Guided Notes Chapter 4
Name_______________________________
Unit 3 ~ Chapter 4 ~ Section 1 ~ Ratios
Ratio: a comparison of two quantities by ______________
You can write a ratio in three ways: ______________
_______________
______________
Equivalent Ratio: two ratios that are _____________
Writing Ratios:
Use the table to answer questions 1-3: Write each ratio as a fraction in simplest form.
1) Tigers wins to Tigers losses
2) Leopards wins to Leopards losses
Panthers
Leopards
Lions
Tigers
Wins
12
9
8
6
Losses
6
9
10
12
3) Lions wins to Tigers wins
Writing Equivalent Ratios:
4
4) Find a ratio equivalent to 5
1
5) Find a ratio equivalent to 8
6) Write the ratio 2 yd to 20 ft as a fraction in simplest form
7) Write the ratio 3 gal to 10 qt as a fraction in simplest form
1
Comparing Ratios:
8) An official U.S. flag has a length-to-width ratio of 19:10. The largest U.S. flag measures 505 ft by 255 ft. Is
this an official U.S. flag?
Tell whether the ratios are equivalent or not equivalent.
9) 7: 3, 128: 54
10) 6.1 to 7, 30.5 to 35
11)
180
240
,
25
34
Write in simplest form.
12)
2.4
13)
16
8.5
15
Bell Ringer9
1) Write the ratio 7 in two other ways.
2) Are
18
9
𝑎𝑛𝑑
16
4
equivalent?
3) To make pancakes, you need 2 cups of water for every 3 cups of flour. Write an equivalent ratio to find
how much water you will need with 9 cups of flour.
2
Challenge:
A bag contains colored marbles. The ratio of red marbles to blue marbles is 1:4. The ratio of blue marbles to
yellow marbles is 2:5. What is the ratio of red marbles to yellow marbles?
Error Analysis
Your math class includes 15 girls and 10 boys. Two new students, a girl and a boy, join the class. Your friend
says the ratio of girls to boys is the same as before. Explain your friends error.
Chemistry
A chemical formula shows the ratio of atoms in a substance. The formula for carbon dioxide, CO2, tells you
that there is 1 atom of carbon (C) for every 2 atoms of oxygen (O). Write a ratio of hydrogen (H) atoms to
oxygen atoms (O) in water, H2O.
3
Chapter 4 ~ Section 2 ~ Unit Rates and Proportional Reasoning
Rate: a ________ that compares two quantities measured in different units
Example: 15 grams of fat in 3 servings
Unit Rate: a rate for ____________ of a given quantity
Example: 15 grams of fat in 3 servings (reduce) =
Unit Cost: a unit rate that gives the ___________________________
Example: $3.50 for 5 apples (reduce) =
Finding a Unit Rate Using Whole Numbers:
1) A package of cheddar cheese contains 15 servings and has a total of 147 grams of fat. Find the unit rate of
grams of fat per serving.
2) Find the unit rate for 210 heartbeats in 3 minutes.
3) Find the unit cost for $42 for 4 shirts.
Finding a Unit Rate Using Decimals and Fractions:
6
1
4) Cindy walks 10 mile in 4 hour. What is her speed in miles per hour?
3
3
5) Find the unit rate for 10 mile in 4 hour.
4
6) Find the unit cost for $3.45 for 3.7 oz.
7) Two sizes of shampoo bottles are shown. Which size is the better buy? Round to the nearest cent.
16oz, $6.19
13.5oz, $3.99
8) Which bottle of apple juice is the better buy: 48 fl oz for $3.05 or 64 fl oz for $3.59?
9) Find the better buy: 8 pens for $3.60 or 12 pens for $4.80.
Bell Ringer1) What is the relationship between a rate and a ratio?
2) What is a unit rate?
3) Find the unit rate: Earn $33 for 3 hours of work.
4) Find each unit cost, then determine the better buy: 3 lb of potatoes for $0.89, or 5 lb of potatoes for $1.59.
