MPM 1DI Ratios and Rates Lesson 8.1

A Ratio:

A Ratio can be written in 3 ways:

Examples

1. Our class has _____________ male and ____________female students.

Note: the order of the numbers is the same as the order of the words!

2. A parking lot has 6 motorcycles, 21 cars, and 7 trucks.

a) What is the ratio of cars to truck?

b) What is the ratio of trucks to motorcycles?

c) What is the ratio of cars to motorcycles?

d) What is the ratio of trucks to cars to motorcycles?

e) What is the ratio of cars to trucks to total

vehicles?

Note: Remember to reduce ratios!

3. a) Write the ratio 12:24 in lowest terms ___________________

b) Write the ratio in 2 different ways: __________________ and _________________

c) If the “12” in part 1 represents the number of apples, and the 24 represents the number of oranges, then

complete this statement…

for every _________________________, there are _________________

d) Write the ratio 4:8:24 in lowest terms _________________

4. Use box to write the following ratios. Reduce where needed!

a) Find the ratio of X’s to O’s

c) Find the ratio of X’s to =’s to O’s

b) Find the ratio of O’s to =’s

d) Find the ratio of O’s to the total number of symbols in the box

XOOX=OO=XOX

=XOOXXXO=X=

OOX=X=O=OOO

A Rate:

A Rate is written:

Examples: Calculate the required rates for the following: (Remember to use units.)

1. Rate of Speed: Terry travels 120km in 2 hours.

What is his average rate of speed?

2. Rate of Pay: Beth earns $56.40 for working an 8

hour shift. What is her rate of pay?

3. The glaciers have melted 35cm in the past 3 years.

What is the rate of melt?

4. Sally walks 7 days a week. This week she walked

a total of 24.5km.

5. Sue gets paid by the hour. She earned $328.80 in

40 hours.

6. The plane flew a distance of 3000km in 7 hours.

7. I can eat 120 jellybeans in 25 minutes.

8. A box of 54 cookies has to feed 9 people

(equally).

9. Use the box to answer the following questions. Write the ratio of:

Count:

total number of letters _____

number of vowels _______

number of consonants _____

a) vowels : consonants

b) total letters : consonants : vowels

c) CAPITAL LETTERS : total letters

d) CAPITAL LETTERS : lower case

letters

e) “For every ONE CAPITAL letter in the sentence above,

there are ________ lower case letters.”

f) My friend can type 2000 words in 50

minutes. Find out her rate of typing.

g) A hockey player makes $1,000,000 a year, and plays 72

games. Find his rate of pay per game.

“My favorite topic

to learn about in math

is RATIO!”

MPM 1DI Direct and Partial Variation

Lesson 8.3

Scenario 1: Tasty Tacos

You begin working at Tasty Tacos with the pay schedule listed in the

table:

1. Fill in the table of values.

2. a) Use 1st differences to determine if this is linear or non-linear?

Linear? Non-Linear?

b) What type of relationship exists in the data points?

3. Graph the data.

4. Choose 2 points on your graph and calculate the slope of

the line.

5. What do you think the slope means in the context of the

problem?

6. From the graph what is the y-intercept. What do you think the y-intercept represents in the context of the

problem?

7. Using the information from question 4 and 6 write an equation that describes the relationship between the

years (x) and the earnings per hour (y).

8. Can you use your equation to calculate what the earnings per hour be after working 15 years?

10. Can you predict after how many years of service would you earn $9.65 per hour?

Years of

Service

Earnings

per hour

1st

Difference

2 $6.85

4 $7.55

6 $8.25

Scenario 2: At the Carnival

Mark is going to a carnival. He is anxiously looking forward to riding as many rides as possible. The tickets

for the rides are $2.50 each.

1. Make a table that shows the number of rides (x) and the total cost

with admission (y) for Mark’s trip to the carnival for 0 to 4 rides.

2. What type of relationship exists in the data points?

2. Graph the data.

3. Choose 2 points on your graph and calculate the slope of

the line.

4. From the graph find the y-intercept.

5. Write an equation that describes the relationship between

the number of rides (x) and the total cost (y).

