A shopkeepper allows a discount of 10% on the marked price of calculator. then by what per cent higher than cost price, should marked the price, so as to gain 20% on selling it at the discount?

Answer

To solve the problem step-by-step, we need to determine the marked price (MP) of the calculator based on the given conditions. Here’s how we can do it:

### Step 1: Understand the Given Information
- Discount on marked price = 10%
- Desired profit = 20%

### Step 2: Define Variables
Let:
- Cost Price (CP) = x
- Marked Price (MP) = y

### Step 3: Calculate Selling Price (SP) after Discount
The selling price (SP) after applying a 10% discount on the marked price can be expressed as:
\[ SP = MP - \text{Discount} \]
\[ SP = y - 0.10y = 0.90y \]

### Step 4: Calculate Selling Price for Desired Profit
To achieve a profit of 20% on the cost price, the selling price can also be expressed as:
\[ SP = CP + \text{Profit} \]
\[ SP = x + 0.20x = 1.20x \]

### Step 5: Set the Two Expressions for SP Equal
Since both expressions represent the selling price, we can set them equal to each other:
\[ 0.90y = 1.20x \]

### Step 6: Solve for Marked Price (MP)
Rearranging the equation to find the marked price (y):
\[ y = \frac{1.20x}{0.90} \]
\[ y = \frac{1.20}{0.90} \times x \]
\[ y = \frac{12}{9} \times x \]
\[ y = \frac{4}{3} \times x \]

### Step 7: Calculate the Percentage Increase from Cost Price to Marked Price
To find the percentage increase of the marked price over the cost price:
\[ \text{Percentage Increase} = \left(\frac{y - x}{x}\right) \times 100\% \]
Substituting for y:
\[ \text{Percentage Increase} = \left(\frac{\frac{4}{3}x - x}{x}\right) \times 100\% \]
\[ = \left(\frac{\frac{4}{3}x - \frac{3}{3}x}{x}\right) \times 100\% \]
\[ = \left(\frac{\frac{1}{3}x}{x}\right) \times 100\% \]
\[ = \frac{1}{3} \times 100\% \]
\[ = 33.33\% \]

### Conclusion
The marked price should be 33.33% higher than the cost price to achieve a 20% profit after allowing a 10% discount.

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