`a:b=3:4`, `c:a=2:9`
Find `a:b:c=?`

Answer

To solve the problem, we need to find the combined ratio \( a:b:c \) given the ratios \( a:b = 3:4 \) and \( c:a = 2:9 \).

### Step-by-Step Solution:

1. **Write the given ratios in fractional form**:
- From the ratio \( a:b = 3:4 \), we can express this as:
\[ \frac{a}{b} = \frac{3}{4} \]
- From the ratio \( c:a = 2:9 \), we can express this as:
\[ \frac{c}{a} = \frac{2}{9} \]

2. **Express \( a \) and \( c \) in terms of \( b \)**:
- From \( \frac{a}{b} = \frac{3}{4} \), we can express \( a \) in terms of \( b \):
\[ a = \frac{3}{4}b \]
- Now, substituting \( a \) into \( \frac{c}{a} = \frac{2}{9} \):
\[ c = \frac{2}{9}a = \frac{2}{9} \left(\frac{3}{4}b\right) = \frac{2 \cdot 3}{9 \cdot 4}b = \frac{6}{36}b = \frac{1}{6}b \]

3. **Now we have expressions for \( a \) and \( c \) in terms of \( b \)**:
- \( a = \frac{3}{4}b \)
- \( c = \frac{1}{6}b \)

4. **Find a common denominator to express all ratios**:
- The denominators are 4 and 6. The least common multiple (LCM) of 4 and 6 is 12.
- Convert \( a \) and \( c \) to have a denominator of 12:
- For \( a \):
\[ a = \frac{3}{4}b = \frac{3 \cdot 3}{4 \cdot 3}b = \frac{9}{12}b \]
- For \( c \):
\[ c = \frac{1}{6}b = \frac{1 \cdot 2}{6 \cdot 2}b = \frac{2}{12}b \]

5. **Now we can express the ratios \( a:b:c \)**:
- \( a = \frac{9}{12}b \)
- \( b = \frac{12}{12}b \)
- \( c = \frac{2}{12}b \)

6. **Combine these into a single ratio**:
- Thus, the ratio \( a:b:c \) can be expressed as:
\[ a:b:c = 9:12:2 \]

### Final Answer:
The combined ratio \( a:b:c \) is \( 9:12:2 \).