Download Ratios and Rates

6.1
Ratios and Rates
Goal: Find ratios and unit rates.
Vocabulary
Ratio:
Equivalent
ratios:
Rate:
Unit rate:
Writing Ratios
You can write the ratio of two quantities, a and b, where b is not
equal to 0, in three ways.
a to b
a
b
a:b
Each ratio is read “the ratio of a to b.” You should write the ratio
in simplest form.
Writing Ratios
Example 1
In a recent baseball season, the Anaheim Angels played 81 home
games. Anaheim won 54 of those games and lost 27. Write the
ratio in three ways.
a. The number of losses to the number of wins
b. The number of losses to the number of games
Solution
a.
Copyright © Holt McDougal. All rights reserved.
b.
Three ways to write the ratio
Three ways to write the ratio
are
are
,
, and
.
,
, and
Chapter 6 • Pre-Algebra Notetaking Guide
.
111
6.1
Ratios and Rates
Goal: Find ratios and unit rates.
Vocabulary
Ratio: A ratio uses division to compare two quantities.
Equivalent Two ratios are called equivalent ratios when they
have the same value.
ratios:
Rate: A ratio of two quantities measured in different units.
Unit rate: A rate that has a denominator of 1 when expressed
in fraction form.
Writing Ratios
You can write the ratio of two quantities, a and b, where b is not
equal to 0, in three ways.
a to b
a:b
a
b
Each ratio is read “the ratio of a to b.” You should write the ratio
in simplest form.
Example 1
Writing Ratios
In a recent baseball season, the Anaheim Angels played 81 home
games. Anaheim won 54 of those games and lost 27. Write the
ratio in three ways.
a. The number of losses to the number of wins
b. The number of losses to the number of games
Solution
a.
Copyright © Holt McDougal. All rights reserved.
1
27
Number of
2
5
4
losses
Number of
wins
b.
Number of
losses
Number of
games
1
27
8
1 3
Three ways to write the ratio
Three ways to write the ratio
1
are 2 , 1 to 2 , and 1 : 2 .
1
are 3 , 1 to 3 , and 1 : 3 .
Chapter 6 • Pre-Algebra Notetaking Guide
111
Checkpoint
1. Use the information given in Example 1. Compare the number
of wins to the number of games using a ratio. Write the ratio in
three ways.
Example 2
Finding a Unit Rate
Vacation On the first day of a family vacation, you and your family
drive 392 miles. The amount of gasoline used is 16 gallons. What
is the average mileage per gallon of gasoline?
Solution
First, write a rate comparing the
to the
. Then write the rate so the denominator
is
.
Divide numerator and
denominator by
.
Simplify.
Answer: The average mileage per gallon of gasoline is
.
Checkpoint Find the unit rate.
220 mi
4h
2. 112
Chapter 6 • Pre-Algebra Notetaking Guide
$115
5 people
3. Copyright © Holt McDougal. All rights reserved.
Checkpoint
1. Use the information given in Example 1. Compare the number
of wins to the number of games using a ratio. Write the ratio in
three ways.
2
, 2 to 3, 2 : 3
3
Example 2
Finding a Unit Rate
Vacation On the first day of a family vacation, you and your family
drive 392 miles. The amount of gasoline used is 16 gallons. What
is the average mileage per gallon of gasoline?
Solution
First, write a rate comparing the total distance traveled to the
number of gallons used . Then write the rate so the denominator
is 1 .
392 miles 16
392 miles
16 gallons 16
16 gallons
24.5 miles
1 gallon
Divide numerator and
denominator by 16 .
Simplify.
Answer: The average mileage per gallon of gasoline is
24.5 miles per gallon .
Checkpoint Find the unit rate.
220 mi
4h
$115
5 people
2. 3. 55 mi
1h
112
Chapter 6 • Pre-Algebra Notetaking Guide
$23
1 person
Copyright © Holt McDougal. All rights reserved.
