How many gallons of a 90% alcohol solution and a 30% alcohol solution must be mixed to get 15 gallons of a 54% alcohol solution?

Question

Hey there! I've tried to solve this using systems of equations, but it didn't work. If you could, please explain it to me step by step

Answer 1

This is a classic *mixture problem* with a classic *rate formula* hidden inside it.

First remember that a percent can't stand alone -- it is always a percent OF something.

What are our unknowns?

Let x = quantity of 90 % solution

Let y = quantity of 30 % solution

Note we have tow unknowns and require two answers.

"Let" is a magic word. It means I am naming this concept with a single letter so I can calculate with it. Be sure to specify your unknown variables with a "let". And be specific; in this case it is quantity, measured in gallons.

Mixture problems always have a *quantity* and a *value*, which are different. Be clear and it will make sense.

For the *value*, we ask how much pure alcohol we have.

Strong solution: Rate 90 % times quantity x = ,90x pure alcohol

Weak solution: Rate 30 % times quantity y = .30y pure alcohol.

When we mix them together we have

quantity: x + y = 15 gallons Equation 1

value: .90x + .30y = .54(15)

(in pure alcohol)) Equation 2

(Note that the final result is 15 gallons of 54 % alcohol so the amount of pure alcohol in the result is .54(15) = 8.1)

Now we have two equations in two variables and we should be able to solve the system.

Multiply equation 2 by 10 to get whole number coefficients, and simplify

9x + 3y = 81

Divide that equation by the common factor of 3 to simplify

3x + y = 27 Equation 2B

x + y = 15 Copy of Equation 1 from above

Now we simply subtract the two equations to eliminate y.

2x + 0 = 12

Divide by 2

x = 6

Substitute x = 6 into Equation 1

6 + y = 15

y = 9

Therefore if we mix 6 gallons of 90 % solution plus 9 gallons of 30 % solution, we will have 15 gallons of 54% solution.

Most mixture problems can be solved easily by using this system. Make a quantity equation for total amount, and a value equation for cost, distance, solute, etc. Then solve.