To determine the fraction of chalk received by the third teacher, first calculate the amount received by the first teacher, which is $\frac{2}{5}$. The remainder after the first teacher is $\frac{3}{5}$, and the second teacher receives $\frac{2}{15}$ of this remainder, equating to $\frac{2}{15} \times \frac{3}{5} = \frac{2}{25}$. Subtracting the amounts received by the first and second teachers from the total gives the third teacher $\frac{13}{25}$ of the chalk.