Ratio & Rates
Reasoning
Name: ______________
1
Ratio
Reasoning
Ratios
Terms
Never
heard of
the term
I think I
know the
term
I know and
can explain
the term
1
1
1
I’ve seen
or heard
of the
term
2
2
2
Ratio
Ratio table
Equivalent
ratios
x-axis
y-axis
Coordinate
plane
Ordered pairs
per
Tape Diagram
3
3
3
4
4
4
1
1
1
2
2
2
3
3
3
4
4
4
1
1
1
2
2
2
3
3
3
4
4
4
2
Big Idea
Relationships of two quantities in such situations may be
described in terms of ratios, rates, percents or proportional
reasoning.
Essential Questions
1. What is a ratio?
2. How do ratio tables help you find equivalent ratios?
3. How does ratio reasoning help you solve real world
problems?
4. What determines the most efficient strategy when solving
proportional reasoning problems?
Standard 6.RP.3a
Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and
plot the pairs of values on the coordinate plane. Use tables to
compare ratios.
3
Ratios
Ratios arise in situations in which two (or more) quantities are related.
Ratios can be indicated in words as “3 to 2”, and “3 for every 2”. For example a
recipe may call for 3 cups of flour to every 2 cups of sugar. Ratios can be expressed
symbolically using the following:
3:2 or
3
2
The order of the numbers is important to the meaning of the ratio. Switching the
numbers changes the relationship. The description of the ratio relationship tells us
the correct order for the numbers in the ratio.
For the above example it shows the ratio of flour to sugar which is 3:2.
What would the ratio 2:3 represent?
Different ratios can be written using the same information. For example, all of the
following ratios can be written for the above situation.

flour to sugar

sugar to flour

flour to total cups

eggs to total cups
4
Practice Writing Ratios
1.
Write the ratio of pentagons to triangles. __________
Write the ratio of triangles to pentagons. __________
Write the ratio of pentagons to all the shapes. __________
Write the ratio of triangles to all the shapes. __________
2.
Write the ratio of bananas to pineapples. ________
Write the ratio of apples to bananas. ________
Write the ratio of pineapples to all the fruit. ________
Write the ratio of apples to pineapples to bananas. ________
Write the ratio of bananas to oranges. ________
Are there any other ratios that could be written from the picture above?
5
Ratio Recording Sheet
Set A: Baggie____
Set B: Baggie____
Set C: Baggie____
Ratios can be written from many different types of information.
Can you think of other sources in which ratios can be written from?
6
1.
A t-shirt manufacturing company surveyed teen-aged girls on their favorite tshirt color to guide their decisions about how many of each color t-shirt they
should design and manufacture. The results of their survey are shown here.
A. Write the ratio of green t-shirts to orange t-shirts. __________
B. Write the ratio of blue t-shirts to all the t-shirts. __________
C. Describe a ratio relationship from the graph for which the ratio is 3:5.
2. What is the ratio of the width of the rectangle to the height of the
rectangle? ____________
3. At the 6th grade school dance, there are 132 boys, 89 girls, and 14 adults.
A. Write the ratio of the number of boys to the number of girls. _________
B. Write the ratio of the number of boys to the number of adults. _________
C. Write the ratio of the total number of students to adults. ___________
4. In the lunch cafeteria, 100 milk cartons were put out for breakfast. At the
end of breakfast 27 remained.
A. What is the ratio of milk cartons taken to total milk cartons? ____________
B. What is the ratio of milk cartons remaining to milk cartons taken? _________
7
Equivalent Ratios
Two ratios are equivalent (are “equal ratios”) if one is obtained from the other by
multiplying or dividing all the measurements by the same (nonzero) number. Thus the
ratio 6:10 is equivalent to 18:30 (multiplying by 3) and also equivalent to 3:5 (dividing
by 2). Of these, the ratio 3:5 is in “simplest form” because the numbers 3 and 5 have
no common factor. When a ratio is in simplest form it is sometimes referred to as
“the value of the ratio.”
3:5
6:10
18:30
Equivalent Ratios
3:5 =
6:10
=
18:30
Double Number Line
0
3
6
18
0
5
10
30
Ratio Tables
3
5
6
10
18
30
3
5
6
10
18
30
Bar Modeling/Tape Diagram
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
8
Ratio tables- are comprised of columns and rows. The columns represent the ratios
and the rows represent the ratios that are equivalent. They can be written both
vertical and horizontal. The ratio relationship always needs to stay equivalent by
multiplying or dividing to get from one ratio to the next.
