Profit obtained on selling an article for ` 310 is equal to the loss incurred on selling that article for ` 230. What will be the loss percentage when selling price is ` 180?

The correct option is 33* 1/3 (which represents $33\frac{1}{3}\%$).

Let's break down the solution step-by-step to understand why this option is correct.

Step 1: Understand the relationship between cost price, selling price, profit, and loss
Let the Cost Price (CP) of the article be $x$.
When an article is sold at a profit, the Profit is given by:
$\text{Profit}=\text{Selling Price (SP)}-\text{Cost Price (CP)}$
When an article is sold at a loss, the Loss is given by:
$\text{Loss}=\text{Cost Price (CP)}-\text{Selling Price (SP)}$

Step 2: Set up the equation using the given information
We are given that the profit obtained by selling the article for ` 310 is equal to the loss incurred by selling it for ` 230.
Using our formulas:
Profit at SP of ` 310 = $310-x$
Loss at SP of ` 230 = $x-230$
Since the profit is equal to the loss:
$310-x=x-230$

Step 3: Solve for the Cost Price ($x$)
Adding $x$ to both sides and adding 230 to both sides, we get:
$2x=310+230$
$2x=540$
$x=270$
So, the Cost Price (CP) of the article is ` 270.

Step 4: Calculate the loss percentage when the selling price is ` 180
Now, we want to find the loss percentage if the new Selling Price (SP) is ` 180.
First, calculate the loss amount:
$\text{Loss}=\text{CP}-\text{New SP}$
$\text{Loss}=270-180=90$
Next, calculate the loss percentage using the formula:
$\text{Loss Percentage}=\left(\frac{\text{Loss}}{\text{CP}}\right)\times 100$
Substituting the values:
$\text{Loss Percentage}=\left(\frac{90}{270}\right)\times 100$
Simplifying the fraction:
$\text{Loss Percentage}=\frac{1}{3}\times 100=\frac{100}{3}\%$
Converting the improper fraction to a mixed fraction:
$\frac{100}{3}=33\frac{1}{3}\%$

Thus, the loss percentage is indeed 33* 1/3.