Question

The paint mixer wants to mix paint that is 30% gloss with paint that is 15% gloss to make 3.75 gallons of paint that is 20% gloss. How many gallons of each paint should the paint mixer mix together?

Answer 1

The paint mixer should mix 1.238 gallon paint that is 30 % gloss and 2.512 gallon paint that is 15 % gloss to get 3.75 gallon of paint that is 20 % gloss.

In this question we must determine the correct volumes of each paint to obtain a 20 % gloss with paint by means of weighted averages:

$c_{r} = x\cdot c_{a} +(1-x)\cdot c_{b}$ (1)

Where:

• $x$ - Proportion of the paint that is 30 % gloss.
• $c_{a}$ - Concentration ratio of the paint that is 30 % gloss.
• $c_{b}$ - Concentration ratio of the paint that is 15 % gloss.
• $c_{r}$ - Resulting concentration ratio

If we know that $c_{a} = 0.3$, $c_{b} = 0.15$ and $c_{r} = 0.2$, then the proportion of the paint that is 30 % gloss is:

$0.2 = 0.3\cdot x +(1-x)\cdot 0.15$

$0.2 = 0.15\cdot x+0.15$

$0.15\cdot x = 0.05$

$x = 0.33$

And the quantity of gallons needed of each paint are:

Paint (30 % gloss)

$V_{a} = 0.33\cdot (3.75\,gal)$

$V_{a} = 1.238\,gal$

Paint (15 % gloss)

$V_{b} = V-V_{a}$

$V_{b} = 3.75\,gal-1.238\,gal$

$V_{b} = 2..512\,gal$

The paint mixer should mix 1.238 gallon paint that is 30 % gloss and 2.512 gallon paint that is 15 % gloss to get 3.75 gallon of paint that is 20 % gloss.

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