By what per cent must the cost price be raised in fixing the sale price in order that there may be a profit of 20% after allowing a commission of 10%?

Options

A. 0.25
B. 133 `1/3` %
C. 33 `1/3` %
D. 0.3

Answer

To solve the problem step-by-step, we will determine how much the cost price (CP) must be raised to achieve a 20% profit after allowing for a 10% commission.

### Step 1: Understand the terms
- **Cost Price (CP)**: The original price of the product.
- **Selling Price (SP)**: The price at which the product is sold.
- **Commission**: A percentage deducted from the selling price, in this case, 10%.
- **Profit**: The percentage gain on the cost price, in this case, 20%.

### Step 2: Define the Selling Price with respect to the Mark Price
Let’s denote the Mark Price (MP) as 100 (for easier calculations).
- After a 10% commission, the Selling Price (SP) will be:
\[ SP = MP - (10\% \text{ of } MP) = 100 - 10 = 90 \]

### Step 3: Define the Selling Price with respect to the Cost Price
To achieve a 20% profit on the Cost Price, we can express this as:
- If CP = 100, then SP = CP + (20\% \text{ of } CP) = 100 + 20 = 120.

### Step 4: Set the Selling Prices equal
From the previous steps, we have two expressions for SP:
1. From the commission: SP = 90
2. From the profit: SP = 120

We need to find a new CP such that:
\[ 90 = CP + (20\% \text{ of } CP) \]

### Step 5: Solve for CP
Let’s denote the new CP as \( CP' \):
\[ SP = CP' + 0.2 \cdot CP' \]
\[ 90 = 1.2 \cdot CP' \]
Now, solve for \( CP' \):
\[ CP' = \frac{90}{1.2} = 75 \]

### Step 6: Calculate the increase in Cost Price
Now, we need to find out how much the original CP must be raised:
- Original CP = 100
- New CP = 75

The increase in CP is:
\[ \text{Increase} = CP' - CP = 75 - 100 = -25 \]

### Step 7: Calculate the percentage increase
To find the percentage increase:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{\text{Original CP}}\right) \times 100 = \left(\frac{-25}{100}\right) \times 100 = -25\% \]

### Final Answer
Thus, the cost price must be raised by **25%** to achieve a profit of 20% after allowing for a 10% commission.

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