Let the number be x. One-third part of x exceeds one-fifth part of x by 6.

Use the difference of fractions: $\frac{1}{3} - \frac{1}{5} = \frac{2}{15}$. So $\frac{2}{15}x = 6$, which means $x = 6 \times \frac{15}{2} = 45$. The required answer is $\frac{1}{3}x = \frac{45}{3} = 15$.

Students often find the value of the number $x$ (which is 45) and mark it as the answer, forgetting that the question asks for the 'one-third part of the number'.

Always verify if the question asks for the variable found or a function of that variable (like 1/3 of the number), as this is a classic way to test attention to detail in competitive exams.