AlgebraLab

Word Lesson: Mixtures

By D Saye

In order to solve problems involving mixture, it is necessary to
 
  • solve one variable equations involving: one step, two steps, multi-steps
  • analyze and understand the problem
  • write and solve an equation for the problem
 
Even though there are two different types of mixes, the process for solving them is the same.
 
  • dry mixture created from two differently priced components such as two types of candy or two types of nuts
  • liquid mixture of a specific strength made from two or more solutions having differing concentrations
 
Suppose the owner of a candy store mixes two types of candies. She decides to create a 20-pound mixture of raspberry-flavored gumdrops and cherry-flavored jelly beans. The gumdrops sell for 0.95perpoundandthejellybeanssellfor0.95 per pound and the jelly beans sell for 1.20 per pound. She plans to sell the mix for $1.10 per pound. How many pounds of each candy should she use in her mix?
 
First, since two quantities are to be mixed together to produce one mixture, we need to recognize that we will set up an equation that shows the following:
 

total cost of gumdrops plus total cost of jelly beans equals the total cost of mixture

To arrive at the equation, it is typically helpful to use a table illustrating the problem such as the following:
 
Type of Candy

Cost of Candy
(unit price)

Amount of Candy
(in pounds)

Total Cost
(in dollars)

gumdrops
jelly beans
mixture
 
The first column shows the types of candy involved, cost is displayed in the second column, amount of each type of candy is listed in the third column, and the fourth column is the product of each cost and each amount for each type of candy. The total cost of the mixture is found by multiplying the cost of each type candy times the amount of each type of candy used in the mixture. The total cost column will be used to write the equation.
 

The candy store owner knows that she wants the total amount of the mixture of candy to be 20 pounds. However, she does not know how many pounds of each type to mix. That is the objective of the problem. So in the “Amount of Candy” column we will use x to represent the amount of gumdrops. Then the “total pounds of candy minus x” will represent the amount of jelly beans: (20 – x). The last column demonstrates that the price of each type candy multiplied times the amount of each type candy represents the total cost of each type candy. The last column is what we use to write the equation.

 
Remember:
 

total cost of gumdrops plus total cost of jelly beans equals the total cost of mixture

So, using the information in the last column:
 
 
We are ready to solve the equation to find the amount of each type of candy the store owner should use in her mixture.
 
First distribute to remove parentheses
Now multiply the equation by 100
Solve for x
pounds
 

Since x represents the amount of gumdrops to be used in the mixture the candy store owner will use 8 pounds of gumdrops. From column three of the table, you can see that the amount of jelly beans to be used is (20 - x). Substituting 8 for the x, we see that the store owner needs to use (20 - 8) which is 12 pounds of jelly beans to create the desired mix.

 
We have now learned the candy store owner will mix 8 pounds of raspberry-flavored gumdrops that cost 0.95perpoundand12poundsofcherry−flavoredjellybeansthatcost0.95 per pound and 12 pounds of cherry-flavored jelly beans that cost 1.20 per pound to create a mixture of 20 pounds of candy that sells for $1.10 per pound.

Directions and/or Common Information:

No audio files were recorded for this set of examples.

A local grocer has decided to mix 100 pounds of cashews and almonds for a holiday special. Cashews typically cost 8perpoundandalmondscost8 per pound and almonds cost 3 per pound. How many pounds of each type of nut must he mix to obtain a mixture that will cost his customers $5 per pound?

A chemist needs a 40% solution of alcohol. He plans to mix 3 liters of a 60% solution with a 25% solution. How many liters of the 25% solution must we mix with the 3 liters of the 60% solution to obtain the desired 40% solution of alcohol?

Directions and/or Common Information:

No audio files were recorded for this set of examples.

The owner of a coffee shop has decided to mix types of teas to create a new blend. He will mix a type of tea that sells for 4perpoundwithatypethatsellsfor4 per pound with a type that sells for 2.40 per pound to produce 80 pounds of mixture that he will sell for 3.60perpound.Howmuchoftheteathatcosts3.60 per pound. How much of the tea that costs 2.40 per pound must he use in the mixture?

  1. 60 pounds
  2. 20 pounds
  3. 30.625 pounds
  4. -360 pounds

How many liters of a 92-octane gasoline should be mixed with 200 liters of a 98-octane gasoline to produce a mixture that is 96-octane gasoline?

This type of problem requires carefully setting up a table using the information given in the problem. Following the development of the table, a linear equation must be written and solved. Using the information written in the last column of the table, set up your equation in this format:
 

total amount of item one plus total amount of item two equals total amount of the mixture

 
After writing the equation and solving for the unknown variable, always check to be certain that the answer obtained answers the question and/or question(s) asked in the problem. It is always advisable to substitute your answers back into the equation to check that your solution is correct.