Example 3
Writing an Equivalent Rate
Water The amount of water used in a certain home is 728 gallons
per week. Write this rate in gallons per day.
Solution
To convert from gallons per week to gallons per day, multiply the rate
1.
by a conversion factor. There are 7 days in 1 week, so
728 gal
1 week
p
Multiply rate by conversion
factor.
p
Divide out common factor
and unit.
Simplify.
Answer: The amount of water used is
Example 4
.
Using Equivalent Rates
Weather Lightning strikes occur about 100 times per second around
the world. About how many lightning strikes occur in 3 minutes?
Solution
1. Express the rate 100 times per second in times per minute.
100 times
1 sec
p
Multiply by conversion
factor. Divide out
common unit.
Simplify.
2. Find the number of times lightning strikes occur around the
world in 3 minutes.
Number of times Rate p Time
Substitute values.
p
Divide out
common unit.
Multiply.
Answer: In 3 minutes, about
around the world.
Copyright © Holt McDougal. All rights reserved.
lightning strikes occur
Chapter 6 • Pre-Algebra Notetaking Guide
113
Example 3
Writing an Equivalent Rate
Water The amount of water used in a certain home is 728 gallons
per week. Write this rate in gallons per day.
Solution
To convert from gallons per week to gallons per day, multiply the rate
1 week
1.
by a conversion factor. There are 7 days in 1 week, so 7 days
728 gal
728 gal
1 week
p 7 days
1 week
1 week
Multiply rate by conversion
factor.
104
7
28 gal
eek
1w
1w
eek
days
7
p Divide out common factor
and unit.
1
104 gal
1 day
Simplify.
Answer: The amount of water used is 104 gallons per day .
Example 4
Using Equivalent Rates
Weather Lightning strikes occur about 100 times per second around
the world. About how many lightning strikes occur in 3 minutes?
Solution
1. Express the rate 100 times per second in times per minute.
60 se
c
100 times
100 times
p 1 sec
1 min
1
sec
6000 times
1 min
Multiply by conversion
factor. Divide out
common unit.
Simplify.
2. Find the number of times lightning strikes occur around the
world in 3 minutes.
Number of times Rate p Time
Substitute values.
6000 times
3
min
p
Divide out
1
min
common unit.
18,000 times
Multiply.
Answer: In 3 minutes, about 18,000 lightning strikes occur
around the world.
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Chapter 6 • Pre-Algebra Notetaking Guide
113
Focus On
Measurement
Converting Rates Between Systems
of Measurement
Use after Lesson 6.1
Goal: Convert rates from one system of measurement to another.
Writing an Equivalent Rate
Example 1
Fruit A supermarket sells apples for $3 per pound. What is this
rate in dollars per kilogram?
Solution
You need to write $3 as
1 lb
. Use the conversion
≈ 1.
factor
$3
≈
1 lb
Multiply rate by conversion
factor. Divide out common unit.
p
Use a calculator to write as
a unit rate.
≈
Answer:
about
per kilogram
Writing an Equivalent Rate
Example 2
Pool The water pump on a public pool can filter 1000 gallons of
water in 5 minutes. What is this rate in kiloliters per minute?
Solution
1. Convert the numerator of
≈
≈
114
Chapter 6 • Pre-Algebra Notetaking Guide
p
1 gal
to
p
.
Multiply rate by
conversion factors.
Divide out common
factor and units.
Simplify.
Copyright © Holt McDougal. All rights reserved.
Focus On
Measurement
Converting Rates Between Systems
of Measurement
Use after Lesson 6.1
Goal: Convert rates from one system of measurement to another.
Writing an Equivalent Rate
Example 1
Fruit A supermarket sells apples for $3 per pound. What is this
rate in dollars per kilogram?
Solution
You need to write $3 as ? dollars . Use the conversion
1kg
1 lb
factor
1 lb
≈ 1.