6. Use the equation to determine the 7. If Mark’s grandmother gave him $50,

cost of 100 tickets. how many rides could he ride?

Conclusions:

There are 2 types of linear relations: ______________________ and _________________________.

Partial Variation: Direct Variation:

Every equation is written:

Rides (x) Cost (y) 1st

Difference

0

4

8

12

16

MPM 1DI Slope as a Rate Lesson 8.2

Problem: Billy earns $10.00/h at his job. He would like to see how time affects his earnings.

Problem: The table shows the distances traveled by an albatross gliding at a constant speed. Can we find

a rate?

Time (s) Distance (m)

1 15

2 30

3 45

4 60

5 75

Problem: The following table shows the cost of a banquet. Can we find a rate for this problem too?

Number

of People Cost ($)

10 700

20 1200

30 1700

40 2200

50 2700

New Definitions:

Independent Variable: Partial Variation:

Dependent Variable: Direct Variation:

Conclusion: The slope of a line tells us __________________________________________. If we find the

slope we can find the ________________________________ over that period.

This graph shows how John's savings account balance

has changed over the course of a year.

John wants to figure out how much he was saving per

month.

Note: We can't count the rise over the run like we did

before because _____________________________

___________________. In most real life problems

the units will not be the same on the x and y axis.

Note: There is a direct relation between __________________________________and the _________ of its

lines.

Study the graph to the left. Two students are on their way to

school.

1. For what values of time are Carlos and Maria stopped?

2. For what values of t do they increase their speed?

3. Which of these speeds increase at the same

rate?

4. Explain how this question relates to the slope

of the lines?

5. For what values of t do Carlos and Maria

decrease their speed?

6. At what time do they achieve their highest

speed?

7. For what values of t do Carlos and Maria

drive at a constant rate?

MPM 1DI Equations for Linear Relations I Lesson 8.4

Before you start remember to write ________________________________________________.

There are 2 types of linear relations: ______________________ and _________________________.

Partial Variation: Direct Variation:

Every equation is written:

Examples: Find an equation to model each situation.

a) To make a home visit and fix the sink a plumber charges a flat fee of $80.00 and an hourly rate of

$25/hour.

b) A shop employee earns $12.50/hour.

c) A college student needs to withdraw $150 from his account each month for rent. He has $1200 at the start

of the term.

d) A taxi company offers $2.50/km with no other fees.

e) A shop will print business cards for $30 set up and $0.25.

Type A: Questions with the Equation Given

A car is expected to decrease in value according to 240002000 tV where V is value of the car and t is

measured in years.

a) What is the starting value of the car? b) At what rate does the value of the car decrease?

c) Graph the relationship. What kind of slope does

the graph have?

Notes for Type A: (Equation given)

Type B: Creating Your Own Equation Using Rate and Initial Value

A car rental company rents cars based on a fixed daily charge and a cost of $0.50/km. The cost of renting a

car for a day is $50.00.

a) Which quantity is the independent variable? b) What is the rate of change in this question?

c) Which quantity is the dependent variable?

d) Write an equation for the rental cost in terms of

distance driven.

e) Graph the relationship.

f) What is the cost of renting a car for a day and driving it i) 50 km? ii) 100 km? iii) 500 km?

Notes for Type B: (Slope & y-intercept given)

MPM 1DI Equations for Linear Relations II Lesson 8.5

Notes: Finding Equations:

a) Given slope b) Given slope and a point c) Given 2 points

& y-intercept

Type C: Creating Your Own Equation Using Rate and a Point

A limousine rental company rents cars based on a fixed charge and a cost of $45/h. The cost of renting a car

for a day and driving it for 3 hours is $200.00.

a) Which quantity is the independent variable? b) What is the rate of change in this question?

c) Write an equation for the rental cost in terms of

time rented.

d) Graph the relationship.

e) What is the cost of renting a car for a day and

renting it i) 5 hours? ii) 1 hour? iii) 10 hours?