Ex. The ratio of cups of grapes to cups of peaches is 5:2. If there are 25 cups of
grapes, how many cups of peaches are there?
Cups grapes
Cups peaches
5
2
10
4
20
8
Fill in the missing pieces of the following ratio tables.
1.
2.
Ice
Cake
Pencils
Cream
1
4
1
5
8
3
5
6
3
18
16
16
3.
Fiction
Non Fiction
Pens
25
4.
Cups of Water
2
Cups of Flour
3
12
36
4
220
18
5.
Mice
Rats
14
20
7
4
24
Look at the completed table above. Where did you have to start to fill in the table?
Make a comparative statement about the relationship between the mice and rats.
9
Double Number Lines- are comprised of 2 number lines that represents the two
parts of a ratio.
Ex 1. The ratio of cups of grapes to cups of peaches is 5:2. If there are 25 cups of
grapes, how many cups of peaches are there?
0
5
10
20
0
2
4
8
Ex 2. The ratio of kids at Jake’s birthday party having cake to ice cream was 4:5.
Create a double number line to find equivalent ratios.
0
0
Ex 3. Is this double number line showing equivalent ratios? Explain.
0
3
9
16
24
0
4
12
20
32
10
Bar Modeling/Tape Diagram- is a visual representation that can be used to help solve
problems.
The ratio of cups of grapes to cups of peaches is 5:2.
Grapes
Peaches
A. What does each unit of the tape diagram represent?
B. What if each unit on the tape diagram represents 1? How many cups of grapes
are there? How many cups of peaches are there?
C. What if each unit on the tape diagrams represents 2? What is the amount of
cups for grapes? Peaches?
D. What is the ratio of cups of grapes to the cups of peaches now?
E. If there are 25 cups of grapes, what is each unit worth? How many cups of
peaches are there?
11
Equivalent Ratios- Equivalent ratios can be expressed by showing what number is
being multiplied or divided by the ratio to get the equivalent ratio.
Ex 1. The ratio of cups of grapes to cups of peaches is 5:2. If there are 25 cups of
grapes, how many cups of peaches are there?
5:2 and 25:____
Ex 2. In a bag mixed walnuts and cashews, the ratio of walnuts to cashews is 5:6.
Determine the amount of walnuts that are in the bag if there are 54 cashews.
5:6 and _____:_____
Ex 3. The morning announcements said that two out of every seven 6th graders in the
school have an overdue library book. Jasmine said, “That would mean 24 of us have
overdue books!” If Jasmine is correct, how many students are in the 6th grade?
Ex 4. Decide whether or not the following pairs of ratios are equivalent.
a.
6:11 and 42:88
b. 7:10 and 49:70
c. 13:8 and 39:27
12
Solving Ratio Word Problems
There are various ways to solve ratio word problems; equivalent ratios, tape diagrams
/bar modeling, ratio tables and double number line. Some problems work better with
certain methods. When looking at a ratio word problem you should always look at
what is given and what you’re looking for to help you determine the best method to
solve the problem.
Ex 1. The ratio of Isabel’s money to Rosalind’s money is 8 : 3. If Isabel has $24 how
much money do the two girls have together?
Equivalent Ratios:
Tape Diagram/Bar Modeling:
Ratio Table:
Double Number Line:
Ex 2. The ratio of boys to girls in the class is 5 : 3. There are 6 more boys than
girls. How many students are there in the class?
Ratio Table:
Tape Diagram/Bar Modeling:
Practice solving using two different methods.
13
1. Brianne made pineapple drinks by mixing pineapple syrup and water in the ratio
2 : 7. If she used 42 liters of water, how much pineapple syrup did she use? (Use
equivalent ratios and a ratio table to solve the problem)
2. David cut a rope 60 m long into two pieces in the ratio 2 : 3. What is the length of
the shorter piece of rope? (Use bar modeling and a ratio table to solve the problem)
Why did we not ask you to use the numerical equivalent ratios strategy? What was
more efficient, the ratio table or the bars?
3. The ratio of Adam’s weight to Johns’ weight is 6 : 5. If Adam weighs 48 kg, find
John’s weight. (Use a double number line to solve the problem)
Choose the strategy that you feel works most efficiently to solve the following
problems.
14
4. The ratio of the number of boys to the number of girls is 2 : 5. If there are 100
boys, how many children are there altogether?