5
Bell RingerCompare. Use <, >, or =.
a)
3
_____
4
−5
3
b)
8
7
12
_____ 28
c)
2
3
11
_____ 15
d)
−1
6
−1
_____ 12
Chapter 4 ~ Section 3 ~ Proportions
Proportion: an equation stating that two ratios are _______________
1
Example: 2 =
4
8
Cross Products: the two products found by ___________ the denominator of each ratio by the numerator of
the other ratio.
Example:
6
=
8
9
12
If two ratios have equal cross products, they form a __________________
Writing Ratios in Simplest Form
1) Determine whether the ratios can form a proportion by writing ratios in simplest form.
a)
1
2
14
, 28
b)
10
12
40
, 56
10
25
c) 24 , 60
d)
18
4.8
12
, 3.6
20
60
Using Cross Products
2) Determine whether the ratios can form a proportion using cross products.
a)
5
9
,
30
54
b)
−7 −55
8
,
65
c)
3
8
6
, 16
d) 1.5 , 4.5
6
Bell Ringer1) What are two methods for determining whether a pair of ratios can form a proportion?
2) Determine whether the ratios can form a proportion.
a)
12
9
16
30
, 12
45
b) 16 , 12
Challenge
1) An astronaut who weighs 174 lbs on Earth weight 29 lbs on the moon. If you weigh 102 lbs on Earth, would
you weigh 17 lbs on the moon? Explain.
2) Determine whether
4𝑛
3
𝑎𝑛𝑑
12𝑛
9
always, sometimes, or never form a proportion. Explain.
3)G Geometry: Is the ratio of b to h the same in both triangles?
h=15cm
h=9cm
b=12cm
b=20cm
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Chapter 4 ~ Section 4 ~ Solving Proportions
Using Unit Rates
1) You know the price of six oranges , but you need eight.
6 oranges is $2.34.
Step 1: Find the unit rate.
Step 2: Multiply to find the cost of 8 oranges.
2) Postcards cost $2.45 for 5 cards. How much will 13 cards cost?
Using Mental Math
3) Solve each proportion using mental math.
a)
𝑧
12
=
21
8
b)
36
10
=
𝑛
40
Using Cross Products
4) Solve each proportion using cross products.
25
a) 38 =
15
𝑥
b)
12
15
=
𝑥
21
c)
16
30
𝑑
= 51
20
d) 35 =
110
𝑚
8
Application- Write and solve a proportion for each situation.
5) If 12 roses cost $21.96, what is the cost of 5 roses?
6) If 3 onions weigh 2.25 lb, how much do 10 onions weigh?
7) Franklin D. Roosevelt was elected president in 1932 with about 22,800,000 votes. The ratio of the number
of votes he received to the number of votes the other candidates received was about 4:3. About how many
votes did the other candidates receive?
Challenge
1) You estimate you will take 75 min to bike 15 mi to a state park. After 30 min, you have traveled 5 mi. Are
you on schedule?
2)
𝑛+2
8
=
7
4
3)
𝑥+4
15
=
𝑥+20
45
9
3
4) A jet takes 5 4 h to fly 2,475 mi from New York City to Los Angeles. About how many hours will a jet flying
at the same average rate take to fly 5,452 mi from Los Angeles to Tokyo?
5) Error Analysis: A videocassette recorder uses 2 m of tape in 3 min when set on extended play. To
determine how many minutes a tape that is 240 m long can record on extended play, one student wrote the
2
proportion3 =
𝑛
. Explain why this proportion is incorrect. Then write a correct proportion.
240
6) Health: Your heart rate is the number of heart beats per minute.
a) What is your heart rate if you count 18 beats in 15 seconds?
b) How many beats do you count in 15 seconds if your heart rate is 96 beats/min?
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1
7) A recipe for fruit salad serves 4 people. It calls for 22 oranges and 16 grapes. You want to serve 11 people.
How many oranges and how many grapes will you need.
Bell Ringer1) The cost of 5 CD’s is $42. At this rate, what is the cost of 7 CD’s?
Solve each proportion
2)
72
45
=
8
𝑛
3)
𝑏
18
=
9
5
4)
19
𝑟
=
152
4
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Chapter 4 ~ Section 5 ~ Similar Figures
When two figures have the same shape, but not necessarily the same size, they are ___________________
In similar triangles, _______________ angles have the ___________________________!
