Let's solve this problem step-by-step:

Given that Hitesh sells an article at \(\frac{4}{13}\) of its marked price and incurs a loss of 30%. Let the marked price be \(M\).

Thus, the selling price (SP) at a loss of 30% is:

\(\text{SP}_1 = \frac{4}{13} \times M\)

Let the cost price (CP) of the article be \(C\). Given a loss of 30%, we have:

\(\text{SP}_1 = C - \frac{30}{100} \times C = 0.7 C\)

Equating the two SP expressions, we get:

\(\frac{4}{13} \times M = 0.7 \times C\)

From this, we find the relationship between \(M\) and \(C\):

\(C = \frac{4M}{13 \times 0.7}\)

Now, if the article is sold at 60% of its marked price, we find the new selling price \(\text{SP}_2\):

\(\text{SP}_2 = 0.6 \times M\)

The profit percentage can be calculated using the formula:

\(\text{Profit \%} = \left( \frac{\text{SP}_2 - C}{C} \right) \times 100\)

Substitute \(\text{SP}_2\) and \(C = \frac{4M}{13 \times 0.7}\) into this formula:

\(\text{Profit \%} = \left( \frac{0.6M - \frac{4M}{13 \times 0.7}}{\frac{4M}{13 \times 0.7}} \right) \times 100\)

Simplify the expression:

\(\text{Profit \%} = \left( \frac{0.6M - \frac{4M}{9.1}}{\frac{4M}{9.1}} \right) \times 100\)

\(= \left( \frac{0.6M \times 9.1 - 4M}{4M} \right) \times 100\)

\(= \left( \frac{5.46M - 4M}{4M} \right) \times 100\)

\(= \left(\frac{1.46}{4}\right) \times 100\)

\(= 36.50\%\)

Therefore, when Hitesh sells the article at 60% of its marked price, he makes a profit of 36.50%. Thus, the correct answer is 36.50%.