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.

Explore how to find the ratio A:C given A:B = 2:3 and 2B:C = 8:5. This step-by-step solution simplifies ratios using algebraic techniques and properties of proportions. Perfect for students in Grades 6-8.

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