1. **State the problem:** We are given two ratios: $a:b=3:4$ and $b:c=4:9$. We want to find the combined ratio $a:b:c$. 2. **Understand the ratios:** The ratio $a:b=3:4$ means $\frac{a}{b}=\frac{3}{4}$. The ratio $b:c=4:9$ means $\frac{b}{c}=\frac{4}{9}$. 3. **Find a common term:** Both ratios share $b$. To combine them, $b$ must be the same in both ratios. 4. **Adjust ratios to have the same $b$:** In $a:b=3:4$, $b=4$. In $b:c=4:9$, $b=4$. Since $b$ is already the same, we can directly combine. 5. **Write combined ratio:** $a:b:c = 3:4:9$. 6. **Final answer:** The combined ratio is $\boxed{3:4:9}$.