If x : y = 4 :5 , then `(3x + y) : (5x + 3y)`=

Options

A. `3:5`
B. `5:3`
C. `17:35`
D. `35:17`

Answer

To solve the problem where \( x : y = 4 : 5 \) and we need to find \( (3x + y) : (5x + 3y) \), we can follow these steps:

### Step 1: Express the ratio in fraction form
Given \( x : y = 4 : 5 \), we can express this as:
\[ \frac{x}{y} = \frac{4}{5} \]

### Step 2: Substitute \( x \) in terms of \( y \)
From the ratio \( \frac{x}{y} = \frac{4}{5} \), we can express \( x \) in terms of \( y \):
\[ x = \frac{4}{5}y \]

### Step 3: Substitute \( x \) in the expressions \( 3x + y \) and \( 5x + 3y \)
Now we substitute \( x \) into the expressions \( 3x + y \) and \( 5x + 3y \).

**Numerator:**
\[ 3x + y = 3\left(\frac{4}{5}y\right) + y = \frac{12}{5}y + y = \frac{12}{5}y + \frac{5}{5}y = \frac{17}{5}y \]

**Denominator:**
\[ 5x + 3y = 5\left(\frac{4}{5}y\right) + 3y = 4y + 3y = 7y \]

### Step 4: Form the ratio
Now we can form the ratio:
\[ (3x + y) : (5x + 3y) = \frac{\frac{17}{5}y}{7y} \]

### Step 5: Simplify the ratio
The \( y \) in the numerator and denominator cancels out:
\[ = \frac{17/5}{7} = \frac{17}{5 \times 7} = \frac{17}{35} \]

### Step 6: Write the final ratio
Thus, we can express the final ratio as:
\[ (3x + y) : (5x + 3y) = 17 : 35 \]

### Final Answer
The final answer is:
\[ \boxed{17 : 35} \]

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