Systems of equations

Question

One mixture of paint has 35% pigment. Another mixture has 10% pigment. If you want 11 gallons of paint that has 23% pigment, how many gallons of each mixture should you use? Express your answer in an ordered pair in the form (gallons of 35% mixture, gallons of 10% mixture). Round the values to the nearest tenth.

Answer 1

Hi Niki.

 

This is a classic system of equations problem, and probably the most hated; mixtures.

 

To solve it, you need to write two different equations: one that accounts for the total volume of paint used, and one that accounts for the total volume of pigment required to achieve your goal.

 

First let's call some variables. I'm going to say:

 

C = number of gallons of 35% pigment paint (C for Concentrated)

D = number of gallons of 10% pigment paint (D for Dilute)

 

We know that in the end we want 11 total gallons of paint, which means:

 

C + D = 11

 

That equation states, in mathematical terms, that the number of gallons of 35% pigment paint (C) plus (+) the number of gallons of 10% pigment paint (D) has to be (=) 11.

 

Now that we've balanced the volume equation, we have to deal with balancing the amount of pigment.

 

Since your concentrated paint has 35% pigment, that means that for every gallon of paint you have, you'll have 0.35 gallons of pigment. Similarly with the dilute pigment, you'll have 0.10 gallons of pigment in every gallon of paint. You want a mixture that ends up being 0.23 pigment.

 

So

 

0.35*C + 0.10*D = (C+D)*.23

 

This is stating that the amount of pigment being added to the mixture from your concentrated paint (0.35*C) plus the amount of pigment being added to the mixture from your dilute paint (0.10*D) must result in a mixture of both paints with a concentration of 23% [ (C+D)*.23 ].

 

Now you have two equations, two unknowns.

 

C + D = 11

0.35*C + 0.10*D = (C+D)*.23

 

Multiply the bottom equation by 100 and substitute 11 for C + D

 

35C + 10D = 11*23

 

Multiply the top equation by -10, then add it to the second equation.

 

-10C - 10D = -110

35C + 10D = 253

 

25C = 143

 

C = 143/25 = 5.72 = 5.7

 

Since C + D = 11

 

5.7 + D = 11

 

D = 5.3

 

Check your answers:

 

If you have 5.7 gallons of 35% paint, that means you'll have added a total of:

 

5.7*0.35 = 1.995 gallons of pigment to the mixture

 

If you have 5.3 gallons of 10% paint, that means you'll have added a total of:

 

5.3*0.10 = 0.53 gallons of pigment to the mixture

 

Totaling:

 

2.53 gallons of pigment.

 

If you have 11 gallons of 23% pigment, that means you have a total of:

 

11*0.23 = 2.53 gallons of pigment.

 

So it checks out. Hope this helps.

Answer 2

0.10(11 - g) + 0.35g = 0.23(11)

 

Can you solve for g?