We are given the ratios:

$\frac{a}{b} = \frac{2}{3} \quad \text{and} \quad \frac{c}{d} = \frac{3}{4}$

We need to find the ratio $\frac{a+c}{b+d}$.

Step 1: Express $a$ and $b$ in terms of a common variable.

From $\frac{a}{b} = \frac{2}{3}$, we can write: $a = \frac{2}{3}b$

Step 2: Express $c$ and $d$ in terms of a common variable.

From $\frac{c}{d} = \frac{3}{4}$, we can write: $c = \frac{3}{4}d$

Step 3: Substitute these into the expression for $\frac{a+c}{b+d}$.

Now, substitute $a = \frac{2}{3}b$ and $c = \frac{3}{4}d$ into the ratio $\frac{a+c}{b+d}$: $\frac{a+c}{b+d} = \frac{\frac{2}{3}b + \frac{3}{4}d}{b + d}$

Step 4: Find a common denominator for the numerator.

To combine the terms $\frac{2}{3}b$ and $\frac{3}{4}d$, we find a common denominator, which is 12. Thus: $\frac{2}{3}b = \frac{8}{12}b \quad \text{and} \quad \frac{3}{4}d = \frac{9}{12}d$ So, the numerator becomes: $\frac{8}{12}b + \frac{9}{12}d = \frac{8b + 9d}{12}$

Step 5: Substitute this into the ratio.

We now have: $\frac{a+c}{b+d} = \frac{\frac{8b + 9d}{12}}{b + d}$

Step 6: Simplify the expression.

Simplifying the complex fraction: $\frac{a+c}{b+d} = \frac{8b + 9d}{12(b + d)}$

This is the simplified ratio. Therefore, the final answer is: $\boxed{\frac{8b + 9d}{12(b + d)}}$

Let me know if you need further clarification or a more detailed breakdown!

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Further Questions to Explore:

1. What would the ratio $\frac{a+c}{b+d}$ be if we had different values for $a$, $b$, $c$, and $d$?
2. How would this ratio change if $\frac{a}{b} = \frac{1}{2}$ and $\frac{c}{d} = \frac{5}{6}$?
3. Can you interpret the ratio $\frac{8b + 9d}{12(b + d)}$ in terms of a real-world situation?
4. How can you use this method to solve problems involving multiple ratios?
5. What happens to this ratio if $b = d$?

Tip: When working with ratios, always express all terms in terms of a common variable to simplify the expression.