What is the sixth term in the geometric sequence $\frac{27}{125}, \frac{9}{25}, \frac{3}{5},\ldots$?  Express your answer as a common fraction.
I notice that this is a geometric sequence, which means that each term is obtained by multiplying the previous term by a constant ratio. To find the ratio, I can divide any term by the previous term. For example, $\frac{9}{25} \div \frac{27}{125} = \frac{9}{25} \cdot \frac{125}{27} = \frac{45}{27} = \frac{5}{3}$. I can check that this ratio is consistent for the other terms: $\frac{3}{5} \div \frac{9}{25} = \frac{3}{5} \cdot \frac{25}{9} = \frac{5}{3}$ as well. So the ratio is $\frac{5}{3}$, and I can use it to find the next terms in the sequence. The fourth term is $\frac{3}{5} \cdot \frac{5}{3} = 1$. The fifth term is $1 \cdot \frac{5}{3} = \frac{5}{3}$. The sixth term is $\frac{5}{3} \cdot \frac{5}{3} = \frac{25}{9}$. 