We are tasked with finding the ratio $x:y$ for $A:C$, given the ratios $A:B = a:8$ and $B:C = 12:c$.

Step-by-step Solution:

1. Express $A$, $B$, and $C$ in terms of common multiples:

   • $A:B = a:8$ means $A = ka$ and $B = k \cdot 8$, where $k$ is a common multiplier.
   • $B:C = 12:c$ means $B = m \cdot 12$ and $C = m \cdot c$, where $m$ is a common multiplier.

2. Equate the two expressions for $B$:

   • From $A:B$: $B = k \cdot 8$.
   • From $B:C$: $B = m \cdot 12$.
   • Therefore, $k \cdot 8 = m \cdot 12$. Simplifying: $k:m = 12:8 = 3:2.$ Hence, $k = 3t$ and $m = 2t$, where $t$ is another multiplier.

3. Express $A$ and $C$ using the common multiplier $t$:

   • $A = ka = 3t \cdot a = 3ta$,
   • $C = mc = 2t \cdot c = 2tc$.

4. Find the ratio $A:C$:

   • The ratio $A:C$ is: $A:C = 3ta : 2tc.$
   • Cancel $t$: $A:C = 3a : 2c.$

Final Answer:

The ratio $x:y$ for $A:C$ is: $x:y = 3a:2c.$

Would you like to dive into any step further or see a numerical example?

---

Related Questions:

1. If $a = 9$ and $c = 4$, what is the numerical value of $x:y$?
2. How do you derive a ratio when given three linked quantities?
3. What happens to the ratios if one of the variables $a$ or $c$ is doubled?
4. Can you explain how ratios are used in scaling problems?
5. How do we check the consistency of ratios in multi-step problems?

Tip:

Always reduce ratios to their simplest form for clarity and accuracy in problems.