5. Josie took a long multiple-choice, end of the year vocabulary test. The ratio of
the number of problems Josie got incorrect to the number of problems she got
correct is 2:9. If there was total of 121 multiple choice questions on the test, how
many did she get correct?
6. Wells College in Aurora, New York was previously an all-girls college. In 2005, the
college began to allow boys to enroll. By 2012, the ratio of boys to girls was 3 to 7.
If there were 200 more girls than boys in 2012, how many boys were enrolled that
year?
15
Tape diagrams/bar modeling can be used to solve any ratio problem. The tapes are
set up based on the ratio. The problem arises when trying to figure out where the
quantity is attached. Below are three problems based on the same ratio, but the
numbers represent different parts of the tape.
1.
The ratio of boys to girls is 5:2 in the classroom. If there are 25 boys, how
many girls are there?
2. The ratio of boys to girls is 5:2 in the classroom. If there is a total of 35
students, how many girls are there?
3. The ratio of boys to girls is 5:2 in the classroom. If there are 15 more boys
than girls, how many boys are there?
16
Ratio Tables and Graphing
The information in a ratio table can be represented visually by plotting the pairs of
values on a coordinate plane. A coordinate plane is created by a horizontal number
line called the x-axis and a vertical number line called the y-axis. An ordered pair is a
set of two numbers that identifies a location that can be graphed on the coordinate
plane. The graph of ordered pairs which are equivalent ratios lies on a line that goes
through the origin.
Ex 1. Dominic works on the weekends and on vacations from school mowing lawns in
his neighborhood. For every lawn he mows, he charges $12.
Complete the table. Then determine ordered pairs and create a labeled graph.
Lawns
Charge
2
4
6
8
10
a. How many lawns will Dominic need to mow in order to make $120? _______
b. How much money will Dominic make if he mowed 9 lawns? _______
17
Ex 2.
1.
How many bracelets did they make if they used 24 beads?
2. Jen and Nikki want to make 12 bracelets to sell at the market. How many
beads will they need? How many charms will they need?
3. How did the ratio table and graph help you answer question number 2?
18
Ex 3. At Books Unlimited, 3 paperbacks cost $18. Complete the table to show the
ratio of books to cost.
Number of Books (n)
Cost (c)
1
3
5
18
7
9
Create ordered pairs from the table, and plot the pairs on the graph below. Label the
axes of the graph and give it a title.
1. What happens to the cost as the number of books increases?
2. Why would knowing the cost of 1 book help you in finding the cost of any
amount of books?
3. What is the cost of 1 book?
19
4. A customer came into the store and wanted to buy 2 books, how much would it
cost?
5. A customer came into the store and wanted to buy 4 books, how much would it
cost?
6. Based on the graph, what is the cost of 10 books?
7. Write an equation to represent the relationship between the number of books
and the cost.
20
Ex 4. Complete the table of values to find the following:
Find the number of cups of sugar needed if for each pie Karrie makes, she has to use
3 cups of sugar.
Pies
1
Cups of
Sugar
3
2
3
4
Use the graph to represent the pies and the cups of sugar needed for 0-8 pies.
a.
Using the graph above, how many cups of sugar would be needed for 10 pies?
21
Ex 5. Kendra’s mom does the laundry at the laundry mat. It costs $3 to do 2 loads of
laundry. Fill in the ratio table.
Loads of
Laundry
Cost ($)
2
3
4
5
6
3
Graph the relationship on the coordinate plane below for doing 0-7 loads of laundry.
a. How much does it cost to do 1 load of laundry?
b. How much would it cost to do 8 loads of laundry?
c. If Kendra’s mom has $16.50, how many loads of laundry can she do?
22
Rates & Measurement
Terms
Never
heard of
the term
I think I
know the
term
1
1
1
1
1
1
1
1
I’ve seen
or heard
of the
term
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
I know and
can
explain
the term
4
4
4
4
4
4
4
4
Rate
Unit Rate
Unit Price
Conversion
Customary
Metric
Corresponds
Double Number
line
Constant speed
Capacity
Centi
Milli
Kilo
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
23
Big Idea
Ratio reasoning can be used to find unit rates in various
situations. Unit rates help us make informed decisions in our
daily lives.
Essential Questions
1. How are unit rates used in the real world?
2. Why is it important to know how to change measurement
units?
Standard 6.RP.2
Understand the concept of a unit rate a/b associated with a ratio a:b
with b not equal to 0, and use rate language in the context of a ratio
relationship.