0.454 kg
$3
$3
1
lb
≈ p 1 lb
1
lb
0.454 kg
Multiply rate by conversion
factor. Divide out common unit.
Use a calculator to write as
a unit rate.
$6.61
1 kg
≈ Answer:
about $6.61 per kilogram
Writing an Equivalent Rate
Example 2
Pool The water pump on a public pool can filter 1000 gallons of
water in 5 minutes. What is this rate in kiloliters per minute?
Solution
1. Convert the numerator of
1000 gal
5 min
≈
≈
114
Chapter 6 • Pre-Algebra Notetaking Guide
1000 gal
5 min
p
0.757 kL
1 min
1000 gal
5 min
3.785 L
1 gal
p
to kiloliters .
1 kL
1000 L
Multiply rate by
conversion factors.
Divide out common
factor and units.
Simplify.
Copyright © Holt McDougal. All rights reserved.
continued
Example 2
2. Convert the
of
≈
to
.
Multiply rate by conversion
factors. Divide out common
factor and units.
≈
Answer:
Simplify.
about
Example 3
.
kiloliters per hour
Using Equivalent Rates
Airplane An airplane travels at 54 kilometers per minute. About
how many miles does the airplane travel in 2 hours?
Solution
1. Convert from kilometers per
≈
Simplify.
2. Find the distance (in
) that the airplane travels in
.
Substitute values. Divide out
common unit.
Multiply.
.
Multiply by conversion factor.
≈
Answer:
Copyright © Holt McDougal. All rights reserved.
.
Write formula for distance.
·
≈
.
Multiply rate by conversion
factor. Divide out common factor
and units.
.
≈
to kilometers per
Divide out common unit.
Use a calculator.
about
miles
Chapter 6 • Pre-Algebra Notetaking Guide
115
continued
Example 2
2. Convert the denominator of
0.757 kL .
≈
0.757 kL
1 min
1 min
≈
Answer:
0.757 kL
1 min
60 min
1 hr
45 kL
1 hr
to hours .
Multiply rate by conversion
factors. Divide out common
factor and units.
Simplify.
about 45
kiloliters per hour
Using Equivalent Rates
Example 3
Airplane An airplane travels at 54 kilometers per minute. About
how many miles does the airplane travel in 2 hours?
Solution
1. Convert from kilometers per minute to kilometers per hour .
54 km
1 min
54 km .
≈
1
min
≈
60 min
1h
3240 km
1h
Multiply rate by conversion
factor. Divide out common factor
and units.
Simplify.
2. Find the distance (in miles ) that the airplane travels in 2 hours .
Distance Rate · Time
3240 km
1h
. 2h
6480 km
≈ 6480 km .
Copyright © Holt McDougal. All rights reserved.
Substitute values. Divide out
common unit.
Multiply.
1 mi
1.609 km
≈ 4027 mi
Answer:
Write formula for distance.
Multiply by conversion factor.
Divide out common unit.
Use a calculator.
about 4027 miles
Chapter 6 • Pre-Algebra Notetaking Guide
115
6.2
Writing and Solving Proportions
Goal: Write and solve proportions.
Proportions
Words A proportion is an equation that states that two ratios
are equivalent.
2
3
8
12
Numbers a
b
c
d
Algebra , where b 0 and d 0
Example 1
Solving a Proportion Using Equivalent Ratios
3
5
x
20
Solve the proportion .
1. Compare denominators.
3
5 2. Find x.
3 5
x
20
Answer: Because 3 ,x
x
20
.
Checkpoint Use equivalent ratios to solve the proportion.
2
9
x
27
1. 116
Chapter 6 • Pre-Algebra Notetaking Guide
5
6
x
36
2. Copyright © Holt McDougal. All rights reserved.
6.2
Writing and Solving Proportions
Goal: Write and solve proportions.
Proportions
Words A proportion is an equation that states that two ratios
are equivalent.
