# Solving Mixture Problems

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Mixture problems are word problems where items or quantities of different values are mixed together.

We recommend using a table to organize your information for mixture problems. Using a table allows you to think of one number at a time instead of trying to handle the whole mixture problem at once.

### Replacing The Solution

Example:

A tank has a capacity of 10 gallons. When it is full, it contains 15% alcohol. How many gallons must be replaced by an 80% alcohol solution to give 10 gallons of 70% solution?

Solution:

Step 1: Set up a table for alcohol. The alcohol is replaced i.e. removed and added.

 original removed added result concentration amount

Step 2: Fill in the table with information given in the question.

A tank has a capacity of 10 gallons. When it is full, it contains 15% alcohol. How many gallons must be replaced by an 80% alcohol solution to give 10 gallons of 70% solution?

Change all the percent to decimals.

Let x = amount of alcohol solution replaced.

 original removed added result concentration 0.15 0.15 0.8 0.7 amount 10 x x 10

Step 3: Multiply down each column.

 original removed added result concentration 0.15 0.15 0.8 0.7 amount 10 x x 10 multiply 0.15 × 10 0.15 × x 0.8 × x 0.7 × 10

Step 4: Since the alcohol solution is replaced, we need to subtract and add.

original – removed + added = result
0.15 × 10 – 0.15 × x + 0.8 × x = 0.7 × 10
1.5 – 0.15x + 0.8x = 7

Isolate variable x
0.8x – 0.15x = 7 – 1.5
0.65x = 5.5

Answer: 8.46 gallons of alcohol solution needs to be replaced.

### Videos

Mixture Problems
Some word problems using systems of equations involve mixing two quantities with different prices. To solve mixture problems, knowledge of solving systems of equations. is necessary. Most often, these problems will have two variables, but more advanced problems have systems of equations with three variables. Other types of word problems using systems of equations include rate word problems and work word problems.

Solving a Mixture Problem using a system of equations.
We set up and solve a mixture problem using a system of equations with two variables. Before solving the problem, a short introduction to what a -solution- with talking about a chemical mixture.
Example: A chemist mixes a 12% acid solution with a 20% acid solution to make 300 milliliters of an 18% acid solution. How many milliliters of each solution does the chemist use?
Example: A grocer sells peanuts for \$3.00 per pound and a mix of fancy nuts for \$5.60 per pound, he wants to make a 20lbs of a new mix that will sell for \$4.00 per pound. How many of each does he need?

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