Mixture Problems

How many ounces of 20% hydrochloric acid solution and 70% hydrochloric acid solution must be mixed to obtain 20 ounces of 50% hydrochloric acid solution?

What are we trying to find in this problem?  We want to know the amount of 20% acid solution needed and we want to know the amount of 70% acid solution needed.  We'll need a variable to represent each of these unknowns:

x = ounces of 20% solution needed
y = ounces of 70% solution needed

Since we have two unknowns, we will have to write a system of two variables to solve for the unknowns.

To help us organize the information in the problem, let's imagine that we're in a laborotory looking at the two bottles of acid we're about to mix together.  One of the bottles is labeled "20%", and it has x ounces of liquid in it.  The other bottle is labled "70%", and it has y ounces of liquid in it.  Here's what the two bottles look like; the amount of acid in each bottle is indicated below the bottles:

 20%
x ounces
 70%
y ounces

If we combine the two bottles of acid, we'll create 20 ounces of 50% acid solution.  Combining (adding together) the two bottles of acid can be shown by adding to our picture:

 20%
x ounces
+
 70%
y ounces
=
 50%
20 ounces

The first of our equations will come from the amount of liquid in the bottles-- adding the liquid together in the two bottles will give us 20 ounces of solution:

x + y = 20

The second of our two equations will come from the amount of pure acid in each bottle.

In the first bottle, 20% of the x ounces of liquid is pure acid, so the amount of pure acid in the first bottle is .20x .
In the second bottle, 70% of the y ounces of liquid is pure acid, so the amount of pure acid in the second bottle is .70y .
In the combined mixture, 50% of the 20 ounces of liquid is pure acid, so the amount of pure acid in the combined mixture is .50(20) .
 pure acid in first bottle + pure acid in second bottle = pure acid in combined mixture
So
.20x + .70y = .50(20)
or
.20x + .70y = 10
Multiply both sides of this equation by 10 to clear the decimals:
2x + 7y = 100
This is the second equation we will use.

Now solve the system of equations

2x + 7y = 100
x + y = 20

Multiply the second equation by -2, then add the two equations together:

2x + 7y = 100
-2x - 2y = -40
5y = 60
y = 12

We will need to use 12 ounces of the 70% acid solution.

To find the amount of 20% acid solution needed, substitute 12 for the y in either equation; we'll use the simpler equation:

x + y = 20
x + 12 = 20
x = 8

So 8 ounces of the 20% acid solution will be needed.