If `x : y = 3 : 2` find `(2x + 3y) : (3x +5y)`

Answer

To solve the problem where we need to find the ratio \( (2x + 3y) : (3x + 5y) \) given that \( x : y = 3 : 2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the given ratio**:
We have \( x : y = 3 : 2 \). This means that \( \frac{x}{y} = \frac{3}{2} \).

2. **Express \( x \) in terms of \( y \)**:
From the ratio, we can express \( x \) as:
\[ x = \frac{3}{2}y \]

3. **Substitute \( x \) into the expressions**:
We need to find \( 2x + 3y \) and \( 3x + 5y \). Let's substitute \( x \) in both expressions:
- For \( 2x + 3y \):
\[ 2x + 3y = 2\left(\frac{3}{2}y\right) + 3y = 3y + 3y = 6y \]
- For \( 3x + 5y \):
\[ 3x + 5y = 3\left(\frac{3}{2}y\right) + 5y = \frac{9}{2}y + 5y = \frac{9}{2}y + \frac{10}{2}y = \frac{19}{2}y \]

4. **Form the ratio**:
Now we can form the ratio:
\[ (2x + 3y) : (3x + 5y) = 6y : \frac{19}{2}y \]

5. **Simplify the ratio**:
To simplify \( 6y : \frac{19}{2}y \), we can eliminate \( y \) from both sides (as long as \( y \neq 0 \)):
\[ 6 : \frac{19}{2} \]
To simplify further, multiply both sides by 2:
\[ 6 \times 2 : 19 = 12 : 19 \]

### Final Answer:
Thus, the ratio \( (2x + 3y) : (3x + 5y) \) is \( 12 : 19 \).

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