ratio, rates, & proportions. ratios a ratio is a comparison of two numbers. o example: tamara...

Ratio, Rates, & Proportions

Ratios• A ratio is a comparison of two numbers.

o Example: Tamara has 2 dogs and 8 fish. The ratio of dogs to fish can be written in three different ways.

*** Be careful with the fraction ratios – they don’t always have identical meanings to other fractions***

• There are two different types of ratioso .Part-to-Part Ratios

Example: Tamara’s dogs to cats is 2 to 8 or 1 to 4

o .Part –to-Whole Ratios Example: Tamara’s dogs to total pets is

2 to 10 or 1 to 5

Try the following Apples to bananas

• There are 12 apples and 3 bananas

• So the ratio of apples to bananas is

4:1, 4 to 1, OR

Oranges to apples

• There are 15 oranges and 12 apples

• So the ratio of oranges to apples is

5:4, 5 to 4, OR

1

4

4

5

1. Roses to iris

3. Daisies to roses

2. Iris to daisies

4. all flowers to roses

A large bouquet of flowers is made up of 18 roses, 16 daisies, and 24 irises. Write each ratio in all three forms in simplest form. Identify which ratios are part-to-part and which ratios are part-to-whole.

18 to 243 to 4

58 to 829 to 9

16 to 188 to 9

24 to 163 to 2

• The ratio is comparing two number. If you changed it to a mixed number it would no longer be a comparison. This is why fraction ratios are tricky 

9

29Explain why the ratio

9

is not written as a mixed number:

Comparing Ratios• Compare ratios by writing in simplest form

oAre these ratios equivalent?250 Kit Kats to 4 M&M’s and 500 Kit Kats to 8 M&M’s

o  12 out of 20 doctors agree and 12 out of 30 doctors agree

o The ratio of students in Ms. B’s classes that had HW was 8 to 2 and 80% of Mrs. Long’s class had HW

=5.624

250 5.62

8

500

5

3

20

12 ≠

5

2

30

12

10

8=

100

80

Rates• A rate is a ratio that compares two different

quantities or measurements.

• Rates can be simplified

• Rates us the words per and foro Example: Driving 55 miles per hour

o Example: 3 tickets for $1

Unit Rates• A unit rate is a rate per one unit. In unit

rate the denominator is always one.o Example: Miguel types 180 words in 4 min.

How many words can he type per minute?

= or words per

minute

rate unit rate word form

45 45

. 1

• Unit rates make it easier to make comparisons.o Example: Taylor can type 215 words in 5 min

How many words can he type per minute?

Who is the faster typist? How much faster? Taylor is 2wpm faster than Miguel

wpm435

215

1. Film costs $7.50 for 3 rolls

3. 90 students and 5 teachers

5. **Snowfall of 12 ¾ inches in 4 ½ hours.

7. Ian drove 30 miles in 0.5 hours

2. Drive 288 miles on 16 gallons of gas.

4. Earn $49 for 40 hours of work

6. Use 5 ½ quarts of water for every 2 lbs of chicken

8. Sarah drove 5 miles in 20 minutes

Try the Following

$2.50 per roll

15 mph

2 ¾ quarts per lb

$1.23 per hour

18 mpg

18 students per teacher

2 5/6 per hour

60 mph

Complex Unit Rates• Suppose a boat travels 30 miles in 2

hourso How do you write this rate?

• Suppose a boat travels 12 miles in 2/3 hourso How do you write this as a rate?

o How do you write this as a division problem?

o Determine the unit rate: 18mph

mphhours

miles15

2

30

3

212 18

2

312

hours

miles

3/2

12

• Suppose a boat travels 8 ¾ miles in 5/8 hours.oHow do you write this as a rate?

oHow do you write this as a division problem?

oDetermine the unit rate: 14 mph

hours

miles

8/5

4/83

8

5

4

38 14

5

8

4

35

•.Complex fractions are fractions that have fractions within them. They are either in the numerator, denominator, or both.

