If `(3a+2b):(5a+3b)=18:29`. Find `a:b`.

Answer

To solve the problem `(3a + 2b):(5a + 3b) = 18:29` and find the ratio `a:b`, follow these steps:

### Step-by-Step Solution:

1. **Set up the equation based on the given ratio:**
\[ \frac{3a + 2b}{5a + 3b} = \frac{18}{29} \]

2. **Cross-multiply to eliminate the fraction:**
\[ 29(3a + 2b) = 18(5a + 3b) \]

3. **Distribute both sides:**
\[ 87a + 58b = 90a + 54b \]

4. **Rearrange the equation to isolate terms involving `a` and `b`:**
\[ 87a + 58b - 90a - 54b = 0 \]
This simplifies to:
\[ -3a + 4b = 0 \]

5. **Rearranging gives:**
\[ 4b = 3a \]

6. **Express `a` in terms of `b`:**
\[ \frac{a}{b} = \frac{4}{3} \]

7. **Write the ratio `a:b`:**
\[ a:b = 4:3 \]

### Final Answer:
Thus, the ratio \( a:b \) is \( 4:3 \).

---