Notes for Type B: (Slope & y-intercept given)

Type D: Creating Your Own Equation Using Two Points

A different limo rental company rents cars based on a fixed daily charge and a cost per hour as well. They

give the information as follows. The cost of renting a car for a day and driving it for 2.5 hours is $200.00 and

the cost of renting a car for a day and driving it 4 hours is $290.00.

a) What is the cost per hour? b) What is the daily charge?

c) Write an equation for the rental cost in terms of

time rented.

d) Graph the relationship.

e) What is the cost of renting a car for a day and

renting it i) 5 hours? ii) 1 hour? iii) 10 hours

Notes for Type D: (2 points given)

MPM 1DI Intersection of 2 Lines

Lesson 8.6

Find the place where 2 lines meet is called ________________________________________.

The point itself is called the ______________________________________________.

Types of solutions:

Examples: Find the point of intersection for:

a) 5

3

2

5

xy

xy

b) 43

14

xy

xy

c) 052

22

yx

xy

Application:

Two city parks are trying to get new customers. The first park, called Centre-City Fun, offers rides at $5.00

each and no entrance fee. The second park, Outer-Spaces , offers rides for $2.00 each with an entrance fee of

$30.00.

i) Write “let statements” and an equation for each

situation.

ii) Graph both equations on the same set of axis.

iii) Which scenario is partial variation and which is

direct?

iv) Find the point of intersection. What does it mean?

200

180

160

140

120

100

80

60

40

20

50 100 150 200 250 300

F

E

DC

A

B

MPM 1DI Graphs and Stories

Lesson 8.7

A graph of distance vs. time can be used to represent a real-life situation. The rate of change in this

relationship is _____________.

Consider the following graph, which represents Aaron the deliveryman’s drive to drop off a package.

Questions:

1) Why are some line segments on the graph steeper

than others?

2) What happens at point B?

3) Over what interval of time is Aaron moving the

fastest? How fast is he going?

4) At what point does Aaron reach the destination?

5) How long did it take to reach the destination?

6) What does the interval from C to D represent?

7) How long did it take Aaron to get back to

headquarters?

8) What was the total time taken to complete the

trip?

9) What was the total travelled distance?

10) What was Aaron’s average speed?

Distance from

headquarters (km)

Time (min)

Maria:

I started to ride my bike to school when I realized that I had left my lunch in the refrigerator. I went back

home and picked up my lunch and drove quickly to school.

Dimitri:

I was anxious to get to school early so I drove very fast but I hit a curb and bent the front wheel. I then had to

walk with my bike to school but I was still on time for my first class.

Chan:

I enjoyed my ride to school this morning and was taking my time, but then I realized how late it was getting

so I had to drive more quickly in order not to be late.

Olga:

Notes:

● If the graph is linear, then the velocity is constant.

● In a non-linear relationship, the velocity changes over time.

● If the velocity is increasing, then there is acceleration.

● If the velocity is decreasing, then there is deceleration.

● A difference table can be used to check the linearity to determine if there is constant velocity,

acceleration, or deceleration.

Review Intersection of Two Lines

1. For the following pairs of lines graph and find the point of

intersection.

a) 92

3

xy b) 13 xy

1 xy 7y

Questions with the Equation Given

2. The cost C, in dollars, of producing music CDs privately is given by C = 2n + 20, where n is the number of

CDs.

i) What is the slope? What does it mean?

ii) What is the y-intercept? What does it mean?

iii) Graph the relation. iv) Is this Direct or Partial Variation? How can you tell?

v) How much would it cost to make 50 CDs?

3. a) A plumber from “Drain All” b) A second plumber from “Leaking Pipes”

charges $60/hour. charges $55 for the visit plus $45/hour.

i) What is the slope? i) What is the slope?

ii) What is the y-intercept? ii) What is the y-intercept?

iii) Write “let statements” and iii) Write “let statements” and

an equation for this relation. an equation for this relation.

iv) Is this Direct or Partial Variation? iv) Is this Direct or Partial Variation?

4. a) For the following graph write a plausible story to describe the different parts of the graph.

5. A tennis court charges a base fee plus 2.50/person. The total

fee for 4 players was $20.00.

i) What is the rate of change?

ii) Find the y-intercept. What does the y-intercept represent?

iii) Write “let statements” and an equation. iv) Graph the relation.

v) Is this relation Direct or Partial Variation?