•Divide complex fractions by multiplying (keep, change, change)

1. Mary is making pillows for her Life Skills class. She bought yards of fabric. Her total cost was $16. What was the cost per yard?

3. Doug entered a canoe race. He rowed miles in hour. What is

his average speed?

3. Mrs. Robare is making costumes for the school play. Each costume requires 0.75 yards of fabric. She bought 6 yards of fabric. How many costumes can Mrs. Robare make?

4. A lawn company advertises that they can spread 7,500 square feet of grass seed in hours. Find the number of square feet of grass seed that can be spread in an hour.

TRY the FOLLOWINGWrite each rate. Then determine the unit rate and write in both fraction and

word form

$5.82 per yard

3000 ft per hour

7mph

8 costumes

4

32

2

122

13

2

1

Comparing Unit RatesDario has two options for buying boxes of pasta. At CornerMarket he can buy seven boxes of pasta for $6. At SuperFoodz he can buy six boxes of

pasta for $5.• He divided 7 by 6 and got 1.166666667 at CornerMarket.

He then divided 6 by 7 and got 0.85714286. He was confused. What do these numbers tell him about the price of boxes of pasta at CornerMarket?

• Decide which makes more sense to you

• Compare the two stores’ prices. Which store offers the better deal?

1.166667 is the number of boxes you can get for $10.85714286 is the price per box

Price per box

CM - $0.86 per box SF - $0.83 per box

Proportions

• Two quantities are proportional if they have a constant ratio or unit rate.

• You can determine proportionality by comparing ratios o Andrew earns $18 per hour for mowing lawns. Is the amount he

earns proportional to the number of hours he spends mowing?• Make a table to show these amounts

• For each number of hours worked, write the relationship of the amount he earned and hour as a ratio in simplest form.

• Are all the rates equivalent?

Earnings ($) 18 36 54 72Time (h) 1 2 3 4

Since each rate simplifies to 18, they are all equivalent. This means the amount of money Andrew earns is proportional to the number of hours

he spends mowing.

o Uptown Tickets charges $7 per baseball game ticket plus $2 processing fee per order. Is the cost of an order proportional to the number of tickets ordered?

• Make a table to show these amounts

• For each number of tickets, write the relationship of the cost of the and the number of tickets ordered.

• Are all the rates equivalent?

Cost ($) 10 17 24 31Tickets Ordered 1 2 3 4

The rates are not equivalent. This means the total cost of the tickets is not proportional to the number of tickets sold.

o Use the recipe to make fruit punch. Is the amount of sugar used proportional to the amount of mix used? Explain.

•Yes, they all reduce to ½

o In July, a paleontologist found 368 fossils at a dig. In August, she found about 14 fossils per day.

•Is the number of fossils the paleontologist found in August proportional to the number of days she spent looking for fossils that month?

No, July average 11.87 fossils per day

Cups of Sugar ½ 1 1 ½ 2Envelopes of Mix 1 2 3 4

Solving Proportions

• A proportion is two equivalent ratios

• When solving proportions we must first ask ourselves – “What are we comparingA lemonade recipe calls for ½ cup of mix for every quart of water. If Jeff

wanted to make a gallon of lemonade, is 2 cups of mix proportional for this recipe?

YES

Determine if the following ratios are proportional?No, 340 ≠ 270 Yes, 72 = 72

27

20

17

10and

24

9

8

3and

4

2

1

2/1

• Proportionality can also be determined between two ratios by simplifying or comparing their cross products

• If they reduce to the same ratio, or their cross products are the same, then they are proportional

• You can also solve proportions for a missing variable by cross multiplying.o Example: Determine if the two ratios are proportional:

12

8

9

6and

8

3

10

4and

15

10

7

2and

b.

c.

Yes 72 = 72 No 32 = 30 No 30 = 70

o Example: Determine the value of x:

o Example: A stack of 2,450 one-dollar bills weighs five pounds. How much do 1,470 one-dollar bills weigh? Set up a proportion – ask ourselves “what are we comparing?”

o Example: Whitney earns $206.25 for 25 hours of work. At this rate, how much will Whitney earn for 30 hours of work?

