A Vendor sells his articles at a certain profit percentage. If he sells his articles at `1//3` of his actual selling price, then he incurs a loss of `40%`. What is his actual profit percentage?

Options

A. `72%`
B. `120%`
C. `80%`
D. none of these

Answer

To solve the problem step by step, we will denote the actual selling price (SP) of the article as \( X \) and the cost price (CP) as \( Y \).

### Step 1: Understanding the Selling Price and Loss
The vendor sells his articles at \( \frac{1}{3} \) of his actual selling price. Therefore, the new selling price (SP') is:
\[ SP' = \frac{X}{3} \]
He incurs a loss of 40% when selling at this price.

### Step 2: Calculating the Loss
A loss of 40% means that the selling price is 60% of the cost price. Therefore, we can express this relationship as:
\[ SP' = CP \times (1 - \text{Loss Percentage}) = Y \times 0.6 \]
Substituting the value of \( SP' \):
\[ \frac{X}{3} = 0.6Y \]

### Step 3: Rearranging the Equation
To find the relationship between \( X \) and \( Y \), we can rearrange the equation:
\[ X = 1.8Y \quad \text{(Multiplying both sides by 3)} \]

### Step 4: Finding the Profit Percentage
The profit percentage is calculated using the formula:
\[ \text{Profit Percentage} = \frac{\text{Profit}}{\text{Cost Price}} \times 100 \]
Where profit is given by:
\[ \text{Profit} = SP - CP = X - Y \]
Substituting \( X = 1.8Y \):
\[ \text{Profit} = 1.8Y - Y = 0.8Y \]

### Step 5: Calculating the Profit Percentage
Now substituting the profit into the profit percentage formula:
\[ \text{Profit Percentage} = \frac{0.8Y}{Y} \times 100 = 80\% \]

### Final Answer
Thus, the actual profit percentage of the vendor is:
\[ \boxed{80\%} \]