Understanding Ratios
To solve the given problem, we need to analyze the different types of ratios mentioned: compound ratio, duplicate ratio, and brigade ratio.

1. Compound Ratio
- The compound ratio of two ratios \(a:b\) and \(c:d\) is given by:
\[
Compound\ Ratio = \frac{a \times c}{b \times d}
\]
- Given the ratio \(4:9\), we can consider this as \(\frac{4}{9}\).

2. Duplicate Ratio
- The duplicate ratio of a ratio \(a:b\) is calculated by squaring both terms:
\[
Duplicate\ Ratio = \left( \frac{a^2}{b^2} \right)
\]
- For the ratio \(3:4\):
\[
Duplicate\ Ratio = \frac{3^2}{4^2} = \frac{9}{16}
\]

3. Brigade Ratio
- The brigade ratio is typically the ratio of two ratios \(a:b\) and \(c:d\), defined as:
\[
Brigade\ Ratio = \frac{a}{c} : \frac{b}{d}
\]
- For the ratios \(2:3\) and \(9:7\):
\[
Brigade\ Ratio = \frac{2 \times 7}{3 \times 9} = \frac{14}{27}
\]

4. Final Calculation
- Now we can combine these ratios to find the overall ratio using the compound formula:
\[
Resulting\ Ratio = \frac{4 \times 9 \times 14}{9 \times 27}
\]
- Simplifying this gives:
\[
Resulting\ Ratio = \frac{504}{243} = \frac{56}{27}
\]

Conclusion
The final answer for the given ratios in the problem is \(56:27\). This method illustrates the use of different types of ratios and their applications in solving complex ratio problems effectively.