2
3
8
12
Numbers a
b
c
d
Algebra , where b 0 and d 0
Example 1
Solving a Proportion Using Equivalent Ratios
3
5
x
20
Solve the proportion .
1. Compare denominators.
3
5 4
2. Find x.
3 4
5
x
20
x
20
Answer: Because 3 4 12 , x 12 .
Checkpoint Use equivalent ratios to solve the proportion.
2
9
5
6
x
27
1. 6
116
Chapter 6 • Pre-Algebra Notetaking Guide
x
36
2. 30
Copyright © Holt McDougal. All rights reserved.
Solving a Proportion Using Algebra
Example 2
x
15
2
5
Solve the proportion . Check your answer.
2
x
5
15
x
p 15
Check:
Write original proportion.
2
5
p Multiply each side by
x
Simplify.
x
Divide.
x
2
15
5
Write original proportion.
2
5
15
2
5
.
for x.
Substitute
Simplify.
.
Checkpoint Use algebra to solve the proportion.
3
7
x
28
4. x
8
49
56
6. 3. 5. Copyright © Holt McDougal. All rights reserved.
8
11
x
6
x
55
14
3
Chapter 6 • Pre-Algebra Notetaking Guide
117
Solving a Proportion Using Algebra
Example 2
x
15
2
5
Solve the proportion . Check your answer.
2
x
5
15
2
x
15 p 15 p 5
15
Multiply each side by 15 .
0
x 3
Simplify.
x 6
Divide.
x
2
15
5
Write original proportion.
5
Check:
Write original proportion.
6
2
5
15
2
5
2
5
Substitute 6 for x.
Simplify. Solution checks .
Checkpoint Use algebra to solve the proportion.
3
7
8
11
x
28
3. x
55
4. 12
x
8
49
56
x
6
5. 14
3
6. 7
Copyright © Holt McDougal. All rights reserved.
40
28
Chapter 6 • Pre-Algebra Notetaking Guide
117
Example 3
Writing and Solving a Proportion
Maple Syrup The sap of maple trees is used to make maple
syrup. It takes 40 gallons of sap to make 1 gallon of maple syrup.
Write and solve a proportion to find the number of gallons of
maple syrup that can be made from 1520 gallons of sap.
Solution
First, write a proportion involving two ratios that compare the
number of gallons of maple syrup to the number of gallons of sap.
Gallons of maple syrup
Gallons of sap
Then, solve the proportion.
p
p
Multiply each side by
x
Simplify.
x
Divide.
.
Answer: About
gallons of maple syrup can be made from
1520 gallons of sap.
Checkpoint
7. Use the information in Example 3. Write and solve a proportion to
find the number of gallons of maple syrup that can be made from
1360 gallons of sap.
118
Chapter 6 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Writing and Solving a Proportion
Example 3
Maple Syrup The sap of maple trees is used to make maple
syrup. It takes 40 gallons of sap to make 1 gallon of maple syrup.
Write and solve a proportion to find the number of gallons of
maple syrup that can be made from 1520 gallons of sap.
Solution
First, write a proportion involving two ratios that compare the
number of gallons of maple syrup to the number of gallons of sap.
Gallons of maple syrup
Gallons of sap
x
1
1520
40
Then, solve the proportion.
1
40
x
1520
1520 p 1520 p 1520
x
40
38 x
Multiply each side by 1520 .
Simplify.
Divide.
Answer: About 38 gallons of maple syrup can be made from
1520 gallons of sap.
Checkpoint
7. Use the information in Example 3. Write and solve a proportion to
find the number of gallons of maple syrup that can be made from
1360 gallons of sap.
34 gallons
118
Chapter 6 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Focus On
Algebra
Direct Variation and Inverse Variation
Use after Lesson 6.2
Goal: Identify direct variations and inverse variations.
