Section 1.2
Mixture Examples
Example 1
Ex: A nurse has 1 liter of a solution that is 20% alcohol. How much pure alcohol must she add to bring
the solution up to 25% concentration?
The easiest way to do this is to make a chart.
Original Solution
Pure Alcohol
New Solution
Volume
1L
x
Concentration
0.20
1.0
Amount of Alcohol
0.20
x
The important thing to remember here is that Volume times Concentration equals the Amount of
alcohol. We make a row for our original solution. It’s volume is 1L, concentration is 0.20, and amount of
alcohol is 1 * 0.20, or 0.20. Then we’re going to add pure alcohol. That’s going to be x. It’s pure alcohol,
so it’s a concentration of 1.0. The amount of alcohol is x * 1.0 = x. Now we need to find what goes in the
row for new solution. Think about it this way: we had 1L of our original solution. We’re adding volume to
it, so how much volume are we adding? We’re adding x liters of pure alcohol. Our new volume will be
1 + x liters.
Original Solution
Pure Alcohol
New Solution
Volume
1L
x
1+x
Concentration
0.20
1.0
Amount of Alcohol
0.20
x
Concentration
0.20
1.0
0.25
Amount of Alcohol
0.20
x
We want the new solution to have concentration 25%.
Original Solution
Pure Alcohol
New Solution
Volume
1L
x
1+x
To get the amount of alcohol in the new solution, we’re going to multiply volume by concentration.
Original Solution
Pure Alcohol
New Solution
Volume
1L
x
1+x
Concentration
0.20
1.0
0.25
Amount of Alcohol
0.20
x
0.25(1 + x)
There’s also another way to get amount of alcohol. We had 0.20 Liters of alcohol to begin with, we
added x liters of alcohol. So, overall we have 0.20 + x.
Original Solution
Pure Alcohol
New Solution
Volume
1L
x
1+x
Concentration
0.20
1.0
0.25
Amount of Alcohol
0.20
x
0.25(1 + x)
0.20 + x
Notice in our last box, where new solution and amount of alcohol intersect, we have two ways of writing
that information. 0.25(1 + x) from multiplying volume by concentration looking down the row and
0.20 + x from adding down the amount of alcohol column. Well, if it’s the amount of alcohol in the new
solution, they have to be equal to each other so we equate the formulas and solve.
0.20 + x = 0.25(1 + x)
Distribute on the right-hand side:
0.20 + x = 0.25 + 0.25x
Now we can move all our x’s to the left-hand side. We do that by subtracting 0.25x.
0.20 + x – 0.25x = 0.25 + 0.25x – 0.25x
0.20 + 0.75x = 0.25
Let’s move the 0.20 to the right-hand side. So subtract 0.20 from both sides.
0.75x + 0.20 – 0.20 = 0.25 – 0.20
0.75x = 0.05
So x = 0.05 divided by 0.75.
0.75x = 0.05
0.75 0.75
x = 0.06666
We need to add 0.67L of pure alcohol to our original solution to obtain a solution that has 25% alcohol.