if a : b = 2 : 3, c:d = 3:4 find (a + c ): (b+d)

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.

Explore how to solve the ratio problem involving a:b = 2:3 and c:d = 3:4 to find the ratio (a+c):(b+d). This step-by-step guide breaks down the algebraic approach to combining ratios and simplifying complex fractions. Ideal for students in Grades 7-9.

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