Ratio, Rates, & Proportions - PowerPoint Presentation

Slide1

Ratio, Rates, & ProportionsSlide2

Ratios

A

ratio is a comparison of two numbers.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

***Slide3

There are

two

different types of ratios.Part-to-Part RatiosExample: Tamara’s dogs to cats is 2 to 8 or 1 to 4.Part –to-Whole RatiosExample: Tamara’s dogs to total pets is 2 to 10 or

1 to 5  Slide4

Try the following

Apples to bananas

There are 12 apples and 3 bananasSo the ratio of apples to bananas is 4:1, 4 to 1, OR Oranges to apples

There are 15 oranges and 12 applesSo the ratio of oranges to apples is 5:4, 5 to 4, OR Slide5

Roses to iris

Daisies to roses

Iris to daisies 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 24

3 to 4

58 to 8

29 to 9

16 to 18

8 to 9

24 to 16

3 to 2Slide6

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 

Explain why the ratio

9

is not written as a mixed number:Slide7

Comparing Ratios

Compare ratios by writing in

simplest formAre these ratios equivalent?250 Kit Kats to 4 M&M’s and 500 Kit Kats to 8 M&M’s  12 out of 20 doctors agree and 12 out of 30 doctors agree

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

=

≠

=Slide8

Rates

A

rate is a ratio that compares two different quantities or measurements.Rates can be simplified

Rates us the words per and for

Example:

Driving 55 miles

per

hour

Example: 3 tickets for $1

 Slide9

Unit Rates

A

unit rate is a rate per one unit. In unit rate the denominator is always one.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

.

1Slide10

Unit rates make it easier to make

comparisons

.Example: Taylor can type 215 words in 5 minHow many words can he type per minute?Who is the faster typist? How much faster? Taylor is 2wpm faster than Miguel

  Slide11

Film costs $7.50 for 3 rolls

90 students and 5 teachers

**Snowfall of 12 ¾ inches in 4 ½ hours.Ian drove 30 miles in 0.5 hoursDrive 288 miles on 16 gallons of gas.

Earn $49 for 40 hours of workUse 5 ½ quarts of water for every 2 lbs

of chicken

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 mphSlide12

Complex Unit Rates

Suppose a boat travels 30 miles in 2 hours

How do you write this rate?Suppose a boat travels 12 miles in 2/3 hoursHow do you write this as a rate?How do you write this as a division problem?

Determine the unit rate: 18mphSlide13

Suppose

a boat travels

8 ¾ miles in 5/8 hours.How do you write this as a rate?How do you write this as a division problem?

Determine the unit rate: 14 mphSlide14

.

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)Slide15

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?

Doug entered a canoe race. He

rowed miles in hour. What is his average speed? 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?

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 FOLLOWING

Write 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 costumesSlide16

Comparing Unit Rates

Dario 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 $1

0.85714286 is the price per box

Price per box

CM - $0.86 per box

SF - $0.83 per boxSlide17

ProportionsSlide18

Two quantities are

proportional

if they have a constant ratio or unit rate.You can determine proportionality by comparing ratios 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

72

Time (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

.Slide19

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

31

Tickets 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.Slide20

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

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

2

Envelopes of Mix

1

2

3

4Slide21

Solving ProportionsSlide22

A proportion is two

equivalent ratios

When solving proportions we must first ask ourselves – “What are we comparing

A 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

Slide23

Proportionality can also be determined between two ratios by

simplifying

or comparing their cross productsIf 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.Example: Determine if the two ratios are proportional:

b.

c.

Yes 72 = 72

No 32 = 30

No 30 = 70Slide24

Example:

Determine the value of x:

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?”

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 = 60

$8.25 per hour

3 pounds

$247.50Slide25

Coordinate Plane ReviewSlide26

The Coordinate Plane

y-axis

Origin

x-axis

Quadrant I

Quadrant II

Quadrant III

Quadrant IVSlide27

Ordered Pair:

is a pair of numbers that can be used to locate a point on a

coordinate planeGraph the You can also solve proportions for a missing variable by cross multiplying.

Example: Determine if the two ratios are proportional:

Yes 72 = 72

No 32 = 30

No 30 = 70Slide28

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

●

●

I

II

III

IVSlide29

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

IISlide30

Steps for GraphingSlide31

Draw and

label

the x and y axis

– don’t forget your arrows

Make a table of values to represent the problem. Be sure to include

the values: -1, 0, 1, and 2

Graph your order pairs

- you need at least 3 points to make a line

Draw a line through the points

– don’t forget your arrows

If the line is straight and goes through the

origin, then the quantities are proportionalSlide32

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 originSlide33

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 originSlide34

SlopeSlide35

.

Slope

is the rate of change between any two points on a lineThe 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 lineSlope is the or  Slide36

Positive Slope

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

Slope = 3Negative Slope

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

Slope =

-2Slide37

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

Negative

Slope = -1

Positive

Slope = 4/3 Slide38

Use the given slope and point to graph each lineSlide39

Use the given slope and point to graph each lineSlide40

Use the given slope and point to graph each lineSlide41

Rate of Change

(Slope)Slide42

.

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) Slide43

Tell whether each graph shows a constant or variable rate of changeSlide44

Tell whether each graph shows a constant or variable rate of changeSlide45

Tell whether each graph shows a constant or variable rate of changeSlide46

Proportional RelationshipsSlide47

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)