Entire Unit 3 Ratios and Proportional Relationships

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UNIT 3: RATIOS AND
PROPORTIONAL RELATIONSHIPS
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LESSON 1: RATIOS
Please fill in your guided notes as you view the
presentation.
 Have fun!!

IN THE REAL WORLD

Baseball How can you compare a baseball
team’s wins to its losses during spring training?
A ratio uses division to compare two numbers.
There are three ways to write a ratio of two
numbers.
WRITING A RATIO
All three ways of writing the ratio of two numbers
are read “the ratio of a to b,” so 18 : 13 is read
“the ratio of eighteen to thirteen.” Two ratios are
equivalent ratios when they have the same
value.
EXAMPLE 1 WRITING A RATIO

1. You can make comparisons about games played by
the Cubs.
a. Wins to losses
wins = 17, losses = 14
Answer: _______
b. Wins to games played
wins = 17, games = 17 + 14 = 31 Answer: _________
YOUR TURN

Use the table above to write the ratio for
the Padres.
1. Wins to losses
2. Wins to games played
3. Losses to wins
EXAMPLE 2 WRITING RATIOS IN SIMPLEST
FORM

Amusement Parks A ride on a roller coaster lasts 2
minutes. Suppose you wait in line for 1 ½ hours to ride
the roller coaster. Follow the steps below to find the ratio
of time spent in line to time spent on the ride.
-Write hours as minutes so that the units are the same.
-Write the ratio of time spent in line to time spent on the
ride.
EXAMPLE 3 COMPARING RATIOS

1.
2.
Music Luis and Amber compared their CD
collections. To determine who has a greater ratio
of rock CDs to pop CDs, write the ratios.
Write ratios as fractions (rock/pop).
Write fractions as decimals and compare.

Luis:

Amber:
YOUR TURN
1.
Does Luis or Amber have a greater ratio of pop
CDs to hip-hop CDs?
1.
Does Luis or Amber have a greater ratio of hiphop CDs to rock CDs?
LESSON 2: RATES

A rate is a ratio of two quantities measured in
different units. A unit rate is a rate that has a
denominator of 1 unit. The three unit rates below
are equivalent. In the third rate, “per” means “for
every.”
EXAMPLE 1 FINDING A UNIT RATE

Kudzu During peak growing season, the kudzu
vine can grow 6 inches in 12 hours. What is the
growth rate of kudzu in inches per hour?
Solution
First, write a rate comparing the inches grown
to the hours it took to grow. Then rewrite the
fraction so that the denominator is 1.

ANSWER The growth rate of kudzu is about 0.5
inch per hour.
YOUR TURN: FIND THE UNIT RATE
1.
2.
3.
1.
$54 in 6 hours
68 miles in 4 days
2 cups in 8 servings
Average Speed If you know the distance
traveled and the travel time for a moving object,
you can find the average rate, or average speed,
by dividing the distance by the time.
EXAMPLE 2 FINDING AN AVERAGE SPEED

Speed Skating A skater took 2 minutes 30
seconds to complete a 1500 meter race. What was
the skater’s average speed?
Rewrite the time so that the units are the
same.
2 min + 30 sec = 120 sec + 30 sec = 150 sec
Find the average speed.
EXAMPLE 3 COMPARING UNIT RATES

Pasta A store sells the same pasta the following
two ways: 10 pounds of bulk pasta for $15.00 and
2 pounds of packaged pasta for $3.98. To
determine which is the better buy, find the unit
price for both types.
ANSWER The bulk pasta is the better
buy because it costs less per pound.
YOUR TURN
1.
It takes you 1 minute 40 seconds to walk 550
feet. What is your average speed?
2.
Which of the following is the better buy: 2 AA
batteries for $1.50 or 6 AA batteries for $4.80?
LESSON 3: SLOPE & UNIT RATES

The slope of a nonvertical line is the ratio of the
rise (vertical change) to the run (horizontal
change) between any two points on the line, as
shown below. A line has a constant slope.
EXAMPLES OF LINES WITH POSITIVE,
NEGATIVE, AND ZERO SLOPES ARE SHOWN

The slope of a vertical line is undefined
EXAMPLE 1 FINDING THE SLOPE OF A
LINE

a.
b.
To find the slope of a line, find the ratio of the
rise to the run between two points on the line
EXAMPLE 2 INTERPRETING SLOPE AS A
UNIT RATE
Slope as a rate – When the graph of a line
represents a real-world situation, the slope of the
line can often be interpreted as a rate.
 Volcanoes The graph represents the distance
traveled by a lava flow over time. To find the
speed of the lava flow, find the slope of the line.