6. The cost of renting a car depends on the distance driven and

a fixed cost for a deposit. The cost for 250km is $65.00. The

cost for 350km is $75.00.

i) Find the cost per km.

ii) Find the deposit cost.

iii) Graph the relation

iv) Write “let statements” and an equation.

Distance

from

home (m)

Time

7. Two gyms stores are competing with each other.

One gym, Family Fitness, offers videos at $10.00 each

visit and

no membership fee. The second gym, Fitness and More ,

offers videos for $4.00 each visit with a membership fee

of $120.00.

i) Write “let statements” and an equation for each

situation.

ii) Graph both equations on the same set of axis.

iii) Find the point of intersection. iv) Explain when each company will be cheaper.

8. a) For the following information make a scatter plot.

b) Draw a line of best

fit.

c) Write the equation for your line

of best fit.

d) If a person studied 5 hours what would you expect the mark to be?

Hours of

Study

Exam

Mark

4 88

1 71

4 87

1 80

0 68

2 76

3 70

0 58

2 84

3 90

3 93

1 74

MPM 1DI Applications of Linear Functions Questions

Part A: Finding the Equations of Linear Relations

Type A: Questions with the Equation Given:

For all the following questions: state if the relation is Partial Variation or Direct Variation.

1. A car is expected to decrease it value according to y=-1360x + 17000 where y is the value of the car after x

years.

a) Find the slope of the line and interpret its meaning.

b) Find the y-intercept and interpret its meaning.

c) Graph the relation.

2. Mohammed makes bicycle tires and is paid according to y= 1.25x, where y is his earnings and x is the

number of tires he makes.

a) Find the slope of the line and interpret its meaning.

b) Find the y-intercept and interpret its meaning.

c) Graph the relation.

3. Jackie sells cars. Her pay is based on the equation y = 0.03x + 350, where y is her weekly earnings and x is

the total value of cars she sells in a week.

a) Find the slope of the line and interpret its meaning.

b) Find the y-intercept and interpret its meaning.

c) How much will Jackie earn if she sells $95 000 worth of cars?

d) Graph the relation.

4. A house is expected to increase in value according to y = 7500x + 125 000, where y is the value of the

house after x years.

a) Find the slope of the line and interpret its meaning.

b) Find the y-intercept and interpret its meaning.

c) Find the value of the house after seven years.

d) Graph the relation.

5. The price, P, of a pizza is given by P = 0.75t + 9, where t is the number of toppings and P is the price of a

pizza without any toppings.

a) Make a table of values using values of t from 0 to 5.

b) Draw the graph. Describe any trends,

c) What is the price of a pizza with 6 toppings?

d) Is it possible to buy a pizza with 2.5 toppings? Explain.

6. The cost, C in dollars, of producing brochures is given by C = 2n + 8 where n is the number of brochures

and 8 is the fixed cost. Fixed costs, such as design, are independent of the number of brochures.

a) Choose values of the variables and draw a graph. Describe the graph.

b) Create two problems based on the graph. Solve your problems.

c) State the slope.

7. For an experiment, a beaker of liquid is heated and its temperature is recorded over time. It is found that

the liquid’s temperature, T in degrees Celsius, is given by T = 2.5t, where t is the time in hours.

a) Choose values for the variables and draw a graph. Describe the graph.

b) Create two problems based on the graph. Solve your problems.

Type B: Questions with the Slope and Y-intercept Given:

For all the following questions: i) write “let” statements

ii) write an equation

iii) graph the relation

iv) state if the relation is Partial Variation or Direct Variation.

1. To fix a car, Sally’s garage charges a flat fee of $20, plus $40/h. How much does it cost for 3 hours work?

2. An author earns a royalty of $0.25 for each book sold. How much does the author make if 2500 books are

sold?

3. A small rocket is launched from a hill 1500m above sea level. It rises at 35 m/s. how long does it take for

the rocket to reach 2375m?

4. Marlon has $1900 in his bank account. He takes $100 out of his account each week. At this rate when will

his account be empty?

5. Create a problem with a relationship for the equation xy 14 . (Make a question to go with your relation.)

6. 5. Create a problem with a relationship for the equation 2510 xy . (Make a question to go with your

relation.)