_______________________ are proportional!
You write _____________________________________ The symbol ~ means “is similar to.”
F
A
83°
83°
40
34
B
44°
53°
50
60
51
C
G
53°
44°
H
75
Polygon: a closed plane figure formed by _____________________ line segments that do not cross
Similar Polygons: corresponding angles have the ______________ and the lengths of the corresponding sides
form ____________________________
You can use proportions to find missing side lengths in similar polygons!
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Finding a Missing Measure
1) Triangle ACT and triangle ODG are similar. Find the value of x.
C
D
32 cm
40 cm
X cm
T
50 cm
24 cm
G
A
30 cm
O
2) The trapezoids are similar. Find the value of x.
D
6
G
L
143°
143°
10
5
3
37°
E
12
O
10
F
N
6
37°
x
M
3) Triangle ABC and triangle ARS are similar. Find the value of x.
A
300 m
75
m
B
R
60 m
C
x
S
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Application:
Geometry: A rectangle with an area of 32 in² has one side measuring 4 in. A similar rectangle has an area of
288 in². How long is the longer side in the larger rectangle?
A triangle with a perimeter of 26 in has two sides that are 8 in long. What is the length of the third side of a
similar triangle which has two sides that are 12 in long?
A 6 ft tall person standing near a flagpole casts a shadow 4.5 ft long. The flagpole casts a shadow 15 ft long.
What is the height of the flagpole?
A 6 ft tall person has a shadow of 5 ft long. A nearby tree has a shadow of 30 ft long. What is the height of the
tree?
14
4 cm
Which is similar to the model?
2 cm
6 cm
4 cm
5 cm
3 cm
4 cm
4 cm
3 cm
8 cm
Use proportions to find which two triangles are similar triangles.
4 cm
Y
4 cm
2 cm
L
A
X
5 cm
5 cm
6 cm
6 cm
Z
B
2 cm
C
M
3 cm
N
15
Bell RingerUse proportions to find the height of the tree.
5 ft
20 ft
4 ft
Challenge
The ratio of the corresponding sides of two similar triangles is 4:9. The sides of the smaller triangle are 10cm,
16cm, and 18 cm. Find the perimeter of the larger triangle.
Closure
1) The angle measures of a triangle are 40, 60, and 80. What are the angle measures of a similar triangle?
2) The side lengths of a triangle are 40cm, 60cm, and 80cm. The smallest side length of a similar triangle is
120 cm. What are the lengths of the other two sides?
3) A 5 ft person near a tree has a shadow 12 ft long. The tree has a shadow of 42 ft long. What is the height of
the tree?
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Bell Ringer- Use proportions to solve the following problems:
1) Three gallons of gasoline cost $3.36. How much do 5 gallons cost?
2) At a rate of 50 mph, a car travels a distance of 600 miles. How far will the car travel at a rate of 40 mph if it
is driven the same amount of time?
3) If the rent for two weeks is $500, how much is the rent for 5 weeks?
Chapter 4 ~ Section 6 ~ Scale Drawings
Scale Drawings: an enlarged or reduced drawing of an object that is similar to the actual object
Scale: the ratio that compares a length in a drawing or model to the corresponding length in the actual object
𝑑𝑟𝑎𝑤𝑖𝑛𝑔 𝑙𝑒𝑛𝑔𝑡ℎ
Scale = 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑎𝑐𝑡𝑢𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ
Example: If a 15 foot boat is 1 inch long on a drawing, you can write the scale of the drawing in the following
ways:
1 in : 15 ft
1 𝑖𝑛𝑐ℎ
15 𝑓𝑒𝑒𝑡
1 in = 15 ft
Using a Scale Drawing:
1) The length of the side of a house is 3 cm on a scale drawing. What is the actual side of the house?
1 cm = 2.5m
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2) The chimney of the house is 4 cm tall on the drawing. How tall is the chimney of the actual house?
1 cm = 2.5m
Finding the Scale Model
3) The length of the model boxcar is 7 in. The actual length of the boxcar is 609 in. What is the scale of the
model? Write in simplest form.
4) The length of a room in an architectural drawing is 10 in. Its actual length is 160 in. What is the scale of the
drawing? Write in simplest form.
