How much percent more than the cost price should a shopkeeper mark his goods so that after allowing a discount of 25% on the marked price, he gains 20%?

Options

A. `70%`
B. `50%`
C. `60%`
D. `55%`

Answer

To solve the problem, we need to determine how much more than the cost price a shopkeeper should mark his goods so that after a 25% discount on the marked price, he still gains 20%.

Let's denote:
- Cost Price (CP) = x
- Marked Price (MP) = y

### Step 1: Calculate the Selling Price (SP) for a 20% gain
To find the selling price that gives a 20% profit on the cost price, we can use the formula:
\[ SP = CP + (20\% \text{ of } CP) \]
\[ SP = x + 0.2x = 1.2x \]

### Step 2: Calculate the Selling Price after a 25% discount on the Marked Price
The selling price after a discount of 25% on the marked price is given by:
\[ SP = MP - (25\% \text{ of } MP) \]
\[ SP = y - 0.25y = 0.75y \]

### Step 3: Set the two expressions for SP equal to each other
Since both expressions represent the selling price, we can set them equal:
\[ 0.75y = 1.2x \]

### Step 4: Solve for the Marked Price (MP) in terms of Cost Price (CP)
Now, we can rearrange the equation to find y in terms of x:
\[ y = \frac{1.2x}{0.75} \]
\[ y = \frac{1.2}{0.75}x \]
\[ y = 1.6x \]

### Step 5: Calculate the percentage increase from Cost Price to Marked Price
To find how much percent more than the cost price the marked price is, we can use the formula:
\[ \text{Percentage Increase} = \left( \frac{MP - CP}{CP} \right) \times 100\% \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left( \frac{1.6x - x}{x} \right) \times 100\% \]
\[ \text{Percentage Increase} = \left( \frac{0.6x}{x} \right) \times 100\% \]
\[ \text{Percentage Increase} = 60\% \]

### Conclusion
The shopkeeper should mark his goods 60% more than the cost price to achieve a 20% gain after allowing a discount of 25% on the marked price.