SOLUTION: Chemistry. A chemist has two solutions: one containing 40% alcohol and another containing 70% alcohol. How much of each should be used to obtain 80 liters of a 49% solution? Im

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Question 235629: Chemistry. A chemist has two solutions: one containing 40% alcohol and another containing 70% alcohol. How much of each should be used to obtain 80 liters of a 49% solution?
Im trying to find a way to solve this using matrices. I'm really unsure about how to go around solving this problem. Thanks
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
The solution that you want will equal 80 liters.

The amount of alcohol that you want in this solution would therefore be .49*80 = 39.2 liters of alcohol.

Let x = the number of liters in the first solution that you have.

Let y = the number of liters in the second solution that you have.

The amount of alcohol in the first solution that you have is therefore .4*x

The amount of alcohol in the second solution that you have is therefore .7*y

You will mix these together to get 39.2 liters of alcohol which represents .49 of your new solution.

We have an equation that says:

.4*x + .7*y = 39.2

We have another equation that says that:

x + y = 80

The first equation tells us how much alcohol is in the new solution.

The second equation tells us how much total solution is in the new solution.

This has to be since the 80 liters are coming from x and y only.

These equations need to be solved simultaneously.

The equations are:

.4*x + .7*y = 39.2
x + y = 80

We can solve by substitution or by elimination.

You can also solve using matrix algebra if you wish.

The equations should be in the form you can use to do that.

I'll use substitution.

Since x + y = 80, I'll solve for y to get:

y = 80 - x

I'll substitute in the equation:

.4*x + .7*y = 39.2 to get:

.4*x + .7*(80-x) = 39.2

Expand parentheses to get:

.4*x + .7*80 - .7*x = 39.2

Simplify to get:

.4*x + 56 - .7*x = 39.2

Combine like terms to get:

-.3*x + 56 = 39.2

Subtract 56 from both sides of the equation to get:

-.3*x = 39.2 - 56 = -16.8

Divide both sides by -.3 to get:

x = 56

Since x = 56 and x + y = 80, this means that:

y + 56 = 80 which means that y = 24

We have:

x = 56
y = 24

Plug these values into the original equations and see if they work out.

The original equations are:

.4*x + .7*y = 39.2
x + y = 80

replace x with 56 and y with 24 to get:

.4*56 + .7*24 = 39.2
56 + 24 = 80

The second equation becomes 80 = 80 so that one is good.

The first equation becomes:

22.4 + 16.8 = 39.2 which becomes:

39.2 = 39.2 so the first part is good two.

The answer to your probem is:

You need 56 liters of the .4 solution and 24 liters of the .7 solution to make 80 liters of a .49 solution.