Understanding the Profit Percentage Problem

This question involves calculating the profit percentage made when an article is sold at its original selling price, given information about a loss incurred at a reduced selling price. We need to determine the relationship between the cost price (CP) and the selling price (SP).

Setting Up the Variables and Equations

Let's define the key variables:

• Let the Cost Price (CP) of the article be represented by $C$.
• Let the original Selling Price (SP) of the article be represented by $S$.

According to the problem statement, there is a 30% loss when the article is sold at 3/5 of the original selling price. This means the new selling price is:

New Selling Price = $\frac{3}{5}S$

A 30% loss implies that the selling price is 70% of the cost price (100% - 30% = 70%). Therefore, we can write the equation:

New Selling Price = $0.70 \times C$

Calculating the Relationship Between Selling Price and Cost Price

By equating the two expressions for the new selling price, we get:

$ \frac{3}{5}S = 0.70 \times C $

Now, we need to express the original Selling Price (S) in terms of the Cost Price (C). We can rearrange the equation:

$ S = \frac{0.70 \times C}{\frac{3}{5}} $

To simplify this, we multiply by the reciprocal of $\frac{3}{5}$, which is $\frac{5}{3}$:

$ S = \frac{0.70 \times C \times 5}{3} $

$ S = \frac{3.5 \times C}{3} $

This shows that the original selling price is $\frac{3.5}{3}$ times the cost price.

Calculating the Profit Percentage

The profit made when selling at the original price (S) is the difference between the selling price and the cost price:

Profit = $S - C$

Substitute the value of S we found:

Profit = $\frac{3.5C}{3} - C$

To subtract, find a common denominator:

Profit = $\frac{3.5C}{3} - \frac{3C}{3}$

Profit = $\frac{3.5C - 3C}{3}$

Profit = $\frac{0.5C}{3}$

Now, we calculate the profit percentage using the formula:

Profit Percentage = $\frac{\text{Profit}}{C} \times 100\%$

Substitute the expression for Profit:

Profit Percentage = $\frac{\frac{0.5C}{3}}{C} \times 100\%$

The 'C' terms cancel out:

Profit Percentage = $\frac{0.5}{3} \times 100\%$

Profit Percentage = $\frac{50}{3}\%$

Final Answer Conversion

To express this percentage as a mixed number, we divide 50 by 3:

$ \frac{50}{3} = 16 \text{ with a remainder of } 2 $

So, the profit percentage is $16\frac{2}{3}\%$. This matches option 4.