Divide $ \dfrac{3y+3}{6y+2} $ by $ 18 $ to get $ \dfrac{ 3y+3 }{ 108y+36 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{3y+3}{6y+2} }{18} & \xlongequal{\text{Step 1}} \frac{3y+3}{6y+2} \cdot \frac{\color{blue}{1}}{\color{blue}{18}} \xlongequal{\text{Step 2}} \frac{ \left( 3y+3 \right) \cdot 1 }{ \left( 6y+2 \right) \cdot 18 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3y+3 }{ 108y+36 } \end{aligned} $$

Divide $ \dfrac{y+1}{36y+12} $ by $ 5y+5 $ to get $ \dfrac{ 1 }{ 180y+60 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{y+1}{36y+12} }{5y+5} & \xlongequal{\text{Step 1}} \frac{y+1}{36y+12} \cdot \frac{\color{blue}{1}}{\color{blue}{5y+5}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 1 \cdot \color{blue}{ \left( y+1 \right) } }{ 36y+12 } \cdot \frac{ 1 }{ 5 \cdot \color{blue}{ \left( y+1 \right) } } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 1 }{ 36y+12 } \cdot \frac{ 1 }{ 5 } \xlongequal{\text{Step 4}} \frac{ 1 \cdot 1 }{ \left( 36y+12 \right) \cdot 5 } \xlongequal{\text{Step 5}} \frac{ 1 }{ 180y+60 } \end{aligned} $$