Question

dig deeper you mix $\frac{1}{4}$ cup of red paint for every $\frac{1}{2}$ cup of blue paint to make 3 gallons of purple paint.
a. how much red paint do you use? how much blue paint do you use?
you use $\boldsymbol{square}$ gallon(s) of red paint and $\boldsymbol{square}$ gallon(s) of blue paint.
b. you decide that you want to make a lighter purple paint. you make the new mixture by adding $\frac{1}{4}$ cup of white paint for every $\frac{1}{4}$ cup of red paint and $\frac{1}{2}$ cup of blue paint. how much red paint, blue paint, and white paint do you use to make $1\frac{1}{2}$ gallons of the lighter purple paint?
you use $\boldsymbol{square}$ gallon(s) of red paint, $\boldsymbol{square}$ gallon(s) of blue paint, and $\boldsymbol{square}$ gallon(s) of white paint.

Explanation:

Step1: Find red:blue ratio

Red:Blue = $\frac{1}{4}:\frac{1}{2} = 1:2$

Step2: Calculate total ratio parts

Total parts = $1+2=3$

Step3: Find red paint for 3 gal

Red paint: $\frac{1}{3} \times 3 = 1$ gallon

Step4: Find blue paint for 3 gal

Blue paint: $\frac{2}{3} \times 3 = 2$ gallons

Step5: New ratio (red:blue:white)

New ratio = $\frac{1}{4}:\frac{1}{2}:\frac{1}{4} = 1:2:1$

Step6: Total new ratio parts

Total new parts = $1+2+1=4$

Step7: Red paint for $1\frac{1}{2}$ gal

Red paint: $\frac{1}{4} \times \frac{3}{2} = \frac{3}{8}$ gallon

Step8: Blue paint for $1\frac{1}{2}$ gal

Blue paint: $\frac{2}{4} \times \frac{3}{2} = \frac{3}{4}$ gallon

Step9: White paint for $1\frac{1}{2}$ gal

White paint: $\frac{1}{4} \times \frac{3}{2} = \frac{3}{8}$ gallon

Answer:

a. You use 1 gallon(s) of red paint and 2 gallon(s) of blue paint.
b. You use $\frac{3}{8}$ gallon(s) of red paint, $\frac{3}{4}$ gallon(s) of blue paint, and $\frac{3}{8}$ gallon(s) of white paint.