Standard 6.RP.3b
Solve unit rate problems including those involving unit pricing and
constant speed.
Standard 6.RP.3d
Use ratio reasoning to convert measurement units; manipulate and
transform units appropriately when multiplying or dividing quantities.
24
A rate is a ratio that compares two quantities expressed in different kinds of units,
like miles and hours or dollars and pounds. When expressing a ratio as a rate, it is
necessary to label each part of the ratio.
Ex. Ratio- 120:2
Rate- 120 miles/2 hours
A unit rate describes how many units of the first type of quantity corresponds to
one unit of the second type of quantity. Examples of a unit rate are miles per hour
and dollars per pound.
List as many situations in which you have seen unit rate used in real life.
Unit Price is the cost per item or cost per unit of measure. Unit price can be
expressed as a ratio. It is also called cost per unit and unit cost.
Label the following with a R- rate, UR- Unit Rate, UP- Unit Price.
______
______
______
______
______
______
$15 for 5 cases
28 miles per gallon
$2.93 for 1 bag of pretzels
23 jumping jacks in 60 seconds
6 bags of chips for $13.98
65 miles per hour
25
Ex 1.
Diet cola was on sale last week: It cost $10 for every 4 packs of diet cola.
a. How much do 2 packs of diet cola cost?
b. How much does 1 pack of diet cola cost?
Packs of Cola
Total cost
4
10
What is the rate of the two quantities?
What is the unit price?
Ex 2. For every 2 minutes in the pool, Janice can swim 6 laps. Complete the following
table showing how many laps Janice completes per minute.
Minutes
Laps
2
6
In the above table, what unit rate did you find?
Could you find a different unit rate? If so what is it?
Is there another way to find the unit rate other than making a table?
26
Ex 3. The Scott family is trying to save as much money as possible. One way to cut
back on the money they spend is by finding deals while grocery shopping; however, the
Scott family needs help determining which stores have the better deals.
At Grocery Mart strawberries cost $2.99 for 2lbs, but at Baldwin Hills Market
strawberries are $3.99 for 3lbs.
a. What is the unit price of strawberries at each grocery store? Round to the
nearest penny, if necessary.
b. If the Scott family wanted to save money, where should they go to buy
strawberries? Why?
Practice:
1. Allison drove her car 90 miles and used 3 gallons of gas.
a. What is the car’s gas mileage in miles per gallon?
b. How much of a gallon does Allison need to drive every mile?
2. Eric wants to buy Skittles at Tops. Five pounds of Skittles costs $7.35. How
much does one pound cost?
27
3. Harry drank 6 glasses of apple juice that contained a total of 360 calories.
What was the unit rate of calories per glass? If Harry drinks only half of a
glass of apple juice, how many calories did he drink?
4. John is buying a new car and wants to buy a car that is very fuel efficient. Car
A can go 420 miles on 20 gallons of gas. Car B can go 360 miles on 15 gallons of
gas. Which car should John purchase?
5. During Sylvia’s trip across the country, she traveled 2,884 miles. Her trip took
7 days. Find the rate to represent the average miles she traveled in 2 days.
6. Mr. Allen drove to New Hampshire over the summer. It took him 8 hours make
the total trip of 400 miles. What was Mr. Allen’s rate of speed?
28
7. Charles made a long distance call that lasted 1 ¼ hours. The telephone service
provider charges $.05 per minute for the call. How much does Charles have to
pay for the call?
8. The 6th grade class is going to Washington D.C. for the class trip. The train
travels at a constant speed. The table shows the distance that the train
travels in various amounts of time.
Distance (mi)
Time (h)
120
2
150
240
3
5
a. Complete the table.
b. How far does the train travel in one hour?
c. How can you use your answer from Part B to determine the distance the train
travels in a given amount of hours?
9. Teagan went to Gamer Realm to buy new video games. Gamer Realm was having
a sale: $65 for 4 video games. Teagan bought 3 games for himself and one
game for his friend, Diego, but does not know how much Diego owes him for the
one game. What does Diego owe him for the video game?
29
Measurement
Customary System of Measurement is the measurement system used most commonly
in everyday life in the United States. The customary system includes units for
measuring length, weight, capacity, area, volume, and temperature.
Can you think of some examples of customary units?
Metric System of Measurement is a base-ten system of measurement. This system
is used in Canada as well as many other countries around the world.
Can you think of some examples of metric units?
