By selling an article at 76% of its marked price, a trader makes a loss of 20%. What will be the profit percentage, if he sells it at 99% of its marked price?

The correct option is:

$4\frac{4}{19}\%$

Step-by-Step Derivation and Explanation:

Let us denote the marked price of the article as $MP$ and the cost price as $CP$.

According to the first condition, the article is sold at 76% of its marked price. This gives us the selling price ($S{P}_1$):

$S{P}_1=0.76\times MP$

At this selling price, the trader makes a loss of 20%. Since a loss of 20% means the selling price is 80% of the cost price, we can write:

$S{P}_1=0.80\times CP$

Equating the two expressions for $S{P}_1$, we get:

$0.80\times CP=0.76\times MP$

From this equation, we can find the relationship between the cost price and the marked price:

$CP=\frac{0.76}{0.80}\times MP=\frac{76}{80}\times MP=\frac{19}{20}\times MP$

Next, the trader sells the article at 99% of its marked price. This gives us the new selling price ($S{P}_2$):

$S{P}_2=0.99\times MP$

To find the profit percentage, we first calculate the profit:

$\text{Profit}=S{P}_2-CP$

Substitute the expressions in terms of $MP$:

$\text{Profit}=0.99\times MP-\frac{19}{20}\times MP$

Since $0.99=\frac{99}{100}$:

$\text{Profit}=\left(\frac{99}{100}-\frac{19}{20}\right)\times MP=\left(\frac{99}{100}-\frac{95}{100}\right)\times MP=\frac{4}{100}\times MP$

Now, we compute the profit percentage relative to the cost price ($CP$):

$\text{Profit Percentage}=\left(\frac{\text{Profit}}{CP}\right)\times 100$

Substitute the values of Profit and $CP$:

$\text{Profit Percentage}=\frac{\frac{4}{100}\times MP}{\frac{19}{20}\times MP}\times 100$

The $MP$ terms cancel out, simplifying the calculation to:

$\text{Profit Percentage}=\left(\frac{4}{100}\times \frac{20}{19}\right)\times 100$

$\text{Profit Percentage}=\frac{4}{100}\times \frac{20}{19}\times 100=\frac{80}{19}\%$

Now, convert the improper fraction $\frac{80}{19}$ into a mixed fraction. Dividing 80 by 19 gives a quotient of 4 and a remainder of 4:

$\frac{80}{19}=4\frac{4}{19}$

Therefore, the profit percentage is:

$4\frac{4}{19}\%$