1. The problem states: Given the ratios $A : B = 7 : 9$ and $B : C = 3 : 4$, find the combined ratio $A : B : C$. 2. To find $A : B : C$, we need a common term for $B$ in both ratios. 3. The $B$ values are $9$ in the first ratio and $3$ in the second ratio. 4. Find the least common multiple (LCM) of $9$ and $3$, which is $9$. 5. Adjust the second ratio $B : C = 3 : 4$ so that $B = 9$ by multiplying both terms by $3$: $$B : C = 3 \times 3 : 4 \times 3 = 9 : 12$$. 6. Now we have the ratios: $$A : B = 7 : 9$$ $$B : C = 9 : 12$$ 7. Combine these to get: $$A : B : C = 7 : 9 : 12$$. Hence, the final ratio is $\boxed{7 : 9 : 12}$.