UNIT 1:
Ratios, Rates, &
Proportions
Review: fractions
A fraction allows you to determine two quantities and their
proportion to each other as part of a whole.
NUMERATOR – number on top (part)
DENOMINATOR – number on bottom (whole)
2
10
For example: You ate four slices of pizza. The pizza was cut into 8
slices. You ate 4 slices of pizza. The pizza was cut into 8 slices. The
4
slices that you ate is represented by . 4 parts of the whole 8
8
eaten.
Review: decimals
A decimal is a part of a whole number broken down over
increments of 10 (exponentially – 10, 100, 1000, 10000, etc.)
For example:
4 slices of pizza of out 8 have been eaten.
4
8
This represents eaten of the pizza eaten.
Use your calculator, what decimal do you get?
4
8
= .5
REVIEW: decimals
Ratios
A ratio allows you to compare two quantities that are of the same
nature, in a given order and have the same type of units.
You can compare for example:
Boys with boys
Girls with girls
Boys with girls
apples with oranges
circles with squares
Length of a rectangle with the width of a rectangle
and much more!!!
Ratios
The ratio of the number a to the number b is represented as:
𝑎
a:b or
𝑏
a is the first term of the ratio and b is the second term of the
ratio.
Find the ratio in this example:
For every 2 steps Billy takes he walks 5 meters.
The first term is  2
The second term is  5
The ratio  2:5
Terms in the Ratio
A term is a number that represents a quantity in a ratio.
Ex.
5 to 8
4:7
Terms
9
2
Find the ratio
What is the ratio of:
a)
All apples to oranges
6 to 3
b)
Green apples : All Fruits
3:9
c)
𝑎𝑙𝑙 𝑎𝑝𝑝𝑙𝑒𝑠
𝑎𝑙𝑙 𝑓𝑟𝑢𝑖𝑡𝑠
6
9
Part-to-part ratios
A Part-to-Part ratio is when you compare 2 different parts of a group to each
other.
The ratio of
circles to squares
4 to 6
circles : squares
4:6
circles
squares
4
6
Part-to-whole ratios
A Part-to-Whole ratio is when you compare part of the group to the whole
group.
The ratio of circles to all shapes can be represented in 3 ways:
4 : 10
or
4 to 10
or
4
10
The ratio of squares to all shapes can be represented in 3 ways:
6 : 10
or
6 to 10
or
6
10
Practice ratios
Christine is 3 years old and weighs 20 kg. Mr. Freckles is 50
years old and weighs 77 kg.
1) What is the ratio of Christine’s age to Mr. Freckles’ age?
2) What is the ratio of Mr. Freckles’ mass to Christine’s mass?
***Write the ratios in all three ways.
Reducing ratios
METHOD 1:
24 Remove the GCF (Greatest Common Factor) from the
Numerator and Denominator. Then divide each term or the
36 ratio by the GCF.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
24 ÷ 12 =
36 ÷ 12 =
2
3
Reducing ratios
METHOD 2:
Divide the Numerator and Denominator by the same
number until you cannot anymore.
40 ÷ 2 = 20 ÷ 2 = 10 ÷ 10 =
120 ÷ 2 = 60 ÷ 2 = 30 ÷ 10 =
1
3
Your turn to reduce a ratio
Reduce 
÷2
÷2
24
108
24
108
24: 108 or
=
÷2
12
54
=
÷2
÷3
6
27
=
÷3
2
9
Scale factor (SF)
The scale factor is the number you multiply or divide each
term in the ration to get the equivalent terms in another
ratio.
What is the scale factor (SF)?
10 : 12
=
30 : 36
4 to 10 = 16 to 40
7
8
=
35
40
SF = 3
SF = 4
SF = 5
Rate
A rate is a comparison of two quantities measured in different
types of units.
*** Unlike ratios, rates always include units.
_25 km_
1 hr
or
_1 photocopy_ or
10 ¢
_12 apples_
$4
Speed is a very common rate
Speed is measured by distance over time
Ex.
_100 meters_ =
10 seconds
(÷ 10) 10 meters
(÷ 10) 1 second
=
10 m/s
Rate example
For Example:
Peter earned $27 for 3 hours worth of work. How much did he
make per hour (Unit = money & Unit = hour)?
$27 ÷ 3h = $9 per hour or
3h ÷ $27 = .11 hours per dollar (18min to earn $1)
Equivalent rates
An equivalent rate is a rate that represents the same
comparison.
_12 apples_ = _3 apples_
$4
$1
_100 km_
2h
= _50 km
1h
Unit rate
Unit rates are rates that have a second unit (or denominator)
that is 1.
Ex.
_2 laps_
or
1 minute
or
_100 km
1 hour
_3 apples_
$1
= 100 km/h
How to convert to a unit rate
1) Write your rate as a fraction (include units!!!)
2) Divide your numerator by your denominator, this is now your
new numerator.
3) Your denominator becomes 1.
Ex. 400 km in 3.5 hours
400 𝑘𝑚
3.5 ℎ𝑜𝑢𝑟𝑠

400 ÷ 3.5 = 114.2857….  114.29 
114.29 𝑘𝑚
1 ℎ𝑜𝑢𝑟
Practice: converting to unit rate
You just got your first summer job! On the first day, you worked
7 hours and made 120 dollars.