How much does Whitney earn per hour?

x

36

10

6

x = 60

$8.25 per hour

3 poundsx

1470

5

2450

pounds

bills

hours

$

3025

25.206 x $247.50

Coordinate Plane Review

The Coordinate Plane

y-axisy-axis

Origin Origin

x-axisx-axis

Quadrant IQuadrant IQuadrant IIQuadrant II

Quadrant IIIQuadrant III Quadrant IVQuadrant IV

• Ordered Pair: is a pair of numbers that can be used to locate a point on a coordinate plane

Ordered Pairs• Ordered Pair: is a pair of numbers that

can be used to locate a point on a coordinate plane.– Example: (3, 2) y - coordinate

x - coordinate

●●

●●

IIIIII

IIIIII IVIV

Graph the following ordered pairs on the coordinate plan and state the quadrants the points are located in•(3, 2)

•(-5, 4)

•(6, -4)

•(-7, 7)

●●

●●

●●

●●

II

I

IV

III

Steps for Graphing

1. Draw and label the x and y axis – don’t forget your arrows

2. Make a table of values to represent the problem. Be sure to include the values: 0, 1, and 2

3. Graph your order pairs- you need at least 3 points to make a line

4. Draw a line through the points – don’t forget your arrows

5. If the line is straight and goes through the origin, then the quantities are proportional

Example: The slowest mammal on Earth is the tree sloth. It moves at a speed of 6 feet per minute. Determine whether the number of feet the sloth moves is proportional to the number of minutes it moves by graphing. Explain your reasoning.

y

x

Number of Minutes 1 2 3

Number of Feet 6 12 18

Yes – it is a straight line

through the origin

Example: The table below shows the number of calories an athlete burned per minute of exercise. Determine whether the number of calories burned is proportional to the number of minutes by graphing. Explain your reasoning.

y

x

Number of Minutes 1 2 3

Number of Feet 4 8 13

No – it is not a straight line and it

doesn’t go through the origin

Slope

• .Slope is the rate of change between any two points on a line

• The sign of the slope tells you whether the line is positive or negative.

• You can find slope of a line by comparing any two points on that line

• Slope is the or  

Positive Slope

• The line goes up 3 (rise) and over 1 (run).

• Slope = 3

Negative Slope

1

3

run

rise

1

3

__

__

xinchange

yinchange

• The line goes down 2 (rise) and over 1 (run).

• Slope = -2

1

2

run

rise

1

2

__

__

xinchange

yinchange

Tell whether the slope is positive or negative . Then find the slope

• Negative• Slope = -1

1

1

run

rise

• Positive• Slope = 4/3

3

4

run

rise

Use the given slope and point to graph each line

Use the given slope and point to graph each line

Use the given slope and point to graph each line

Rate of Change (Slope)

• .Rate of change (slope) describes how one quantity changes in relation to another.

• For graphs, the rate of change (slope) is constant ( a straight line)

Tell whether each graph shows a constant or variable rate of change

Constant Variable

Tell whether each graph shows a constant or variable rate of change

ConstantVariable

Tell whether each graph shows a constant or variable rate of change

Constant Variable

Proportional Relationships

• A proportional relationship between two quantities is one in which the two quantities vary directly with one another (change the same way). This is called a direct variation.

• For graphs, the rate of change (slope) is constant ( a straight line)

Determine whether each table represents a direct variation by comparing the ratios to check for a

common ratio

251

25 25

2

50 25

3

75 25

4

100

This table represents a direct

variation and is proportional

0555.36

2 0769.

52

4 882.0

68

6

This table does NOT represent a direct variation

and is NOT proportional

Determine whether each table represents a direct variation by comparing the ratios to check for a

common ratio

33.12

4 375.

16

6 4.

20

8

This table does NOT represent a direct variation

and is NOT proportional

4.5.12

5 4.

25

10 4.

