We are given the ratios $A:B = 2:3$ and $2B:C = 8:5$. Let's find the ratio $A:C$.

Step 1: Express $A:B$ and $2B:C$

From the ratio $A:B = 2:3$, we can express this as: $\frac{A}{B} = \frac{2}{3} \quad \text{or} \quad A = \frac{2}{3}B$

From $2B:C = 8:5$, we can express this as: $\frac{2B}{C} = \frac{8}{5} \quad \text{or} \quad C = \frac{5}{8} \cdot 2B = \frac{5}{4}B$

Step 2: Express $A:C$

Now we need to find the ratio $\frac{A}{C}$.

Substitute $A = \frac{2}{3}B$ and $C = \frac{5}{4}B$ into the ratio $\frac{A}{C}$: $\frac{A}{C} = \frac{\frac{2}{3}B}{\frac{5}{4}B} = \frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$

Final Answer:

The ratio $A:C = 8:15$.

Would you like more details on this solution or have any further questions?

Here are five related questions for further practice:

1. If $A:B = 3:4$ and $B:C = 5:6$, what is $A:C$?
2. Given $A:B = 5:7$ and $B:C = 9:11$, find $A:C$.
3. If $3A:B = 9:4$ and $B:C = 8:7$, what is $A:C$?
4. If $A:B = 7:2$ and $4B:C = 10:9$, find $A:C$.
5. Given $A:B = 3:5$ and $6B:C = 12:13$, what is $A:C$?

Tip: When working with ratios, always try to express everything in terms of one variable (like B here), then simplify.