Question

A paint mixer wants to mix paint that is 40% gloss with paint that is 25% gloss to make 5.25 gallons of paint that is 30% gloss. How many gallons of each paint should the paint mixer mix together?A. 2 1/2 gallons of 40% gloss and 2 3/4 gallons of 25% glossB. 1 1/2 gallons of 40% gloss and 3 3/4 gallons of 25% glossC. 1 3/4 gallons of 40% gloss and 3 1/2 gallons of 25% glossD. 2 1/4 gallons of 40% gloss and 3 gallons of 25% gloss

Answer 1

Step 1

Write the system of equations required to solve the problem

$\begin{gathered} 0.4x+0.25y=0.3(5.25)----(1) \\ x+y=5.25_{}---(2) \\ \text{where ;} \\ x\text{ represents the 40}\%\text{ gloss gallon} \\ y\text{ represents the 25}\%\text{ gloss gallon} \end{gathered}$

Step 2

Find the value of x using the substitution method

Find the value of y from equation 2

$\begin{gathered} x+y=5.25 \\ y=5.25-x \end{gathered}$

Substitute for y as seen above in equation 1

$\begin{gathered} 0.4x+0.25y=0.3(5.25) \\ 0.4x+0.25(5.25-x)=1.575_{} \\ 0.4x+1.3125-0.25x=1.575 \\ 0.4x-0.25x=1.575-1.3125 \\ 0.15x=0.2625 \\ \frac{0.15x}{0.15}=\frac{0.2625}{0.15} \\ x=1.75\text{ or }\frac{7}{4}\text{ or 1}\frac{3}{4} \end{gathered}$

Step 3

Find the value of y by substituting for x in equation 2

$\begin{gathered} x+y=5.25_{} \\ 1.75+y=5.25 \\ y=5.25-1.75 \\ y=\frac{7}{2}or\text{ 3.5 or 3}\frac{1}{2} \end{gathered}$

Therefore,

$\text{There are 1}\frac{3}{4\text{ }}\text{ gallons of 40\% gloss and 3}\frac{1}{2}\text{ gallons of 25\% gloss}$

The answer is, therefore, option C