Mixture Problems

A light green latex paint that is 20% yellow paint is combined with a darker green latex paint that is 45% yellow paint. How many gallons of each paint must be used to create 15 gallons of a green paint that is 25% yellow paint?

Let \(x\) be the number of gallons of the 20% yellow paint and let \(y\) be the number of gallons of the 40% yellow paint. This means that we want those two numbers to add up to 15: \(x+y=15\).

Now if we want 15 gallons of 25% yellow paint, that means we want \(0.25⋅15=3.75\) gallons of pure yellow pigment. The expression \(0.20⋅x\) represents the amount of pure yellow pigment in the \(x\) gallons of 20% yellow paint. The expression \(0.45⋅y\) represents the amount of pure yellow pigment in the \(y\) gallons of 45% yellow paint. Combing the last two adds up to the 3.75 gallons of pure pigment in the final mixture:

\(0.20x+0.40y=3.75\)

The system is: \(\left\{\begin{aligned}

x+y &=15 \\

0.20 x+0.45 y &=3.75

\end{aligned}\right.\)

We can isolate one variable and use substitution to solve the system:

\(x=15−y\)

Now solve for \(y\).

\(\begin{aligned}

0.20(15-y)+0.45 y &=3.75 & & \\

3-0.20 y+0.45 y &=3.75 & & \text { Distributive Property } \\

3+0.2 y &=3.75 & & \text { Add like terms. } \\

0.25 y &=0.75 & & \text { Subtract } 3 . \\

y &=3 & & \text { Divide by } 0.25

\end{aligned}\)

Now we can plug in \(y=3\) into \(x+y=15\):

\(x+y=15 \Rightarrow x+3=15 \Rightarrow x=12\)

This means 12 gallons of 20% yellow paint should be mixed with 3 gallons of 45% yellow paint in order to get 15 gallons of 25% yellow paint.