How much per cent above the cost price should a shopkeeper mark his goods so that after allowing a discount of 10% on the marked price, he gains 8%?

Answer

To solve the problem, we need to determine how much percentage above the cost price a shopkeeper should mark his goods so that after allowing a 10% discount on the marked price, he still makes an 8% profit on the cost price.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the cost price (CP) of the goods be \( CP \).
- Let the marked price (MP) be \( x \) percent above the cost price. Therefore, \( MP = CP + \frac{x}{100} \times CP = CP \left(1 + \frac{x}{100}\right) \).

2. **Calculate Selling Price (SP):**
- The shopkeeper allows a discount of 10% on the marked price. Thus, the selling price (SP) after the discount is:
\[ SP = MP - 10\% \text{ of } MP = MP \times \left(1 - \frac{10}{100}\right) = MP \times 0.90 \]
- Substituting the value of MP:
\[ SP = CP \left(1 + \frac{x}{100}\right) \times 0.90 \]

3. **Calculate Desired Selling Price for 8% Profit:**
- The shopkeeper wants to gain an 8% profit on the cost price. Therefore, the desired selling price (SP) should be:
\[ SP = CP + 8\% \text{ of } CP = CP \times \left(1 + \frac{8}{100}\right) = CP \times 1.08 \]

4. **Set the Two Expressions for SP Equal:**
- Now we have two expressions for SP:
\[ CP \left(1 + \frac{x}{100}\right) \times 0.90 = CP \times 1.08 \]
- We can cancel \( CP \) from both sides (assuming \( CP \neq 0 \)):
\[ \left(1 + \frac{x}{100}\right) \times 0.90 = 1.08 \]

5. **Solve for x:**
- Divide both sides by 0.90:
\[ 1 + \frac{x}{100} = \frac{1.08}{0.90} \]
- Calculate \( \frac{1.08}{0.90} \):
\[ \frac{1.08}{0.90} = 1.2 \]
- Now, we have:
\[ 1 + \frac{x}{100} = 1.2 \]
- Subtract 1 from both sides:
\[ \frac{x}{100} = 0.2 \]
- Multiply both sides by 100:
\[ x = 20 \]

6. **Conclusion:**
- The shopkeeper should mark his goods **20%** above the cost price.