5) You want to make a scale model of a sailboat that is 51 ft long and 48 ft tall. You plan to make the model 17
in long. Which equation can you use to find x, the height of the model?
a)
48
51
=
17
𝑥
b)
17
51
=
𝑥
48
c)
48
17
=
𝑥
51
d)
𝑥
17
=
51
48
6) If the sailboat is 15 ft wide, how wide should the model be? Write a proportion and fill in the information
you know.
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Finding the Scale of a Map
1
5
7) The map key shows that a map distance of 4 in represents an actual distance of 8mi. Find the actual
distance represented by 1 in to write the scale of the map.
1
2
8) Find the actual distance represented by 1in to write the scale of the map with a key 4 in = 5 mi.
9) A model boat is 3.5 ft long. The scale model is 1:10. What is the actual length of the boat?
10) A living room is 15 ft wide and 18 ft long. The scale of a floor plan of the house is 1 in : 10 ft. Find the
width and the length of the living room on the floor plan.
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Challenge
A special effects artist has made a scale model of a dragon for a movie. In the movie, the dragon will appear to
be 16 ft tall. The model is 4 in tall. What scale has the artist used?
The same scale is used for a model of a baby dragon, which will appear to be 2 ft tall.
What is the height of the model?
Closure
1) The actual length of a wheelhouse of a mountain bike is 260 cm. The length of the wheelbase in the scale
drawing is 4 cm. Find the scale of the drawing.
2) An architect’s model of a house is 44in high. The actual house is 20 ft high and 45 ft wide. What should the
width of the architect’s model be?
20
2
4
3) The scale on the map shows that a map distance of 5 in represents an actual distance of 5 mi. Find the
actual distance represented by 1 in to write the scale of the map.
Bell Ringer- Use proportions to solve the following problems:
1) The scale of a map is 1 cm: 25 km. Find the actual distance for a map of 6.2 cm.
2) The actual length of a machine part is 40 in. The length of the machine part in a scale drawing is 5 in. Find
the scale of the drawing.
3) A scale drawing of a playground is 4.7 cm wide and 6.2 cm long. The playground is 15 m wide. How long is
it?
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Chapter 4 ~ Section 7 ~ Proportional Relationships
Constant of Proportionality: the value of the ratio of quantities in a ________________________ relationship
Using a Table to Determine a Proportional Relationship
1) The table below shows the distance Keisha traveled during a bike race. Is there a relationship between
time and distance?
Hours
0
2
4
5
7
Miles
0
13
26
32.5
45.5
2) The table below shows the distances Dave rode in the same bike race. Is there a proportional relationship
between his time and distance?
Hours
Miles
0
0
3
18.6
6
35.2
8
49.6
9
56.8
Using a Graph to Find a Unit Rate
3) The graph on the right displays the data about Kiesha’s bike-athon. What is Keisha’s speed in miles per
hour?
(7 , 45.5)
Distance (mi)
(5 , 32.5)
(4, 26)
(2, 13)
(1, r)
Time (h)
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4) Use the graph on the right. What is Damon’s reading speed in pages per day?
Pages
(3, 45)
(2, 30)
Time (days)
Using a Ratio to Identify a Unit Rate
5) The table below shows a proportional relationship between the number of minutes and the amount the customer
pays for cell phone service.
Minutes, m
Price, p (dollars)
100
10
500
50
1000
100
1500
150
a) Identify the constant of proportionality.
b) Use the constant of proportionality to write an equation to find the price for m minutes.
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Bell Ringer- Find the constant of proportionality for each table below
a) yards of cloth per blanket
Yards (y)
16
Blanket(s) 8
32
16
b) pay per hour
40
20
Hours (h)
2
10
16
Pay (p)
$11
$55
$88
ChallengeJasmine baby sat for 3 ½ h one day and 4 h 20 min the next day. She earned $47. Write an equation using the constant
of proportionality to describe the relationship between e earnings and h hours worked.
Closure
1) Does the table below represent a proportional relationship? Explain your reasoning.
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2) The graph on the smart board shows the number of laps Ken swims each day. How many laps does Ken swim each
day?
3) A lake that is 8 fathoms deep is 48 feet deep. Write an equation to find the number of feet x in f fathoms.
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