How many gallons

Question

How many gallons each of 25% and 35% alcohol should be mixed to obtain 20 gallons of 32% alcohol?

Answer 1

This is a system of 2 linear equations problem. Because there are two unknowns (how many gallons of the 25% alcohol we have and how many gallons of the 35% alcohol we have) we need to create two equations.

1) Let's start by establishing our variables:

x is the number of gallons of the 25% alcohol 
y is the number of gallons of the 35% alcohol

2) Then we need to come up with the two equations:

 

Equation 1: x + y = 20

*Where did this come from?

• This equation represents the amount of liquid.
• If we take the amount of gallons of each and add them we should end up with 20 gallons total in our final mixture

 

 

Equation 2: .25x + .35y = .32(20)

*Where did this come from?

• This equation represents the amount of alcohol only.
• .25x represents the amount of alcohol in x gallons of the 25% alcohol, .35y represents the amount of alcohol in y gallons of the 35% alcohol, .32(20) represents the amount of alcohol in the final mixture which is 20 gallons. The total amount of alcohol in the final mixture should be equivalent to the amount of alcohol in each solution since we are mixing (adding) them together. (*remember that you need to move the decimal two places when using percentages)
• It is important to note that you must multiply the .32 times 20 at the end. This is the most common mistake that I see students make. You must make sure that your equation is alcohol = alcohol (not alcohol = %)

 

3) Solve the system of equations using method of your choice (elimination or substitution)

I choose to solve the system using substitution.

Here are our two equations again

 

x + y = 20

.25x + .35y = .32(20)

 

Start by solving the first equation for one of the variables so we can substitute it into the second equation. Solving the first equation for y, we get:

 

y = 20 - x

 

Substituting in 20-x for y in the second equation we get:

 

.25x + .35(20-x) = .32(20)

 

Simplify and solve for x.

 

.25x + .35(20) + .35(-x) = .32(20)
.25x + 7 - .35x = 6.4

-.1x + 7 = 6.4

-.1x = -.6

x = 6

 

Since we defined x to be the amount of the 25% alcohol, that means 6 gallons of the 25% alcohol were used in the mixture. This means the remaining amount represents the 35% alcohol. To find the amount of the 35% alcohol we can just subtract 6 from 20, which is 14. So,

 

y = 14

This means the amount of the 35% alcohol is 14 gallons, while the amount of the 25% alcohol is 6 gallons.

To check our answer, plug x and y into the second equation to make sure they work.

 

.025x + .35y = .25(6) + .35(14) = 1.5 + 4.9 = 6.4√

.32(20) = 6.4√

 

Answer:

 

6 gallons of the 25% solution

14 gallons of the 35% solution