The percentage of loss when an article is sold at Rs. 50 is the same as that of the profit when it is sold at Rs. 70. The above mentioned percentage of profit or loss on the article is

Options

A. `10%`
B. `12%`
C. `16(2)/(3)%`
D. `8(1)/(3)%`

Answer

To solve the problem, let's denote the cost price (CP) of the article as \( x \).

### Step 1: Set up the equations for loss and profit
When the article is sold for Rs. 50, there is a loss. The loss can be expressed as:
\[ \text{Loss} = \text{Cost Price} - \text{Selling Price} = x - 50 \]
The percentage of loss is given by:
\[ \text{Percentage of Loss} = \left(\frac{\text{Loss}}{\text{Cost Price}}\right) \times 100 = \left(\frac{x - 50}{x}\right) \times 100 \]

When the article is sold for Rs. 70, there is a profit. The profit can be expressed as:
\[ \text{Profit} = \text{Selling Price} - \text{Cost Price} = 70 - x \]
The percentage of profit is given by:
\[ \text{Percentage of Profit} = \left(\frac{\text{Profit}}{\text{Cost Price}}\right) \times 100 = \left(\frac{70 - x}{x}\right) \times 100 \]

### Step 2: Set the percentages equal to each other
According to the problem, the percentage of loss when sold at Rs. 50 is equal to the percentage of profit when sold at Rs. 70. Therefore, we can set the two equations equal to each other:
\[ \left(\frac{x - 50}{x}\right) \times 100 = \left(\frac{70 - x}{x}\right) \times 100 \]

### Step 3: Simplify the equation
We can cancel out 100 from both sides:
\[ \frac{x - 50}{x} = \frac{70 - x}{x} \]
Now, cross-multiply:
\[ (x - 50) \cdot x = (70 - x) \cdot x \]
This simplifies to:
\[ x - 50 = 70 - x \]

### Step 4: Solve for x
Now, add \( x \) to both sides:
\[ x + x - 50 = 70 \]
This simplifies to:
\[ 2x - 50 = 70 \]
Now, add 50 to both sides:
\[ 2x = 120 \]
Finally, divide by 2:
\[ x = 60 \]

### Step 5: Calculate the percentage of loss and profit
Now that we have the cost price \( x = 60 \), we can calculate the percentage of loss and profit.

1. **Percentage of Loss when sold at Rs. 50:**
\[ \text{Loss} = 60 - 50 = 10 \]
\[ \text{Percentage of Loss} = \left(\frac{10}{60}\right) \times 100 = \frac{1000}{60} \approx 16.67\% \]

2. **Percentage of Profit when sold at Rs. 70:**
\[ \text{Profit} = 70 - 60 = 10 \]
\[ \text{Percentage of Profit} = \left(\frac{10}{60}\right) \times 100 = \frac{1000}{60} \approx 16.67\% \]

### Conclusion
Thus, the percentage of loss when sold at Rs. 50 and the percentage of profit when sold at Rs. 70 is both approximately \( 16.67\% \).