YOUR TURN
1.
Plot the points (3, 4) and (6, 3). Then
find the slope of the line that passes
through the points.
2. In Example 2, suppose the line starts at
the origin and passes through the point
(3, 6). Find the speed of the lava flow.
EXAMPLE 3 USING SLOPE TO DRAW A
LINE

Draw the line that has a
slope of –3 and passes
through (2, 5).
1. Plot (2,5)
2. Write the slope as
a fraction
3. Move 1 unit to the
right, and 3 units
down to plot the
second point
4. Draw a line
through the two
points
YOUR TURN
3. Draw the line that has a slope of 1/3 and passes
through (2, 5).
LESSON 4: WRITING AND SOLVING
PROPORTIONS


Sports A person burned about 150 calories while skateboarding for 30
minutes. About how many calories would the person burn while skateboarding
for 60 minutes? In Example 1, you will use a proportion to answer this
question.
A proportion is an equation that states that two ratios are equivalent.
Algebra =
, where b and d are nonzero numbers.
EXAMPLE 1 USING EQUIVALENT RATIOS


Sports A person burned about 150 calories while
skateboarding for 30 minutes. About how many
calories would the person burn while
skateboarding for 60 minutes? In Example 1, you
will use a proportion to answer this question.
To find the number C of calories the person
would burn while skateboarding for 60 minutes,
solve the proportion = .
Answer: Ask yourself, what number can you multiply
by 30 to get 60? The answer is 2. So, multiply 150 x
2 to get 300. So the person would burn about 300
calories while skateboarding for 60 minutes.
EXAMPLE 2 SOLVING PROPORTIONS USING
ALGEBRA

Solve the proportion =
.
.
YOUR TURN

1.
Use equivalent ratios
to solve the
proportion.

5.
6.
2.
7.
3.
8.
4.
Use algebra to solve
the proportion.
EXAMPLE 3 WRITING AND SOLVING A
PROPORTION


Setting Up a Proportion There are different
ways to set up a proportion. Consider the
following problem.
Yesterday you bought 8 bagels for $4. Today
you want only 5 bagels. How much will 5
bagels cost?
EXAMPLE 3 WRITING AND SOLVING A
PROPORTION

Empire State Building

Step 1: Write a proportion

Step 2: Solve the proportion
At maximum speed,
the elevators in the Empire State Building
can pass 80 floors in 45 seconds. Follow the
steps below to find the number of floors
that the elevators can pass in 9 seconds.
LESSON 5: SOLVING PROPORTIONS USING
CROSS PRODUCTS
In the Real World
 Science At space camp, you can sit in a
chair that simulates the force of gravity on
the moon. A person who weighs 105 pounds
on Earth would weigh 17.5 pounds on the
moon. How much would a 60 pound dog
weigh on the moon? You’ll find the answer
in Example 2.
In the proportion , the products 2 • 6 and 3 • 4
are called cross products. Notice that the
cross products are equal. This suggests the
following property.
CROSS PRODUCTS PROPERTY
Cross Products Property
Words The cross products of a proportion are equal.
EXAMPLE 1 SOLVING A PROPORTION
USING CROSS PRODUCTS
Use the cross products property to solve.
EXAMPLE 2 WRITING AND SOLVING A
PROPORTION

Science At space camp, you can sit in a
chair that simulates the force of gravity on
the moon. A person who weighs 105 pounds
on Earth would weigh 17.5 pounds on the
moon. How much would a 60 pound dog
weigh on the moon?
EXAMPLE 3 WRITING AND SOLVING A
PROPORTION
Penguins At an aquarium, the ratio of
rockhopper penguins to African penguins is
3 to 7. If there are 50 penguins, how many
are rockhoppers?
 First, determine the ratio of rockhoppers to
total penguins. Then, set up proportion.

YOUR TURN

In John’s class, the ratio of boys to girls is 5
to 8. If there are 39 students in his class,
how many are girls?
LESSON 6: SCALE DRAWINGS AND
MODELS

The floor plan below is an example of a scale
drawing. A scale drawing is a diagram of an
object in which the dimensions are in
proportion to the actual dimensions of the
object.
The scale on a scale
drawing tells how
the drawing’s
dimensions and the
actual dimensions
are related. The scale
“1 in. : 12 ft” means
that 1 inch in the
floor plan represents
an actual distance of
12 feet.
EXAMPLE 1 USING THE SCALE OF A MAP


Maps Use the map of
Maine to estimate the
distance between the
towns of China and New
Sweden.
Solution From the map’s
scale, 1 centimeter
represents 65 kilometers.
On the map, the distance
between China and New
Sweden is 4.5 centimeters.
Find the distance in
kilometers.
EXAMPLE 2 FINDING A DIMENSION ON A
SCALE MODEL


White House A scale model of the White
House appears in Tobu World Square in
Japan. The scale used is 1 : 25. The height of
the main building of the White House is 85
feet. Find this height on the model.
Write and solve a proportion to find the height h
of the main building of the model of the White
House.
EXAMPLE 3 FINDING THE SCALE


Dinosaurs A museum is creating
a full-size Tyrannosaurus rex
from a model. The model is 40
inches in length, from the nose to
the tail. The resulting dinosaur
will be 40 feet in length. What is
the model’s scale?
Write a ratio. Make sure that both
measures are in inches. Then simplify
the fraction.
ANSWER The model’s scale is 1 : 12.
YOUR TURN
1.
The model of the Eiffel Tower in Tobu World Square
is 12 meters high. The scale used is 1 : 25.
Estimate the actual height of the Eiffel Tower.
2.
On a map of Colorado, the distance from Rico to
Lizard Head Pass on Route 145 is about 9.5 cm.
From the map scale, 1 cm represents 2 km.
Estimate the actual distance between the
towns.
3.
The caboose on a model train is 6.75 inches long.
The full-size caboose is 36 feet. What is the
model’s scale?

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