Vocabulary
Direct variation
Inverse variation
Constant of variation
Identifying the Type of Variation
Example 1
Crafts A student club is buying ribbon to make crafts. The table
gives the cost y of buying an amount x of a particular type of ribbon.
Amount (inches), x
5
10
15
20
Cost (dollars), y
2
4
6
8
a. Determine whether the data represents direct variation or
inverse variation.
b. Identify the constant of variation and interpret it for this
situation.
c. Find the value of y when x 40.
Solution
a. Compare the
2
5
for all the data pairs (x, y).
The
are all
, so the data represents
.
Copyright © Holt McDougal. All rights reserved.
Chapter 6 • Pre-Algebra Notetaking Guide
119
Focus On
Algebra
Direct Variation and Inverse Variation
Use after Lesson 6.2
Goal: Identify direct variations and inverse variations.
Vocabulary
Direct variation Paired data (x, y) represent direct variation when
the ratios yx for x ≠ 0 all equal a nonzero constant k.
Inverse variation Paired data (x, y) represent inverse variation
when the products xy all equal a nonzero
constant k.
Constant of variation In both direct variation and inverse variation,
the constant k is called the constant of
variation.
Example 1
Identifying the Type of Variation
Crafts A student club is buying ribbon to make crafts. The table
gives the cost y of buying an amount x of a particular type of ribbon.
Amount (inches), x
5
10
15
20
Cost (dollars), y
2
4
6
8
a. Determine whether the data represents direct variation or
inverse variation.
b. Identify the constant of variation and interpret it for this
situation.
c. Find the value of y when x 40.
Solution
a. Compare the ratios
2
5
2
5
4
10
2
5
y
x
for all the data pairs (x, y).
6
15
2
5
8
20
2
5
The ratios are all equal , so the data represents direct
variation .
Copyright © Holt McDougal. All rights reserved.
Chapter 6 • Pre-Algebra Notetaking Guide
119
continued
Example 1
b. The constant of variation is
. For this situation, the constant
of variation represents the
per
of the ribbon.
The constant of variation is
c.
.
Write
.
each side by
.
Simplify.
Identifying the Type of Variation
Example 2
Crafts The club in Example 1 has a budget for ribbon. For a ribbon
that costs x dollars per inch, the table gives the amount y of the
ribbon that can be purchased using the entire budget.
Cost (dollars per inch), x
3
6
Amount (inches), y
24
12
9 12
8
6
a. Determine whether the data represents direct variation or
inverse variation.
b. Identify the constant of variation and interpret it for this situation.
c. Find the value of y when x 2.
Solution
a. Compare the
.
for all the data pairs (x, y).
.
,
The products are
.
,
,
.
, so the data represents
.
b. The constant of variation is
. For this situation, the constant
of variation represents the
c. 2y Write
of
.
.
each side by
.
Simplify.
120
Chapter 6 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
continued
Example 1
b. The constant of variation is
2
.
5
For this situation, the constant
of variation represents the cost per inch of the ribbon.
The constant of variation is $.40 per inch .
c.
y
40
2
5
Write proportion .
y
40 . 4
40 2
0
5
Multiply each side by 40 .
y 16
Example 2
Simplify.
Identifying the Type of Variation
Crafts The club in Example 1 has a budget for ribbon. For a ribbon
that costs x dollars per inch, the table gives the amount y of the
ribbon that can be purchased using the entire budget.
Cost (dollars per inch), x
3
6
Amount (inches), y
24
12
9 12
8
6
a. Determine whether the data represents direct variation or
inverse variation.
b. Identify the constant of variation and interpret it for this situation.
c. Find the value of y when x 2.
Solution
a. Compare the products
3 . 24 72 ,
xy for all the data pairs (x, y).
6 . 12 72 , 9 . 8 72 , 12 . 6 72
The products are all equal , so the data represents inverse
variation .
b. The constant of variation is 72 . For this situation, the constant
of variation represents the budget of $72 .
c. 2y 72
2y
2
72
2
y 36
120
Chapter 6 • Pre-Algebra Notetaking Guide
Write equation .
