In order to make a specific shade of green paint, a painter mixes \( \frac{1}{2} \) of a gallon of blue paint with \( \frac{4}{5} \) of a gallon of yellow paint. Write an equation to represent the number of gallons of yellow paint, \( y \). needed for any amount of blue paint, \( b \).

To represent the relationship between the amount of blue paint, \( b \), and the amount of yellow paint, \( y \), we can set up a proportion based on the amounts used in this specific shade. From the mixture, we can see that \( \frac{1}{2} \) gallon of blue paint corresponds to \( \frac{4}{5} \) gallon of yellow paint. The relationship can be expressed as: \[ \frac{b}{\frac{1}{2}} = \frac{y}{\frac{4}{5}} \] To find \( y \) in terms of \( b \), we can cross-multiply and solve for \( y \): \[ b \cdot \frac{4}{5} = y \cdot \frac{1}{2} \] Multiplying both sides by 2 gives: \[ y = \frac{8}{5}b \] This equation shows that for any amount of blue paint, the amount of yellow paint needed is \( \frac{8}{5} \) times the amount of blue paint. Now you're all set to create your perfect shade of green!