In order that there may be a profit of 20% after allowing a discount of 10% on the marked price, the cost price of an article has to be increased by

Options

A. 0.3
B. `33(1)/(3)%`
C. 0.33
D. `33(2)/(3)%`

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Define the Marked Price
Let the marked price (MP) of the article be \( X \).

### Step 2: Calculate the Selling Price after Discount
Since there is a discount of 10% on the marked price, the selling price (SP) can be calculated as:
\[ SP = MP - \text{Discount} = X - 0.10X = 0.90X \]

### Step 3: Calculate the Cost Price for a 20% Profit
To achieve a profit of 20%, we can use the formula for selling price in relation to cost price (CP):
\[ SP = CP + \text{Profit} = CP + 0.20 \times CP = 1.20 \times CP \]
From this, we can express the cost price in terms of the selling price:
\[ CP = \frac{SP}{1.20} \]
Substituting the selling price we found in Step 2:
\[ CP = \frac{0.90X}{1.20} \]

### Step 4: Simplify the Cost Price Expression
Now, we simplify the expression for CP:
\[ CP = \frac{0.90X}{1.20} = \frac{90X}{120} = \frac{3X}{4} \]

### Step 5: Determine the Increase in Cost Price
We need to find out how much the cost price needs to be increased. The increase in cost price can be calculated as:
\[ \text{Increase} = MP - CP = X - \frac{3X}{4} \]
This simplifies to:
\[ \text{Increase} = \frac{4X}{4} - \frac{3X}{4} = \frac{X}{4} \]

### Step 6: Calculate the Percentage Increase in Cost Price
To find the percentage increase in cost price, we use the formula:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{CP}\right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left(\frac{\frac{X}{4}}{\frac{3X}{4}}\right) \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} = \left(\frac{1}{3}\right) \times 100 = \frac{100}{3} \approx 33.33\% \]

### Final Answer
Thus, the cost price of the article has to be increased by approximately \( 33.33\% \).

---