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Pre-Algebra
Unit Rates and Proportional Relationships
Learning Goal: We will compute unit rates and identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams and verbal descriptions of proportional relationships.
Vocabulary:
ü Unit rate – compares a quantity to one. Unit rates can be determined from proportional graphs, tables, equations, and verbal descriptions.
Example 1: Unit Rates and Proportional Graphs.
Myrtle drives the same number of miles to and from work each day, as shown on the graph. Based on the graph, what is the unit rate of miles driven per day?
ON YOUR OWN:
Adam bought a new video game. He graphed the points he earned. What is his rate of points per minute?
Example 2: Unit Rates and Tables
The table shows the cost of grapes in the produce aisle at the grocery store. Based on the table, what is the price per pound of grapes?
ON YOUR OWN:
An oval-shaped walking path at a local park is ¾ of a mile long. Four walkers recorded the number of laps they walked and the time it took them in the table.
What is the unit rate for each walker?
Example 3: Unit Rates and Equations
Ty earns a certain amount of money per hour at his job. The equation below shows how much money he earned last week in h hours.
$12h=$324
What is the unit rate in the equation above?
Example 4: Unit Rates and Verbal Descriptions
A pudding recipe requires 23 of a cup of milk for every 134 cups of sugar. What is the unit rate of sugar to milk in the pudding recipe?
ON YOUR OWN:
A bakery uses 1025 ounces of icing for every 14 of a cake. What is the unit rate in ounces of icing per cake?
HOMEWORK:
1. The graph represents a parrot’s heartbeat.
Based on the graph, what does the point (1, 300) represent?
2. Fill in the box. The cost to manufacture the KZR 250 model of motorcycle at the Kawahama factory is shown in the table below. Using the information in the table, determine the constant of proportionality (unit rate), and use it to complete the table.
3. Stefanie is painting her bedroom. She can paint 1213 square feet in 25 of an hour. How many square feet can she paint in one hour?
4. A punch recipe requires 35 of a cup of pineapple juice for every 212 cups of soda. What is the unit rate of soda to pineapple juice in the punch?
5. A blueprint for a house shows the height of the house to be 318 inches. The actual house is 2113 feet tall. What is the unit rate in feet per inch?
6. At Clean Cut Manufacturing, Jethro made a graph representing the number of riding lawn mowers manufactured, y, based on the number of hours the assembly line is operating, x. The graph is a straight line that passes through the origin and the point (1, 13). Which of the following graphs and statements fit the situation?
a) / b)c) The assembly line manufactures 13 riding lawn mowers per hour of operation.
d) It takes 13 hours of operation for the assembly line to manufacture each riding lawn mower.
e) The y-intercept of the graph is 13.
f) The slope of the graph is 1.
7. Kaitlyn is interviewing applicants for an open job position in her department. During the interview process, applicants are required to type as much of a given handwritten document as possible in five minutes. The results for the three applicants are shown below. What is the unit rate, in pages typed per hour, for each applicant?
8. On a trip to Italy, Chandra traded her U.S. dollars for Euros, based on the graph. Based on the graph, what is the unit rate of Euros per dollar?
9. At West Painting, they get about three calls a day asking for an estimate of the cost for having the interior of a house painted. To write up an estimate for the cost of a job, they need to know how much paint a job will take. If they average painting 34 of a room with 25 of a gallon of paint, they can paint, on average, how many rooms per gallon?
10. Blake has entered a mountain bike competition at Starvation Flats Bike Park. Placement in the competition is determined by a person’s average speed over four laps around the course. The course itself is 25 of a mile long. If Blake completed the four laps in 12 minutes, then his average speed around the track is how many miles per hour?
11. Tanya is training a turtle for a turtle race. For every 16 of an hour that the turtle is crawling, he can travel 17 of a mile. At what unit rate is the turtle crawling?
Proportional Relationships
Learning Goal:
· We will explain what a point on the graph of a proportional relationship means and determine whether two quantities are proportionally related.
· We will represent proportional relationships by equations.
A(directly) proportional relationship exists between two variables, x and y, if there is a nonzero constant, m, such that y=mx.· The equation y=mx is a linear equation, which means its graph is
a ______.
· When x=0, y=______.
· The graph of a proportional relationship contains the point ______.
· By solving for m, it can be seen m=yx.
· m is called the constant of proportionality and/or the unit rate.
Example 1: Graphs of Proportional Relationships
Which of the following graphs represents a proportional relationship?
ON YOUR OWN:
Which of the following graphs represents a proportional relationship?
Example 2: Tables of Proportional Relationships
Determine if the following table represents a proportional relationship.
ON YOUR OWN:
Does the table below represent a proportional relationship?
Vocabulary:
ü Proportional relationship – a linear relationship whose graph is a straight line passing through the origin. The equation for a proportional relationship is of the form y=mx, where m is the unit rate.
Example 3:
A small package delivery company delivers the same number of packages each day, as shown on the graph below.
What does the point (2, 70) represent?
ON YOUR OWN:
A high school band is selling tickets to its concert. The ticket prices are shown in the graph below.
Based on the graph, what does the point (4, 40) represent?
Example 4:
A line passing through which of the following pairs of coordinates represents a proportional relationship?
A. (1, 3) and (1, 6)
B. (2, 3) and (4, 6)
C. (1, 2) and (3, 4)
D. (3, 1) and (9, 6)
ON YOUR OWN:
A line passing through which of the following pairs of coordinates represents a proportional relationship?
Example 5:
The total amount of money, M, that a phonoe store earns each day in profits is proportional to the number of phones, p, that the store sells per hour, h. Which equation represents the relationship between the total amount the store earns and the number of phones sold?
ON YOUR OWN:
The total amount of money, M, that Bob earns each week at his job is proportional to the wages, w, that he earns per hour for working h hours. Which equation represents the relationship between the total amount Bob earns and hours worked?
HOMEWORK:
1. Steve tutors his fellow math students. Last week, he tutored 6 hours and earned $120. What is the unit rate relative to his earnings?
2. Sandra sets the cruise control in her car to 72 miles per hour when she goes on road trips. Which equation can be used to find how many hours, h, it will take her to drive m miles at her constant speed?
3. A class is going on a field trip to see a movie. Each ticket is $6.00. The number of students, n, who will receive a $6.00 ticket is proportional to the cost, C, of all the tickets. Which equation could be used to find the total cost of all the tickets?
4. Philip’s Soup Kitchen charges the same price, $8.95, for a bowl of any of their soups. The Ragdale Family Reunion was being held at a park across the street from Philip’s Soup Kitchen. For lunch at the reunion, people had a choice of a bologna sandwich or a bowl of soup from Philip’s Soup Kitchen. If R people ended up ordering a bowl of soup for lunch, then what equation can be used to determine the total price, P, that was charged for the soup?
5. Which of the following tables represents a proportional relationship?
6. A group of friends went bowling after school. The table below shows the number of games each person bowled and the number of strikes they made during those games.
Each person’s relation of games bowled to strikes is shown on the coordinate plane below. Determine which of the friends had the same ratio of games bowled to strikes, and select their corresponding points on the coordinate plane.
7. The graph below shows how many miles an athlete biked per day.
Based on the graph, what does the point (2.5, 50) represent?
8. Does the table below represent a proportional relationship?
9. The amount Kathleen has to pay for a doctor visit and the amount the insurance company pays are in a proportional relationship. Based on the graph, what does the point (20, 30) represent?