Type C: Questions with the Slope and a Point Given:

1. Autoloaners rent cars based on a fixed daily charge and a cost of $0.55/km. the cost of renting a car for one

day and driving it for 250km is $187.50.

a) Which variable is the dependent variable?

b) What is the rate of change n this situation?

c) Write an equation for the rental cost in terms of the distance driven.

d) Graph the equation.

e) What is the cost of renting a car for one day if it is driven 50 km? 130 km? 500 km?

2. A clown will come to children’s parties for a base fee plus $5.25 per child. The total fee for a party with

ten children is $102.50.

a) What is the rate of change in this situation?

b) Write the equation for the cost of hiring the clown, C, in terms of the number of children, n.

c) Graph the equation.

d) What does the y-intercept represent in this situation?

3 A taxi charges a base fee plus $0.75/km. A 10-km trip costs $8.70.

a) What is the rate of change in this situation?

b) Write an equation for the cost of hiring a taxi in terms of length of the trip.

c) Graph the equation.

d) What does the y-intercept represent in this situation?

Type D: Questions with 2 Points Given:

1. The cost of having a luncheon catered at the Dew Drop Inn depends on the number of meals served plus a

fixed fee to rent the dining room. The cost for 20 meals is $170. The cost for 60 meals is $310.

a) Find the cost per meal served.

b) Find the cost to rent the dining room.

c) Write an equation for the total cost of the luncheon in the form y = mx + b.

d) Find the cost of a luncheon for 85 people.

2. The LeBlanc family is driving home. They are using cruise control and their speed is constant. After 3 h,

they are 350 km from home. After 5 h, they are 130 km from home.

a) What is the independent variable? dependent variable?

b) Represent the given information as two points on a graph.

c) Write an equation for the line that passes through the two points.

d) What do the slope and the y-intercept mean in terms of this situation?

Part B: Finding the Point of Intersection.

1. Bill wants to earn extra money selling lemonade in front of his house. It costs $1.20 to start his business

and each cup of lemonade costs $0.06 to make. He plans to sell the lemonade for $0.10 a cup.

a) Find an equation that represents his costs.

b) Find an equation that represents his revenue.

c) Graph both equations on the same set of axis.

d) What does the point of intersection mean in this case?

e) Does Bill make a profit or lose money for i) 20 cups sold? ii) 35 cups sold? iii) 50 cups sold

2. Movies To Go rents videos for $2.50 each and has no membership fee. Videorenters rents videos for $2

each but has a $10 membership fee.

a) Write an equation for each situation.

b) Graph both equations on the same set of axes. Find the point of intersection.

c) What does the point of intersection mean in this case?

d) What advice would you give to someone who is deciding which video store to use?

3. Ontario Express charges $5 plus $1/kg to send a package out of the province. Day by Day Delivery

charges $3.50 plus $1.25/kg.

a) Write equations to model this situation.

b) Graph both equations on the same set of axes.

c) What does the point of intersection mean in this situation?

d) When is Ontario Express less expensive than Day by Day Delivery? more expensive?

4. Students can choose from two different monthly cafeteria beverage plans.

Plan A: Pay $0.75 per glass; Plan B: Pay $10 plus $0.50 per glass

a) Write the algebraic equations for both plans.

b) Find the cost of 20 beverages under each plan.

c) Find the cost of 40 beverages under each plan.

d) Which plan is better if you drink 30 beverages a month? 50 beverages a month

Part C: Understanding Graphs.

1. For each graph, explain what the y-intercept, x-intercept, and the slope mean in the context of each

relationship.

2. A stress test is a method of evaluating the health of a patient’s

cardiovascular system. A technician monitors the patient’s pulse while he

or she rides an exercise bike or runs on a treadmill. If the patient’s pulse

rate exceeds a safe maximum rate, the patient is at risk of a heart attack.

The maximum rate is based on the patient’s age as shown in this graph.

a) What does the y-intercept represent in this situation?

b) What does the slope of the graph represent?

c) Write an equation for the line.

d) Use the equation to find the maximum heart rate in a t 12 stress test for

a 58-year-old patient.