Transcript Ratio, Rates, & Proportions

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 ratios
o .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
Oranges to apples
• There are 12 apples • There are 15 oranges
and 3 bananas
and 12 apples
• So the ratio of
• So the ratio of
apples to bananas is
oranges to apples is
4
5
4:1, 4 to 1, OR
5:4,
5
to
4,
OR
1
4
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.
1. Roses to iris
18 to 24
3 to 4
2. Iris to daisies
24 to 16
3 to 2
3. Daisies to roses
16 to 18
8 to 9
4. all flowers to roses
58 to 8
29 to 9
9
29
Explain why the ratio 9
is not written as a mixed number:
• 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
Comparing Ratios
• Compare ratios by writing in simplest form
o Are these ratios equivalent?
250 Kit Kats to 4 M&M’s and 500 Kit Kats to 8 M&M’s
250
 62.5
4
=
500
 62.5
8
o 12 out of 20 doctors agree and 12 out of 30 doctors
12 2
12 3

≠ 30  5
20 5
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
8
10
=
80
100
Rates
• A rate is a ratio that compares two
different quantities or measurements.
• Rates can be simplified
• Rates us the words per and for
o 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?
=
.
rate
45
1
unit rate
or
45 words per minute
word form
• 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?
215
 43 wpm
5
 Who is the faster typist? How much faster?
Taylor is 2wpm faster than Miguel
Try the Following
1. Film costs $7.50 for 3 rolls 2. Drive 288 miles on 16 gallons
of gas.
$2.50 per roll
18 mpg
3. 90 students and 5 teachers
4. Earn $49 for 40 hours of work
18 students per teacher
$1.23 per hour
5. **Snowfall of 12 ¾ inches in
6. Use 5 ½ quarts of water for
4 ½ hours.
every 2 lbs of chicken
2 5/6 per hour
2 ¾ quarts per lb
7. Ian drove 30 miles in 0.5
8. Sarah drove 5 miles in 20
hours
minutes
60 mph
15 mph
Complex Unit Rates
• Suppose a boat travels 30 miles in 2 hours
o How do you write this rate? 30 miles  15mph
2hours
• Suppose a boat travels 12 miles in 2/3 hours
o How do you write this as a rate?
12 miles
2 / 3hours
o How do you write this as a division problem?
2
12 
3
3
 18
2
o Determine the unit rate: 18mph
12 
• Suppose a boat travels 8 ¾ miles in 5/8
hours.
o How do you write this as a rate?
83 / 4miles
5 / 8hours
o How do you write this as a division problem?
3 5
8 
4 8
35 8
  14
4 5
o Determine the unit rate: 14 mph
• 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)
TRY the FOLLOWING
Write each rate. Then determine the unit rate and write in both fraction and
word form
1. Mary is making pillows for her Life
3. Mrs. Robare is making costumes for the
3
Skills class. She bought yards of 2
school play. Each costume requires
4
fabric. Her total cost was $16.
0.75 yards of fabric. She bought 6
What was the cost per yard?
yards of fabric. How many costumes
can Mrs. Robare make?
$5.82 per yard
3. Doug entered a canoe race. He
rowed miles in hour. What is
1
1
3 speed?
his average
2
2
7mph
4.
8
costumes
A lawn company advertises that
they can spread 7,500 square feet
of grass seed in hours. Find the
1
number of square feet of grass 2
2
seed that can be spread in an
hour.
3000 ft per hour
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?
1.166667 is the number of boxes you can get for $1
0.85714286 is the price per box
• Decide which makes more sense to you
Priceprices.
per box
• Compare the two stores’
Which store offers the better deal?
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 $18 per hour for mowing lawns. Is the amount he
earns proportional to the number of hours he spends mowing?
• Make a table to show these amounts
Earnings ($)
Time (h)
18
1
36
2
54
3
72
4
• 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?
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 $2
processing fee per order. Is the cost of an order proportional to
the number of tickets ordered?
• Make a table to show these amounts
Cost ($)
Tickets Ordered
10
1
17
2
24
3
31
4
• For each number of tickets, write the relationship of the cost of the
and the number of tickets ordered.
• Are all the rates equivalent?
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.
Cups of Sugar
Envelopes of Mix
½
1
1
2
1½
3
2
4
• 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
Solving Proportions
• 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?
1/ 2 2

YES
1
4
Determine if the following ratios are proportional?
10
20 No, 340 ≠ 270
and
17
27
3
9
and
8
24
Yes, 72 = 72
b.
• Proportionality can also be determined between two ratios
c.
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:
6
8
and
9
12
Yes 72 = 72
4
3
and
10
8
No 32 = 30
2
10
and
7
15
No 30 = 70
o Example: Determine the value of x:
6 36

10
x
x = 60
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?”
bills
2450 1470
3 pounds

pounds
5
x
o Example: Whitney earns $206.25 for 25 hours of work. At this rate,
how much will Whitney earn for 30 hours of work?
$
206 .25 x

$247.50
hours
25
30
 How much does Whitney earn per hour?
$8.25 per hour
Coordinate Plane Review
The Coordinate Plane
Quadrant II
Quadrant I
x-axis
y-axis
Origin
Quadrant III
Quadrant IV
• Ordered Pair: is a pair of numbers that can be used to
locate a point on a coordinate plane
• Graph the
• You can also solve proportions for a missing variable by
cross multiplying.
o Example: Determine if the two ratios are proportional:
6
8
and
9
12
Yes 72 = 72
4
3
and
10
8
No 32 = 30
2
10
and
7
15
No 30 = 70
Ordered Pairs
• Ordered Pair: is a pair of numbers that can
be used to locate a point on a coordinate
plane.
y - coordinate
– Example:
x - coordinate
(3, 2)
II
I
●
III
IV
Graph the following ordered pairs on the coordinate
plan and state the quadrants the points are located in
• (3, 2)
I
●
• (-5, 4)
II
●
●
• (6, -4)
IV
• (-7, 7)
II
●
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: -1,
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.
Number of Minutes
1
2
3
Number of Feet
6
12
18
y
Yes – it is a
straight line
through the origin
x
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.
Number of Minutes
1
2
3
Number of Feet
4
8
13
y
No – it is not a
straight line and it
doesn’t go
through the origin
x
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).
rise 3

run 1
change_ in _ y 3

change_ in _ x 1
• Slope = 3
Negative Slope
• The line goes down 2
(rise) and over 1 (run).
rise  2 change_ in _ y  2


run
1 change_ in _ x
1
• Slope = -2
Tell whether the slope is positive or negative . Then find the slope
rise  1

run
1
• Negative
• Slope = -1
rise 4

run 3
• Positive
• Slope = 4/3
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
Tell whether each graph shows a constant or variable rate of change
Tell whether each graph shows a constant or variable rate of change
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)