A ratio of two parts blue to six parts yellow to make two gallons of lime green. A gallon of lime green sells for ?$27. Find the cost of 1 quart of blue pain.

Question

A hardware store mixes paints in a ratio of two parts blue to six parts yellow to make two gallons of lime green. A ratio of five parts blue to three parts yellow makes two gallons of ocean blue. A gallon of lime green sells for ?$27, and a gallon of ocean blue sells for ?$30. Find the cost of 1 quart of blue paint and the cost of 1 quart of yellow paint.

Answer 1

There are several tricky parts to this question.

 

Let's assign variables for our colors:

b = blue

y = yellow

g = lime green

o = ocean blue

 

Look at the ratio of gallons blue to gallons yellow to make 2 gallons lime green

2b:6y = 2g

This means 2/8 of the 2 gallons of lime green are from blue

(1/4)(2) = 1/2 gallon of blue in 2 gallons lime green

So:  1 gal lime green contains 1/4 gal blue

 

6/8 of the 2 gallons of lime green are from yellow

3/4 of the 2 gallons of lime green are from yellow

(3/4)(2) = 3/2 gallons of yellow in 2 gallons of lime green

So:  1 gal lime green contains 3/4 gal yellow

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Now look at the ratio of blue to yellow to make ocean blue:

5b:3y = 2o

 

5/8 of the 2 gallons of ocean blue is from blue

(5/8)(2) = 5/4 gallons of blue in 2 gallons ocean blue

So:  1 gal ocean blue contains 5/8 gal blue

 

(3/8) of the 2 gallons of ocean blue are from yellow

(3/8)(2) = 3/4

There is 3/4 gallons of yellow in the 2 gallons of ocean blue

So:  There is 3/8 gal yellow in 1 gal ocean blue

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We know that 1 gal of lime grn sells for $27

1 gal of lime green will contain 1/4 gal blue and 3/4 gal yellow

(1/4)b + (3/4)y = 27    Equation 1

 

We know that 1 gal of ocean blue sells for $30

(5/8)b + (3/8)y = 30   Equation 2

 

We now have two equations with 2 unknowns that

will give us the cost of a gallon of blue and a gallon of yellow

 

(1/4)b + (3/4)y = 27

(5/8)b + (3/8)y = 30

 

Let's clear the fractions.  Multiply eqn 1 by 4 and eqn 2 by 8

 

b + 3y = 108

5b + 3y = 240   Subtract eqn 1 from equation 2

 

4b + 0y = 32

b = $8 per gallon

 

Plug back into first equation:

8 + 3y = 108

3y = 100

y = 100/3 or $33.33 per gallon

 

The last trick is to realize the problem asked for the cost of

a quart of blue and a quart of yellow.  There are 4 quarts in each

gallon so:

 

1 quart of blue costs 8/4 = $2

1 quart of yellow costs (100/3)/4 = 100/12 = $8.33

 

Hope this helps.  It's a bit long and complicated.  Let me know

if you have questions.