What you should learn

To solve mixture problems

To solve problems involving uniform motion

NCTM Curriculm Standards 2, 4, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Weighted Average

Mixture Problem

Uniform Motion

Rate Problems

Introduction: In Mr. Calloway's American History class, semester grades are based on five unit exams, one semester exam, and a long-term project. Each unit exam is worth 15% of the grade, the semester exam is worth 20%, and the project is worth 5% of the grade.

Parker's scores for the semester are given in the table below. If 100% is a perfect score for each item, find Parker's average for the semester.

In order to find Parker's average for the semester, you may want to add the percentages adn then divide by the number of items, 7. But this assumes that each has the same weight. So, you need to find the weighted average of the scores.

Definition of Weighted Average: The weighted average M of a set of data is the sum of the product of each number in the set and its weight divided by the sum of all the weights.

 

You can find the weighted average for the application above by multiplying each score by the percentage of the semester grade that it represents and then dividing by the sum of the weights, or 100.

M = [15(79 + 83 + 96 + 91 + 89) + 20(90) + 5(95)]/100

= 8845/100 or 88.45

So, Parker's average for the semester is about 88%.

Mixture problems involve weighted averages. In a mixture problem, the weight is usually a price or a percentage of something.

Exercise 1: Suppose the Central Perk coffee shop sells a cup of espresso for $2.00 and a cup of cappuccino for $2.50. On Friday, Rachel sold 30 more cups of cappuccino than espresso, and she sold $178.50 worth of expresso and cappuccino. How many cups of each were sold?

Explore

Let e represent the number of cups of espresso sold. Then e + 30 represents the number of cups of cappuccino sold.

Plan

Make a chart of the information.

2e + 2.5(e + 30) = 178.50

2e + 2.5e + 75 = 178.50

4.5e + 75 = 178.50

4.5e = 103.50

e = 23

There were 23 cups of espresso sold. There were 23 + 30, or 53 cups of cappuccino sold.

 

 

Sometiems mixture problems are expressed in terms of percents.

Exercise 2: An advertisement for an orange drink claims that the drink contains 10% orange juice. Jamel needs 6 quarts of the drink to serve at a party, and he wants the drink to contain 40% orange juice. How much of the 10% drink and pure orange juice should Jamel mix to obtain 6 quarts of a mixture that contains 40% orange juice?

Motion problems are another application of weighted averages. When an object moves at a constand speed, or rate it is said to be in uniform motion. The formula d = rt is used to solve uniform motion or rate problems. In the formula, d represents distance, r represents rate, and t represents time. You can also use equations and charts when solving motion problems.

Exercise 3: On Friday, Shenae and her brother Rafiel went to visit their grandparents for the weekend. Luckily, traffic was light and they were able to make the 50-mile trip in exactly one hour. On Sunday, they weren't so lucky. The trip home took exactly two hours. What was their average speed for the round trip?

To find the average speed for each leg of the trip, rewrite d = rt as r = d/t.

Going

r = d/t = 50 miles/1 hour = 50 miles per hour

Returning

r = d/t = 50 miles/2 hours = 25 miles per hour

You may think that the average spped of the trip would be (50 + 25)/2 or 37.5 miles per hour. However, Shenae did not drive at these speeds for equal amounts of time. You can find the weighted average for their trip.

Round Trip

M = [50(1) + 25(2)]/3 = 100/3 or 33 1/3

Their average speed was 33 1/3 miles per hour.

Exercise 4: Two city buses leave their station at the same time, one heading east and the other heading west. The eastbound bus travels at 35 miles per hour, and the westbound bus travels at 45 miles per hour. In how many hours will they be 60 miles apart? (Hint-use problem solving techniques and what you learn earlier).

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 9 - 21 odd, 23 - 29

Alternative Homework: Enriched: 8 - 20 even, 21 - 29

Extra Practice: Students book page 766 Lesson 4-7

Extra Practice Worksheet: Click Here.

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