Transcript Equivalent Ratios

Equivalent
Ratios
Lesson 2.03
After completing this lesson, you will be able to say:
• I can make tables of equivalent ratios.
•
I can use tables to find missing values and
compare ratios.
•
I can solve unit rate problems by reasoning
about ratios and rates.
Equivalent Ratios
Equivalent ratios:
Ratios that have the same simplest form
and express the same relationship
between two quantities.
Equivalent ratios are ratios that express the
same relationship between two quantities
Example of Equivalent Ratios
The ratios 2:8, 4:16, 6:24, and 8:32 are
examples of equivalent ratios.
Ratios that are equivalent have the same simplest form.
The simplest form for these ratios is 1:4
Ratio Tables
Given ratio 2:6.
• You can begin by first making a table and creating some empty spaces that
you know you will have to fill in later.
• Your table can be horizontal or vertical–it does not matter.
Horizontal Table
Vertical Table
Number of Subs
Number of subs
2
Bag of chips
6
2
Bags of chips
6
Ratio Tables – Using Addition
+2
+2
+2
Number of subs
2
4
6
8
Bag of chips
6
12
18
24
+6
+6
Notice that to find each value, you do
something different in each row.
+6
Number of subs: The pattern is to add 2 each time, because that is the first
term of the ratio.
2+2= 4; 4+2= 6; 6+2= 8
Bags of chips: The pattern is to add 6 each time, because that is the second
term of the ratio.
6+6 = 12; 12 + 6 = 18; 18 + 6 = 24
This shows that the ratios 2:6, 4:12, 6:12, and 8:24 are all equivalent.
Ratio Tables – Using Multiplication
Another way to create a ratio table is by using multiplication
Using multiplication you can use any number you want. Start by
multiplying the top row by a number, then multiply the bottom row by
the same number.
x2
x2
x2
Number of subs
2
4
8
16
Bag of chips
6
12
24
48
x2
x2
x2
Number of subs:
2 x 2 = 4; 4 x 2 = 8; 8 × 2 = 16
Bags of chips:
6 × 2 = 12; 12 × 2 = 24; 24 × 2 = 48
This shows that the ratios 2:6,4:12, 8:24, and 16:48 are all equivalent.
Ratio Tables – Using Rate Reasoning
You can also find the unit ratio to help you fill in the table
To make a table using rate reasoning, you have to calculate the unit rate.
Start by determining what is being multiplied by 2 to get 6, you can do this by
dividing 6 by 2
6 divided by 2 = 3
The unit rate shows that there are 3 bags of chips per sub
Then you can use the unit rate and multiply both columns by that number to find
equivalent ratios
Number of Subs
X3
X3
Bags of chips
1
3
2
6
6
18
18
54
X3
X3
Unit rate
A rate expressed such that it reveals
how much of the first quantity there is
for just one unit of another, such as 2
feet per second or $6 per hour.
Using rate tables to find Unit Rate
• Ellie raised $150 for walking 6 laps.
• Use a ratio table and unit rate to show her progress
Laps
Amount Raised
1
$25
Your goal is to find out how much Ellie
made per lap. This will be the unit rate.
Calculate the unit rate by dividing $150
(amount raised) by 6 (number of laps).
$150
= $25 𝑝𝑒𝑟 𝑙𝑎𝑝
6 𝑙𝑎𝑝𝑠
2
3
4
5
6
$150
Now you know your unit rate is $25 per
unit, which means for every lap Ellie
completes, she earns $25.
Fill in the rest of the table to see Ellie's
progress.
Check your work
Laps
Amount Raised
1
$25
2
$50
3
$75
4
$100
5
$125
6
$150
Determining Unknown Values
How can we use equivalent ratios and unit rates to help calculate missing values:
After their walk, the walkers will receive oranges as a snack. The event
organizers figured out that they would need to order slightly more oranges
just in case more people show up. Therefore, they purchased them at a
ratio of 3 to 4. For every 3 walkers registered for the event, the organizers
will order 4 oranges. How many oranges would be needed for 24 walkers?
There are two ways this problem can be solved.
Determine Unknown Values
Method 1
Walkers
3
Oranges
4
24
Determine what number is being multiplied by 3 to get 24 in the
walkers' row. You do the same for the second row. By using ratio
reasoning, you multiply the 4 by 8, as well.
Walkers
3x8
24
Oranges
4x8
32
This tells us that for 24 walkers, the organizers need to purchase 32
oranges.
Determining Unknown Values
Method 2
Walkers
3
Oranges
4
24
3
Calculate the unit rate, 3 ÷ 4 =
4
Do this same operation for 24 walkers using the unit rate
24 ÷
3
4
=
24
1
4
3
x =
96
= 32
3
This tells us that for 24 walkers, the organizers need to purchase
32 oranges.
Comparing Ratios
After the walkathon, Ellie’s team decides that they would like to go mini-golfing to
celebrate their efforts. They meet at Ellie’s house to decide which place offers the
better deal. Ellie finds the following deals:
Mini-Adventure Golf: $24 for 6 people
Marvelous Mini-Golf: $18 for 4 people
Which would be the better deal if there were 12 people playing?
Find the Unit Rate for each location
Based on the unit rate, Mini-Adventure Golf has a better deal, but is it the
best deal?
Comparing Unit Rates
Use the unit price to complete the rest of the table until you
arrive at 12 people
Mini-Adventure Golf
Marvelous Mini-Golf
Price
People
Price
People
$4.00
1
$4.50
1
$16.00
4
$18.00
4
$32.00
8
$36.00
8
$48.00
12
$54.00
12
It will cost $48 for 12 people at Mini-Adventure Golf and $54 for 12
people at Marvelous Mini-Golf. So Ellie and her friends decide to go
to Mini-Adventure.
Now that you completed this lesson, you should
be able to say:
• I can make tables of equivalent ratios.
•
I can use tables to find missing values and
compare ratios.
•
I can solve unit rate problems by reasoning
about ratios and rates.