Even after a discount of `q%` on marked price a trader gains by `p%` What is the markup percentage over the cost price?

Options

A. `(p+q)/((q-p))xx100`
B. `(p+q)/((100-p))xx100`
C. `(p+q)/((100-p))xx100`
D. not possible

Answer

To solve the problem of finding the markup percentage over the cost price when a trader gains by \( p\% \) after giving a discount of \( q\% \) on the marked price, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables:**
- Let the Cost Price (CP) be \( 100 \) (this is a convenient choice to simplify calculations).
- Let the Marked Price (MP) be \( x \).

2. **Calculate Selling Price (SP):**
- The trader gives a discount of \( q\% \) on the marked price. Therefore, the Selling Price (SP) can be calculated as:
\[ SP = MP - \text{Discount} = x - \left(\frac{q}{100} \times x\right) = x \left(1 - \frac{q}{100}\right) \]

3. **Calculate Selling Price in terms of Profit:**
- The trader gains \( p\% \) on the Cost Price. Therefore, the Selling Price can also be expressed as:
\[ SP = CP + \text{Profit} = 100 + \left(\frac{p}{100} \times 100\right) = 100 + p \]

4. **Set the Two Expressions for SP Equal:**
- Since both expressions represent the Selling Price, we can set them equal to each other:
\[ x \left(1 - \frac{q}{100}\right) = 100 + p \]

5. **Rearranging to Find Marked Price (MP):**
- Rearranging the equation gives:
\[ x = \frac{100 + p}{1 - \frac{q}{100}} \]

6. **Calculate Markup Percentage:**
- The markup percentage is calculated as:
\[ \text{Markup} = \frac{MP - CP}{CP} \times 100 = \frac{x - 100}{100} \times 100 = x - 100 \]
- Substitute the value of \( x \):
\[ \text{Markup} = \frac{100 + p}{1 - \frac{q}{100}} - 100 \]

7. **Simplifying the Markup Expression:**
- To simplify:
\[ \text{Markup} = \frac{100 + p - 100(1 - \frac{q}{100})}{1 - \frac{q}{100}} = \frac{100 + p - 100 + q}{1 - \frac{q}{100}} = \frac{p + q}{1 - \frac{q}{100}} \]

8. **Final Expression for Markup Percentage:**
- The final markup percentage is:
\[ \text{Markup Percentage} = p + q \]

### Conclusion:
The markup percentage over the cost price is \( p + q \).