Divide each side by 2 .
Simplify.
Copyright © Holt McDougal. All rights reserved.
6.3
Solving Proportions Using
Cross Products
Goal: Solve proportions using cross products.
Vocabulary
Cross
product:
Determining if Ratios Form a Proportion
Example 1
Tell whether the ratios form a proportion.
12 20
21 35
4 8
26 42
a. , b. , Solution
4
8
26
42
a.
p
Write proportion.
p
Form cross products.
Multiply.
Answer: The ratios
a proportion.
12
20
21
35
b.
p
Write proportion.
p
Form cross products.
Multiply.
Answer: The ratios
a proportion.
Cross Products Property
Words The cross products of a proportion are equal.
2
5
6
15
Numbers Given that , you know that
a
b
c
d
Algebra If , where b 0 and d 0, then
Copyright © Holt McDougal. All rights reserved.
.
.
Chapter 6 • Pre-Algebra Notetaking Guide
121
6.3
Solving Proportions Using
Cross Products
Goal: Solve proportions using cross products.
Vocabulary
A cross product of two ratios is the product of the
Cross
product: numerator of one ratio and the denominator of the
other ratio.
Determining if Ratios Form a Proportion
Example 1
Tell whether the ratios form a proportion.
12 20
21 35
4 8
26 42
a. , b. , Solution
a.
4
8
26
42
Write proportion.
4 p 42 26 p 8
168
Form cross products.
Multiply.
208
Answer: The ratios do not form a proportion.
b.
12
20
21
35
Write proportion.
12 p 35 21 p 20
420
Multiply.
420
Answer: The ratios
Form cross products.
form
a proportion.
Cross Products Property
Words The cross products of a proportion are equal.
2
5
6
15
Numbers Given that , you know that 2 p 15 5 p 6 .
a
b
c
d
Algebra If , where b 0 and d 0, then ad bc .
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Chapter 6 • Pre-Algebra Notetaking Guide
121
Example 2
Writing and Solving a Proportion
Earnings You earn $68 mowing 4 lawns. How much would you
earn if you mowed 7 lawns?
Solution
Money earned
Lawns mowed
p
p
Cross products property
Multiply.
Divide each side by
x
Simplify.
Answer: If you mowed 7 lawns, you would earn
.
.
Checkpoint Tell whether the ratios form a proportion.
9
39
15
65
1. 12
45
6
28
2. Use the cross products property to solve the proportion.
14
42
x
6
3. 122
Chapter 6 • Pre-Algebra Notetaking Guide
4
9
16
x
4. Copyright © Holt McDougal. All rights reserved.
Example 2
Writing and Solving a Proportion
Earnings You earn $68 mowing 4 lawns. How much would you
earn if you mowed 7 lawns?
Solution
x
68
7
4
68 p 7 4 p x
Money earned
Lawns mowed
Cross products property
476 4x
Multiply.
476
4x
4
4
Divide each side by 4 .
119 x
Simplify.
Answer: If you mowed 7 lawns, you would earn $119 .
Checkpoint Tell whether the ratios form a proportion.
9
39
15
65
12
45
1. 6
28
2. yes
no
Use the cross products property to solve the proportion.
14
42
4
9
x
6
3. 2
122
Chapter 6 • Pre-Algebra Notetaking Guide
16
x
4. 36
Copyright © Holt McDougal. All rights reserved.
6.4
Similar and Congruent Figures
Goal: Identify similar and congruent figures.
Vocabulary
Similar
figures:
Corresponding
parts:
Congruent
figures:
When naming similar
figures, list the letters
of the corresponding
vertices in the same
order. For the
diagram at the right,
it is not correct to say
CBA ~ EFD,
because C and E
are not corresponding
angles.
