Pens are bought at 5 for Rs. 4 and sold at 4 for Rs. 5. Find the gain per cent.

Options

A. `0.55%`
B. `45.26%`
C. `56.25% `
D. `60%`

Answer

To solve the problem of finding the gain percentage when pens are bought and sold at different rates, we can follow these steps:

### Step 1: Calculate the Cost Price (CP) of one pen
Given that 5 pens are bought for Rs. 4, we can find the cost price of one pen as follows:

\[ \text{Cost Price of 1 pen} = \frac{\text{Total Cost Price}}{\text{Number of Pens}} = \frac{4}{5} \text{ Rs.} \]

### Step 2: Calculate the Selling Price (SP) of one pen
Given that 4 pens are sold for Rs. 5, we can find the selling price of one pen as follows:

\[ \text{Selling Price of 1 pen} = \frac{\text{Total Selling Price}}{\text{Number of Pens}} = \frac{5}{4} \text{ Rs.} \]

### Step 3: Calculate the Gain
Gain is calculated as the difference between the selling price and the cost price:

\[ \text{Gain} = \text{Selling Price} - \text{Cost Price} = \frac{5}{4} - \frac{4}{5} \]

To perform this subtraction, we need a common denominator. The least common multiple of 4 and 5 is 20:

\[ \text{Gain} = \left(\frac{5 \times 5}{20}\right) - \left(\frac{4 \times 4}{20}\right) = \frac{25}{20} - \frac{16}{20} = \frac{9}{20} \text{ Rs.} \]

### Step 4: Calculate the Gain Percentage
The gain percentage is calculated using the formula:

\[ \text{Gain Percentage} = \left(\frac{\text{Gain}}{\text{Cost Price}}\right) \times 100 \]

Substituting the values we have:

\[ \text{Gain Percentage} = \left(\frac{\frac{9}{20}}{\frac{4}{5}}\right) \times 100 \]

To simplify this, we can multiply by the reciprocal of the cost price:

\[ \text{Gain Percentage} = \left(\frac{9}{20} \times \frac{5}{4}\right) \times 100 = \frac{9 \times 5}{20 \times 4} \times 100 \]

Calculating this gives:

\[ \text{Gain Percentage} = \frac{45}{80} \times 100 = \frac{45 \times 100}{80} = \frac{4500}{80} = 56.25\% \]

### Final Answer
The gain percentage is **56.25%**.

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