30
Measurement Conversions
Using a conversion factor is essential in order to convert one unit to another. For
example if you are converting between inches and feet you need to know how many
inches are in one foot. (1 foot = 12 inches) In order to convert measurements you
may choose to use equivalent ratios, ratio tables or a double number lines.
A conversion factor is a fraction/ratio equal to 1 since the value described in the
numerator and denominator is the same.
For example:
12 𝑖𝑛𝑐ℎ𝑒𝑠
1 𝑓𝑜𝑜𝑡
or 12 in:1 ft is a conversion factor since the numerator and
denominator name the same amount.
Since the ratio is equivalent to 1, the identity property of multiplication allows an
amount to be multiplied by the ratio.
List other conversion factors that you know.
Why is
12 𝑖𝑛𝑐ℎ𝑒𝑠
1 𝑓𝑜𝑜𝑡
≠
12
1
?
31
Ex 1. Mark is making Gatorade to bring to his soccer team. He fills a 32 quart
container with Gatorade. How many gallons of Gatorade did Mark make?
Ex 2. Hector measured the weight of his dog to be 62.5 pounds. How many ounces
does Hector’s dog weigh?
Measurements can be converted between measurement systems. A customary unit
can be converted to a metric unit and a metric unit to a customary unit. For example
inches can be converted to centimeters.
1 inch = 2.54 centimeter
0.454 kilograms = 1 pound
Customary
Metric
Metric
Customary
Ex 1. My table measures 200 cm across. What does it measure in inches? Round to
the nearest tenth.
Ex 2. Fred ate at Red Robin and did the “Red Challenge” and ate a 3lb burger in half
an hour to win the challenge. What was the weight of the burger in kilograms?
32
Practice. You may use your conversion chart to help solve.
1. Sarah is measuring her little brother, Jonny’s height. He is 4 feet 5 inches tall,
how many inches tall is he?
2. Karen is making punch for a party. She is making 80 cups of punch. How many
gallons of punch is that?
3. Alex was riding his skateboard and jumped off a ramp 22 inches high. How many
centimeters did Alex jump?
4. Amy is buying chocolate chips to make cookies. They come in a bag of 36 ounces.
How many pounds of chocolate chips is Amy buying?
33
5. Lake Erie is about 1,302 feet deep at its deepest point. How many yards is that?
6. Jill and Ericka make 1.32 gallons of lemonade for their lemonade stand. How many
liters of lemonade will they be able to sell?
7. Claudia’s skis 5.5 miles on a snowy winter day. How many kilometers is that?
8. Beau buys a 3.5 pound bag of trial mix for a hike. He wants to make two-ounce
bags for his friends he is hiking with. How many two-ounce bags can he make?
9. Chad sprinted to a friend’s house next door and it was 334.645 inches away from
his house. How many meters did Chad sprint?
34
Metric Measurement
Measurement prefixes can be added to different units of measurement to change
their value. The three main units of measurement are meters, liters and grams.
Prefix
*base
unit
Meaning
ones
(1)
Length
Mass
Capacity
meter
gram
liter
kilo-
thousand
(1,000)
1 kilometer =
______________
1 kilogram =
1 kiloliter =
______________ ______________
deci-
tenths
(0.1)
1 meter =
____ decimeters
1 gram =
____ decigrams
1 liter =
____ deciliters
centi-
hundredths
(0.01)
1 meter =
____ centimeters
1 gram =
____ centigrams
1 meter =
____ centiliters
milli-
thousandths
(0.001)
1 meter =
____ millimeters
1 gram =
____ milligrams
1 meter =
____ milliliters
1. Mary bought 4,200 milliliters of water on a hot summer day. How many liters is
that?
2. Carlos has a 1.2 meter long piece of wood. He wants to cut it into 3 equal lengths.
How long should each piece be in millimeters?
35
3. Bill was trying to find the density of a block of wood and found the mass of the
wood to be 360 centigrams. How many grams is the piece of wood?
4. Ted picked out a pumpkin that weighed 52,400 grams. How many kilograms did
Ted’s pumpkin weight?
5. Claudia’s skis are 150 centimeters long. How many meters is this?
6. Joanie wants to frame a rectangular picture that is 1.7 meters by 0.9 meters.
Joanie has 500 centimeters of wood to use for the frame. Does Joanie have
enough to frame the photo? Prove your answer.
7. Mike is buying a snowboard that is 153 centimeters. How many meters long is his
snowboard?
36