If the cost price of 8 pens is same as the selling price of 6 pens, then the gain percent is

Question

Accepted Answer

To determine the gain percent, we need to establish a relationship between the cost price (CP) and the selling price (SP) of the pens.

Let CP be the cost price of one pen and SP be the selling price of one pen.

According to the problem statement:
The cost price of 8 pens is the same as the selling price of 6 pens.
This can be written as an equation:
$8 	imes CP = 6 	imes SP$

To find the gain percent, we need to express SP in terms of CP, or vice versa, and then use the formula for gain percent.
From the equation $8 	imes CP = 6 	imes SP$, we can find the ratio of SP to CP:
$\frac{SP}{CP} = \frac{8}{6}$
$\frac{SP}{CP} = \frac{4}{3}$

This means $SP = \frac{4}{3} 	imes CP$.
Since $SP > CP$, there is a gain.

The formula for gain percent is:
Gain Percent = $\frac{SP - CP}{CP} 	imes 100\%$

Substitute $SP = \frac{4}{3} 	imes CP$ into the formula:
Gain Percent = $\frac{\frac{4}{3} CP - CP}{CP} 	imes 100\%$
Gain Percent = $\frac{(\frac{4}{3} - 1) CP}{CP} 	imes 100\%$
Gain Percent = $\frac{(\frac{4-3}{3})}{1} 	imes 100\%$
Gain Percent = $\frac{\frac{1}{3}}{1} 	imes 100\%$
Gain Percent = $\frac{1}{3} 	imes 100\%$
Gain Percent = $33\frac{1}{3}\%$

Therefore, the gain percent is $33\frac{1}{3}\%$.

**Justification for Option 3 being correct:**
As calculated above, by setting up the relationship between the cost price of 8 pens and the selling price of 6 pens, we found that the selling price of one pen is $\frac{4}{3}$ times its cost price. This leads to a gain of $\frac{1}{3}$ of the cost price, which, when expressed as a percentage, is $33\frac{1}{3}\%$. This perfectly matches Option 3.

**Explanation for why other options are incorrect:**

*   **Option 1: $11\frac{1}{2}\%$**
    This option represents a gain of $11.5\%$. Our calculation clearly shows a gain of $33\frac{1}{3}\%$, which is significantly higher than $11.5\%$. Therefore, this option is incorrect.

*   **Option 2: $13\frac{1}{3}\%$**
    This option represents a gain of approximately $13.33\%$. Our derived gain percent is $33\frac{1}{3}\%$. Since $13\frac{1}{3}\%$ is not equal to $33\frac{1}{3}\%$, this option is incorrect.

*   **Option 4: 25%**
    This option represents a gain of $25\%$. If the gain were $25\%$, then $SP = CP + 0.25CP = 1.25CP = \frac{5}{4}CP$. However, our calculation shows $SP = \frac{4}{3}CP$. Since $\frac{4}{3} 
eq \frac{5}{4}$, this option is incorrect. A $25\%$ gain would imply that the cost price of 8 pens is equal to the selling price of $8 	imes \frac{4}{5} = 6.4$ pens, not 6 pens.

The final answer is $\boxed{	ext{33\frac{1}{3}\%}}$