How much did you make per hour? (This is unit rate!)
Comparison of Ratios and of Rates
Two methods to compare ratios and rates:
1) We find the same common denominator or the same basis
of comparison.
2
Ex:
3
5
7
< , since
70 𝑤𝑜𝑟𝑑𝑠
4 𝑚𝑖𝑛
>
14
21
95 𝑤𝑜𝑟𝑑𝑠
6 𝑚𝑖𝑛
<
15
21
, since
210 𝑤𝑜𝑟𝑑𝑠
12 𝑚𝑖𝑛
>
190 𝑤𝑜𝑟𝑑𝑠
12 𝑚𝑖𝑛
Practice method 1 comparing
rates and ratios
Comparison of Ratios and of Rates
2) We calculate their quotients (divide the numerator by the
denominator).
Ex:
600 𝑚
5𝑠
50
20
<
>
60
25
500 𝑚
4𝑠
, since
50
20
= 2.5 and
60
25
, since 120 m/s < 125 m/s
= 2.4
Practice method 2 comparing
rates and ratios
Comparing rates
It is very important to know how to compare rates!
So how do you do this? Simple!
Change your rates to unit rates
Ex. Which store charges more for pears?
Store A: $12 for 10 pears
Store B: $8 for 6 pears
Important conversions : time
Hours to minutes to seconds
X 60
X 60
1 hr = 60 minutes = 3600 seconds
60 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
1 ℎ𝑜𝑢𝑟
60 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
1 𝑚𝑖𝑛𝑢𝑡𝑒
A) How many hours is 240 minutes?
B) How many minutes is 480 seconds?
Important conversions : distance
kilometers to meters
meters to kilometers
÷ 1000
X 1000
1 km = 1000 m
1000 m = 1 km
1 𝑘𝑚
1000 𝑚
A) How many meters is 2.5 kilometers?
B) How many kilometers is 550 meters?
Practice: comparing rates
Which car is traveling faster?
Car A:
100 km / hour
Car B:
2 hours to travel 220 km
Car C:
40 km in 30 minutes
Equivalent ratios and rates
If two ratios or rates have the same quotient (result once
divided), they are equivalent.
Ex:
1) the
8
ratios
5
10 𝑘𝑚
1 ℎ𝑟
are equivalent since
8
=
5
1.6 and
and
40 𝑘𝑚
4 ℎ𝑟𝑠
are equivalent since
= 10 km/hr and
40 𝑘𝑚
4 ℎ𝑟𝑠
= 10 km/hr
2) the rates
10 𝑘𝑚
1 ℎ𝑟𝑠
and
24
15
24
15
= 1.6
Proportions
When two ratios or rates are compared and determined to be
equivalent, they are proportional.
Proportions involves comparing, multiplying and dividing.
So! If the ratio of a to b is equal to the ratio of b to c then
𝑎
𝑏
𝑐
𝑑
a:b = c:d or = is a proportion.
A proportion has 4 terms:
Means
a: b = c:d
Extremes
Proportion: property of the crossproducts
In every proportion, the product of the end terms is equal to
the product of the middle terms.
So, from the
𝑎
proportion
𝑏
𝑐
𝑑
= , we find the equality ad = bc
Cross-products
4
2
From the proportion =
16
8
𝑎
𝑏
=
,we know that 4 x 8 = 2 x 16
32 = 32
𝑐
𝑑
Proportions practice
Are the following proportions true or false?
a)
12
18
=
10
15
b)
21
49
=
24
56
c)
1.4
3.5
=
3.4
8.5
d)
2.5
7.5
=
0.5
2.5
A new proportion given a
proportion
Given a proportion, we get a new proportion by
if
𝑎
𝑏
- interchanging the middle terms
if
𝑎
𝑏
= , then =
- inverting the ratios
if
𝑎
𝑏
= , then =
- interchanging the end terms
𝑐
𝑑
𝑑
𝑏
𝑐
𝑎
𝑐
𝑑
𝑎
𝑐
𝑏
𝑑
𝑐
𝑑
𝑏
𝑎
𝑑
𝑐
= , then =
A new proportion given a
proportion
Given the proportion
1)
2)
3)
5
3
=
10
6
, find three new proportions.
Finding a missing term in a
proportion
Use cross-products!
Ex:
3
24
=
15
𝑥
3
24
=
15
𝑥
15 times 24 divided by 3
15 × 24
=𝑥
3
120 = 𝑥
Find the missing proportion
An employee’s salary is proportional to the duration of his work.
For 3 hours of work, he receives $24 and for 12 hours, $96.
How much will de receive for 15 hours of work?
Number of
hours
3
12
15
Salary ($)
24
96
x
HINT! Use cross-products!
Proportions: challenge question
7 pounds of strawberries makes 8 fruit roll-ups. If every batch
of fruit roll-up requires the same amount of strawberries, how
many pounds of strawberries are required to make 20 fruit rollups?
A) 0.875
B) 2.5
C) 17.5
D) 23
*** HINT: Set up a ratio and use cross-product!!!