If `x : y = 3 : 4` find `(3x + 4y) : (5x + 6y)`.

Answer

To solve the problem, we need to find the ratio \( (3x + 4y) : (5x + 6y) \) given that \( x : y = 3 : 4 \).

### Step-by-Step Solution:

1. **Express x and y in terms of a common variable (k):**
Since \( x : y = 3 : 4 \), we can express \( x \) and \( y \) as:
\[ x = 3k \quad \text{and} \quad y = 4k \]
where \( k \) is a positive constant.

**Hint:** Use a common variable to express the quantities in the given ratio.

2. **Substitute x and y into the expressions \( 3x + 4y \) and \( 5x + 6y \):**
Now, we substitute \( x \) and \( y \) into the expressions:
\[ 3x + 4y = 3(3k) + 4(4k) = 9k + 16k = 25k \]
\[ 5x + 6y = 5(3k) + 6(4k) = 15k + 24k = 39k \]

**Hint:** Replace \( x \) and \( y \) in the expressions with their values in terms of \( k \).

3. **Form the ratio \( (3x + 4y) : (5x + 6y) \):**
Now, we can form the ratio:
\[ (3x + 4y) : (5x + 6y) = 25k : 39k \]

**Hint:** When forming the ratio, notice that \( k \) cancels out.

4. **Simplify the ratio:**
Since \( k \) is common in both terms, we can simplify the ratio:
\[ 25k : 39k = 25 : 39 \]

**Hint:** Cancel out common factors in the ratio to simplify it.

### Final Answer:
The ratio \( (3x + 4y) : (5x + 6y) \) is \( 25 : 39 \).