Overview:
Calculating the values for weighted averages and mixtures is common to many fields, from education to chemistry. The calculations for both require attention to the amount of each quantity and its relationship to the whole.
Weighted Averages:
Suppose a student takes 6 math tests with an average of 87. The final is worth 2 regular tests, and the score on that test is 95. What is their average for the class? In order to solve the problem, the average of 87 is weighted at 6/8 and the average of 95 is weighted at 2/8, and then they are added together. Using equivalent fractions, 6/8 is equal to .75, and .75 times 87 is 65.25, and 2/8 is equal to .25, and .25 times 95 is 23.75. The final average is 65.25 +23.75, or an average of 89.
Varying Rates:
Suppose a trip takes one speed going and another speed returning. The distance will be the same in both cases, but the percentage of time it takes will be different, and will also be a weighted averages type of problem. The distance between Metropolis and Gotham City is 240 miles. The Batmobile drove within the speed limit at 60 mph to get from Metropolis to Gotham City. However, traffic was bad on the way back, and the Batmobile could only go 40 mph. What is the average speed for the round trip? It took 4 hours to get there (because 240/60 is 4) and 6 hours to get back (because 240/40 is 6), for 10 hours total. Using weighted averages, 4/10 (.60) + 6/10(40) is equal to 48 miles per hour on average. (It would be much faster to fly.)
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Figure 1: Holy weighted averages, Batman! Will we ever get through this traffic? |
Mixtures:
Mixtures are solved in a similar method to weighted averages. Cologne is a solution of perfume oil diluted with alcohol. It is less expensive because the alcohol is much less expensive than the perfume oil. Suppose 5 ounces of perfume oil at 100.00 per ounce are mixed with 25 ounces of alcohol at 3.00 per ounce. How much is the resulting mixture worth per ounce? The total amount of cologne is 30 ounces, so 5/30 (100.00) + 25/30 (3.00) = 16.67 + 2.50, or about $19.17 per ounce.
Multiple Mixtures:
The weighted average of more than two quantities is similar to finding the weighted average of two quantities. The Candy Crunch Company is always looking for new confections. Suppose there is a mixture of 7 ounces of popcorn at 1.00 an ounce, 2 ounces of peanuts at 3.00 an ounce, and 1 ounce of chocolate chips at 2.00 an ounce. The resulting mixture of popcorn-nut-chocolate goodness will sell for .7(1.00) + .2(3.00) +.1(2.00), or 1.50 an ounce.
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Figure 2: The weighted average with more than two ingredients is similar to the weighted average with 2 ingredients. |
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