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***
Rates
• A rate is a ratio that compares two
different quantities or measurements.
• Rates can be simplified
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
Proportions
I can…
• Determine if two ratios form a
proportion
What is a Proportion?
• A proportion is two equivalent ratios
• Example: 2=
3
9
6
Proportions
• If two ratios are proportional they
will have the same unit rate
• Are the following ratios
proportional?
2
8
4
12
7
21
=
=
5
25
9 = 27
14 42
Proportions
• We can also determine if tables are
proportional by looking at unit rate.
x
2
4
6
7
8
y
4
8
12
14
16
• Always put y when finding unit rate
x
4 =2 8
12
14 =2 16 =2
=2
=2
2
4
6
8
7
• 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
• Yes, they all reduce to ½
1
2
1½
3
2
4
Solving Proportions
I can…
• Solve a proportion using cross
multiplcation
• 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 areI located in
●
• (3, 2)
II
• (-5, 4)
• (6, -4)
• (-7, 7)
●
●
IV
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)