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 ID: 555194
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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)