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Mixture problems are commonly considered to be those in which various types of ingredients are mixed together and the ingredients have different concentrations of some key items (like salt, gold, copper, etc.)
The point of many of the applications that you study at this level is to come up with a process that will be reliable when the number of ingredients an concentrations gets larger. The answer is not anywhere near as important as the process of getting there. So, here we will concentrate on process.
Example 1: A dietician at General Hospital wants a wants a patient to have a meal that has 70 grams (gr) of protein, 100 grams of carbohydrates, and 800 milligrams (mg) of calcium. The hospital only has three food items on the menu - chicken, potatoes, and milk. Each serving of chicken has 30 gr of protein, 35 gr of carbs, and 200 mg of calcium. Each serving of potatoes has 4 gr of protein, 31 gr of carbs, and 12 mg of calcium. Each serving of milk has 9 gr of protein, 13 gr of carbs, and 225 mg of calcium. How many servings of each food should the dietician provide for the patient?
Step 1: Identify what you are trying to find and use that to define your variables (unknowns). There will often be an explicit statement in the problem that will help you here. In this problem notice that the very last sentence asks How many servings of each food should the dietician provide for the patient? So, we are looking for the number of servings of chicken, the number of servings of potatoes, and the number of servings of milk.
Define the unknowns.
Let
x represent the number of servings of chicken
y represent the number of servings of potatoes
z represent the number of servings of milk
Step 2: Identify the constraints. That is, what must be accomplished? The problem helps with that too. It states that the dietician wants a patient to have a meal that has 70 grams (gr) of protein, 100 grams of carbohydrates, and 800 milligrams (mg) of calcium. So we will write three equations that describe the amount of protein, carbolhydrates, and calcium.
Let's look at the protein. We must have a total of 70 gr of protein and it will come from the servings of chicken, potatoes, and milk. Specifically, each serving of chicken has
30 gr of protein, each serving of potatoes has
4 gr of protein, and each serving of milk has
9 gr of protein. The contraint equation for protein will be
30x + 4y + 9z = 70
Now look at the carbs. We must have a total of 100 gr of carbs and it will come from the servings of chicken, potatoes, and milk. Specifically, each serving of chicken has
35 gr of protein, each serving of potatoes has
31 gr of protein, and each serving of milk has
13 gr of protein. The contraint equation for carbohydrates will be
35x + 31y + 13z = 100
Finally, must have a total of 800 mg of calcium and it will come from the servings of chicken, potatoes, and milk. Specifically, each serving of chicken has
200 mg of calcium, each serving of potatoes has
12 mg of calcium, and each serving of milk has
225 mg of calcium. The contraint equation for calcium will be
200x + 12y + 225z = 800
Step 3: Use one of the methods discussed in your class to solve this sytem of linear equations and answer the original question.
30x + 4y + 9z = 70
35x + 31y + 13z = 100
200x + 12y + 225z = 800
Step 4: Answer the original question. The patient will need to receive 1.65 servings of chicken, .5 servings of potatoes, and 1.06 servings of milk.
Now using the first example as a template, let's try another problem that is set up in a similar manner. This problem comes to you from Sullivan & Sullivan's textbook, College Alegbra - Concepts through Functions.
Example 2: A Florida juice company completes preparation of its products by sterilizing, filling, and labeling bottles. Each case of orange juice requires 9 minutes for sterilizing, 6 minutes for filling, and 1 minute for labeling. Each case of lemon juice requires 10 minutes for sterilizing, 4 minutes for filling, and 2 minutes for labeling. Each case of lime juice requires 12 minutes for sterilizing, 4 minutes for filling, and 1 minute for labeling. If the company runs the sterilizing machine for 398 minutes, the filling machine for 164 minutes, and the labeling machine for 58 minutes, how many cases of each type of juice are prepared?
Step 1: Identify what you are trying to find and use that to define your variables. The last sentence contains the question how many cases of each type of juice are prepared?
Define the unknowns.
Let
x represent the number of cases of orange juice
y represent the number of cases of lemon juice
z represent the number of cases of lime juice
Step 2: Identify the constraints. The problem states that states that the company can only run the sterilizing machine for 398 minutes, the filling machine for 164 minutes, and the labeling machine for 58 minutes. So we will write three equations that describe the number of minutes allowed on each a machine.
| |
# min. per case of orange |
# min. per case of lemon |
# min. per case of lime |
Total minutes |
| Sterilizing |
9 |
10 |
12 |
398 |
| Filling |
6 |
4 |
4 |
164 |
| Labeling |
1 |
2 |
1 |
58 |
| Sterilizing: |
9x+10y+12z = 398 |
| Filling: |
6x+4y+4z =164 |
| Labeling: |
x+2y+z = 58 |
Step 3: Use one of the methods discussed in your class to solve this sytem of linear equations and answer the original question.
Step 4: Answer the original question. You can prepare 6 cases of orange juice, 20 cases of lemon juice, and 12 cases of lime juice.
© 2009 Jo Steig
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