5.37

15

This table represents a direct

variation and is proportional

• The equations of such relationships are always in the form y = mx and when graphed produce a line that passes through the origin.

Proportional Linear Function

Equation: y = 2x

Non- Proportional Linear Function

Equation: y = 2x – 1

• In the equation y=mx, m is the slope of the line, and its also called the unit rate, the rate of change, or the constant of proportionality of the function.

• The number of computers built varies directly as the numbers of hours the production line operates. What is the ratio of computers build to hours of production

Try the following

1

25

2

50

25:1 or 25 computers built per

hour

• A charter bus travels 210 miles in 3½ hours. Assuming that the distance traveled is proportional to the time traveled, how far will the bus travel in 6 hours?

  hours

miles

65.3

210 x

360 miles

• Determine whether the linear function is a direct variation. If so stat the constant of variation.

hours

miles58

2

116

Yes this is a direct variation because there is a constant of variation of 58

mph

583

174 58

4

232 58

5

290

• Janelle planted ornamental grass seeds. After the grass breaks the soil surface, its height varies directly with the number of days. What is the rate of growth?

 days

inches75.0

2

5.1

0.75 inches per day

75.04

3

• The amount Dusty earns is directly proportional to the number of newspapers he delivers. How much does Dusty earn for each newspaper delivery?

 papers

$50.0

4

2

Dusty earns $0.50 per paper he

delivers

50.08

4

• Ten minutes after a submarine is launched from a research ship, it is 25 meters below the surface. After 30 minutes, the submarine has descended 75 meters. At what rate is the submarine driving?

  ute

meters

min5.2

10

25

The submarine is descending at a rate of 2.5 meters per minute

5.230

75

• The Stratton family rented 3 DVD’s for $10.47. The next weekend, they rented 5 DVD’s for $17.45. What is the rental fee for a DVD?

  DVD

$49.3

3

47.10

The rental fee for DVD’s is $3.49 per DVD

49.35

45.17

• Five Gala apples cost $2

• Tess rides her bike at 12 mph

Fill in the table for each proportional relationship

Apples 0 5 10 15Cost   $2    

Hours 0 1 2 3

Miles   12    

Apples 0 5 10 15Cost  0 $2 $4 $6

Hours 0 1 2 3

Miles  0 12 24 36

• An Elm tree grows 8 inches each year.

Draw the graph of the proportional relationship between the two quantities

Years

He

igh

t (i

n.)

8

4

0

• David adds $3.00 to his savings account each week

Draw the graph of the proportional relationship between the two quantities

2

1

0

Sa

vin

gs

Acc

ou

nt

Ba

lan

ce (

$)

Weeks

Slope, Rate of Change,

Graphs & Tables

• The table shows the amount of money a Booster Club made washing cars for a fundraiser. Use the information to find the rate of change in dollars per car.– Write and equation that shows the money

raised m of washing c cars.

 car

$8

10

80

Since it cost $8 per car the equation would be m = 8c

85

40 8

15

120

• The table shows the number of miles a plane traveled while in flight. Use the information to find the approximate rate of change in miles per minute.

utes

miles

min

The approximate rate of change is 9.7 miles per minute

66.930

290 66.9

60

580 66.9

90

870

• The graph represents the distance traveled while driving a car on the highway. Use the graph to find the rate of change in miles per hour.– Pick two points on the line

x

ymph60

1

60

12

60120

• Write an equation that shows the distance m for the time h.

 x

y

mph301

30

12

3060

m = 30h

• The table below shows the relationship between the number of seconds y it takes to hear the thunder after a lightning strike and the distance x you are from the lightning.

• SKIP THIS QUESTION

Distance (y) 0 1 2 3 4 5

Seconds (x) 0 5 10 15 20 25

• Graph the data. Then find the slope of the line. Explain what the slope represents.

 W

ate

r L

oss

cm)

Weeks

• Graph the data. Then find the slope of the line. Explain what the slope represents.

 T

em

per

atu

re

(˚F

)

Time

• Finish the rest for Homework