1. **State the problem:** Given the ratios $\frac{a}{b} = \frac{3}{5}$ and $\frac{b}{c} = \frac{4}{5}$, find the ratio $a : b : c$ in simplest form. 2. **Express variables based on given ratios:** From $\frac{a}{b} = \frac{3}{5}$, we write $a = \frac{3}{5}b$. From $\frac{b}{c} = \frac{4}{5}$, we write $b = \frac{4}{5}c$. 3. **Substitute to express all variables in terms of $c$:** Substitute $b = \frac{4}{5}c$ into $a = \frac{3}{5}b$: $$ a = \frac{3}{5} \times \frac{4}{5}c = \frac{12}{25}c $$ 4. **Form the ratio $a : b : c$ using expressions in $c$:** $$ a : b : c = \frac{12}{25}c : \frac{4}{5}c : c $$ Since $c$ is common, simplify: $$ \frac{12}{25} : \frac{4}{5} : 1 $$ 5. **Eliminate denominators for simplicity:** Multiply all terms by 25 to get whole numbers: $$ 12 : 20 : 25 $$ 6. **Check for further simplification:** The greatest common divisor (GCD) of 12, 20, and 25 is 1, so the ratio is already in simplest form. **Final answer:** $$ a : b : c = 12 : 20 : 25 $$