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:
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:
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:
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:
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|
.20x + .70y = .50(20)
Multiply both sides of this equation by 10 to clear the decimals:
This is the second equation we will use.
Now solve the system of equations
Multiply the second equation by -2, then add the two equations together:
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:
So 8 ounces of the 20% acid solution will be needed.