The per cent profit earned by selling an article at ₹5400 is equal to the per cent loss incurred on selling the same article at ₹3600. What will be the per cent profit if the article is sold at ₹5000?

Options

A. 7
B. 15
C. `9 1/6`
D. `11 1/9`

Answer

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

### Step 1: Understand the problem
We need to find the cost price (CP) of the article based on the information given about selling prices (SP) and the relationship between profit and loss.

### Step 2: Set up the equations
Let the cost price of the article be \( x \).

1. When the article is sold at ₹5400, the profit is:
\[ \text{Profit} = SP - CP = 5400 - x \]
The percentage profit is given by:
\[ \text{Percentage Profit} = \frac{5400 - x}{x} \times 100 \]

2. When the article is sold at ₹3600, the loss is:
\[ \text{Loss} = CP - SP = x - 3600 \]
The percentage loss is given by:
\[ \text{Percentage Loss} = \frac{x - 3600}{x} \times 100 \]

### Step 3: Set the profit equal to the loss
According to the problem, the percentage profit at ₹5400 is equal to the percentage loss at ₹3600:
\[ \frac{5400 - x}{x} \times 100 = \frac{x - 3600}{x} \times 100 \]
We can simplify this by canceling out 100 from both sides:
\[ \frac{5400 - x}{x} = \frac{x - 3600}{x} \]

### Step 4: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ (5400 - x) \cdot x = (x - 3600) \cdot x \]
This simplifies to:
\[ 5400x - x^2 = x^2 - 3600x \]

### Step 5: Rearrange the equation
Rearranging the terms, we have:
\[ 5400x + 3600x = 2x^2 \]
\[ 9000x = 2x^2 \]
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 9000 = 2x \]
Thus,
\[ x = \frac{9000}{2} = 4500 \]

### Step 6: Calculate the profit when sold at ₹5000
Now we know the cost price (CP) is ₹4500. If the article is sold at ₹5000, the profit is:
\[ \text{Profit} = 5000 - 4500 = 500 \]

### Step 7: Calculate the percentage profit
The percentage profit is given by:
\[ \text{Percentage Profit} = \frac{\text{Profit}}{\text{CP}} \times 100 = \frac{500}{4500} \times 100 \]
Calculating this gives:
\[ \text{Percentage Profit} = \frac{500}{4500} \times 100 = \frac{1}{9} \times 100 \approx 11.11\% \]

### Final Answer
The percentage profit when the article is sold at ₹5000 is approximately \( 11 \frac{1}{9} \% \).

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