By what `%` must the cost of goods be marked up so that even after a discount of `20%` the same amount is recovered as before the discount?

Options

A. 20
B. 25
C. 40
D. 10

Answer

To solve the problem of determining the percentage markup required so that after a 20% discount, the selling price equals the original cost price, we can follow these steps:

### Step 1: Define Variables
Let:
- \( CP \) = Cost Price
- \( MP \) = Marked Price (the price after markup)

### Step 2: Express Selling Price After Discount
When a discount of 20% is applied to the marked price, the selling price \( SP \) can be expressed as:
\[ SP = MP - (20\% \text{ of } MP) = MP \times (1 - 0.20) = MP \times 0.80 \]

### Step 3: Set Up the Equation
According to the problem, we want the selling price after the discount to equal the cost price:
\[ SP = CP \]
Substituting the expression for \( SP \) from Step 2:
\[ MP \times 0.80 = CP \]

### Step 4: Solve for Marked Price
Rearranging the equation to find \( MP \):
\[ MP = \frac{CP}{0.80} \]

### Step 5: Calculate Markup
The markup amount can be calculated as:
\[ \text{Markup} = MP - CP \]
Substituting the expression for \( MP \):
\[ \text{Markup} = \frac{CP}{0.80} - CP \]
To combine the terms, we can express \( CP \) as \( \frac{CP \times 0.80}{0.80} \):
\[ \text{Markup} = \frac{CP}{0.80} - \frac{CP \times 0.80}{0.80} = \frac{CP - CP \times 0.80}{0.80} = \frac{CP \times 0.20}{0.80} \]

### Step 6: Calculate Percentage Markup
The percentage markup is given by:
\[ \text{Percentage Markup} = \left( \frac{\text{Markup}}{CP} \right) \times 100 \]
Substituting the expression for Markup:
\[ \text{Percentage Markup} = \left( \frac{\frac{CP \times 0.20}{0.80}}{CP} \right) \times 100 = \left( \frac{0.20}{0.80} \right) \times 100 \]
Calculating this gives:
\[ \text{Percentage Markup} = \frac{0.20}{0.80} \times 100 = 25\% \]

### Conclusion
Thus, the cost of goods must be marked up by **25%** so that even after a discount of 20%, the same amount is recovered as before the discount.

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