratio:
\[ SP = \frac{3}{2} CP \]
Now substituting this into the profit formula:
\[ Profit = \frac{3}{2} CP - CP = \frac{3}{2} CP - \frac{2}{2} CP = \frac{1}{2} CP \]

5. **Calculating Gain Percentage**:
The gain percentage is calculated using the formula:
\[ Gain\% = \left( \frac{Profit}{CP} \right) \times 100 \]
Substituting the profit we found:
\[ Gain\% = \left( \frac{\frac{1}{2} CP}{CP} \right) \times 100 = \frac{1}{2} \times 100 = 50\% \]

6. **Final Answer**:
The gain percentage is \( 50\% \), which can also be expressed as \( 0.5 \) in decimal form.

### Summary:
The gain percentage when the cost price of 12 pens is equal to the selling price of 8 pens is \( 50\% \) or \( 0.5 \).