If cost prices of 100 note books is same as selling price of 80 note books. What is the percent gain ?

Options

A. `15%`
B. `30%`
C. ` 25%`
D. `20%`

Answer

To solve the problem step by step, we need to find the percentage gain based on the given information about the cost price (CP) and selling price (SP) of notebooks.

### Step-by-Step Solution:

1. **Define Variables:**
Let the cost price (CP) of 100 notebooks be \( x \) rupees.

2. **Calculate CP of One Notebook:**
The cost price of one notebook is:
\[ \text{CP of one notebook} = \frac{x}{100} \text{ rupees} \]

3. **Relate CP and SP:**
According to the problem, the cost price of 100 notebooks is equal to the selling price (SP) of 80 notebooks. Therefore:
\[ \text{SP of 80 notebooks} = x \text{ rupees} \]

4. **Calculate SP of One Notebook:**
The selling price of one notebook is:
\[ \text{SP of one notebook} = \frac{x}{80} \text{ rupees} \]

5. **Calculate Gain:**
Gain is defined as the selling price minus the cost price. Thus, the gain for one notebook is:
\[ \text{Gain} = \text{SP of one notebook} - \text{CP of one notebook} \]
Substituting the values we calculated:
\[ \text{Gain} = \frac{x}{80} - \frac{x}{100} \]

6. **Find a Common Denominator:**
To subtract these fractions, we need a common denominator. The least common multiple of 80 and 100 is 400. Thus, we rewrite the fractions:
\[ \text{Gain} = \frac{5x}{400} - \frac{4x}{400} = \frac{5x - 4x}{400} = \frac{x}{400} \]

7. **Calculate Gain Percentage:**
Gain percentage is calculated using the formula:
\[ \text{Gain Percentage} = \left( \frac{\text{Gain}}{\text{CP}} \right) \times 100 \]
Substituting the values we have:
\[ \text{Gain Percentage} = \left( \frac{\frac{x}{400}}{\frac{x}{100}} \right) \times 100 \]
Simplifying this:
\[ \text{Gain Percentage} = \left( \frac{x}{400} \times \frac{100}{x} \right) \times 100 = \frac{100}{4} = 25\% \]

8. **Conclusion:**
Therefore, the percentage gain is \( 25\% \).