If `a:b=2:3`, then `(5a+b):(3a+2b)` is

Options

A. `13:12`
B. `15:17`
C. `12:13`
D. `13:11`

Answer

To solve the problem, we start with the given ratio \( a:b = 2:3 \). We need to find the ratio \( (5a + b):(3a + 2b) \).

### Step-by-step Solution:

1. **Express \( a \) and \( b \) in terms of a common variable**:
Since \( a:b = 2:3 \), we can let:
\[ a = 2k \quad \text{and} \quad b = 3k \]
where \( k \) is a positive constant.

2. **Substitute \( a \) and \( b \) into the expression**:
We need to find:
\[ (5a + b):(3a + 2b) \]
Substituting the values of \( a \) and \( b \):
\[ 5a + b = 5(2k) + 3k = 10k + 3k = 13k \]
\[ 3a + 2b = 3(2k) + 2(3k) = 6k + 6k = 12k \]

3. **Form the ratio**:
Now we can form the ratio:
\[ (5a + b):(3a + 2b) = 13k : 12k \]

4. **Simplify the ratio**:
The \( k \) cancels out:
\[ 13k : 12k = 13 : 12 \]

Thus, the final answer is:
\[ \boxed{13:12} \]