P buys some toffees at 6 for a rupee and sells them at 4 for a rupee. Find his gain percent.

Answer

To solve the problem of finding the gain percent when P buys toffees at 6 for a rupee and sells them at 4 for a rupee, we can follow these steps:

### Step 1: Determine the Cost Price (CP) and Selling Price (SP)
- **Cost Price (CP)**: P buys 6 toffees for 1 rupee. Therefore, the cost price of 1 toffee is:
\[ CP = \frac{1 \text{ rupee}}{6 \text{ toffees}} = \frac{1}{6} \text{ rupee per toffee} \]

- **Selling Price (SP)**: P sells 4 toffees for 1 rupee. Therefore, the selling price of 1 toffee is:
\[ SP = \frac{1 \text{ rupee}}{4 \text{ toffees}} = \frac{1}{4} \text{ rupee per toffee} \]

### Step 2: Calculate the Gain
- To find the gain, we subtract the cost price from the selling price for one toffee:
\[ \text{Gain} = SP - CP = \frac{1}{4} - \frac{1}{6} \]

- To perform the subtraction, we need a common denominator. The least common multiple of 4 and 6 is 12.
- Convert \(\frac{1}{4}\) to twelfths:
\[ \frac{1}{4} = \frac{3}{12} \]
- Convert \(\frac{1}{6}\) to twelfths:
\[ \frac{1}{6} = \frac{2}{12} \]

- Now, subtract:
\[ \text{Gain} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12} \text{ rupee} \]

### Step 3: Calculate the Gain Percent
- The gain percent is calculated using the formula:
\[ \text{Gain Percent} = \left( \frac{\text{Gain}}{CP} \right) \times 100 \]

- Substitute the values:
\[ \text{Gain Percent} = \left( \frac{\frac{1}{12}}{\frac{1}{6}} \right) \times 100 \]

- Simplifying the fraction:
\[ \frac{\frac{1}{12}}{\frac{1}{6}} = \frac{1}{12} \times \frac{6}{1} = \frac{6}{12} = \frac{1}{2} \]

- Now, calculate the gain percent:
\[ \text{Gain Percent} = \left( \frac{1}{2} \right) \times 100 = 50\% \]

### Final Answer
P's gain percent is **50%**.

---