Properties of Similar Figures
TABC S TDEF
C
The symbol S indicates that two
figures are similar.
4
1. Corresponding angles of similar
figures are congruent.
aA c aD, aB c aE, aC c aF
2. The ratios of the lengths of
corresponding sides of similar
figures are equal.
3
B
5
F
6
E
A
8
10
D
AB
BC
AC
1
DE
EF
DF
2
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Chapter 6 • Pre-Algebra Notetaking Guide
123
6.4
Similar and Congruent Figures
Goal: Identify similar and congruent figures.
Vocabulary
Similar Two figures are similar figures if they have the same
figures: shape but not necessarily the same size.
Corresponding
parts:
Corresponding parts of figures are sides or
angles that have the same relative position.
Congruent Two figures are congruent if they have the same
figures:
shape and the same size.
When naming similar
figures, list the letters
of the corresponding
vertices in the same
order. For the
diagram at the right,
it is not correct to say
CBA ~ EFD,
because C and E
are not corresponding
angles.
Properties of Similar Figures
TABC S TDEF
C
The symbol S indicates that two
figures are similar.
4
1. Corresponding angles of similar
figures are congruent.
aA c aD, aB c aE, aC c aF
2. The ratios of the lengths of
corresponding sides of similar
figures are equal.
3
B
5
F
6
E
A
8
10
D
AB
BC
AC
1
DE
EF
DF
2
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Chapter 6 • Pre-Algebra Notetaking Guide
123
Example 1
Identifying Corresponding Parts of Similar Figures
Given T XYZ S TUVW, name the
corresponding angles and the
corresponding sides.
X
W
V
Solution
Corresponding angles:
U
Y
Z
Corresponding sides:
Checkpoint
1. Given STUV S WXYZ, name the corresponding
angles and the corresponding sides.
U
V
T
S
W
X
Z
Example 2
Because all the ratios
of the lengths of
corresponding sides
of the figure in
Example 2 are equal,
you can use any pair
of lengths of
corresponding sides
to write the ratio. To
check the solution,
choose another pair
of lengths of
corresponding sides.
124
Finding the Ratio of Corresponding Side Lengths
Given ABCD S QRST, find the ratio of
the lengths of the corresponding sides
of ABCD to QRST.
Write a ratio comparing the lengths of a
pair of corresponding sides. Then substitute
the lengths of the sides and simplify.
AD
QT
Y
A
9.6
8
D
B
8
6.4 C
12
R
10
T
10
8
S
Answer: The ratio of the lengths of the corresponding sides is
Chapter 6 • Pre-Algebra Notetaking Guide
.
Copyright © Holt McDougal. All rights reserved.
Example 1
Identifying Corresponding Parts of Similar Figures
Given T XYZ S TUVW, name the
corresponding angles and the
corresponding sides.
X
W
V
Solution
Corresponding angles:
U
Y
Z
aX c aU, aY c aV, aZ c aW
Corresponding sides:
&& and UV
&
&, YZ
&
& and VW
&&, XZ
&
& and UW
&&
XY
Checkpoint
1. Given STUV S WXYZ, name the corresponding
angles and the corresponding sides.
Corresponding angles: aS c aW,
aT c aX, aU c aY, aV c aZ
U
V
T
S
W
X
&
& and WX
&&,
Corresponding sides: ST
&
& and XY
&&, UV
&
& and YZ
&
&, VS
&
& and ZW
&&
TU
Example 2
Because all the ratios
of the lengths of
corresponding sides
of the figure in
Example 2 are equal,
you can use any pair
of lengths of
corresponding sides
to write the ratio. To
check the solution,
choose another pair
of lengths of
corresponding sides.
124
Z
Y
Finding the Ratio of Corresponding Side Lengths
Given ABCD S QRST, find the ratio of
the lengths of the corresponding sides
of ABCD to QRST.
Write a ratio comparing the lengths of a
pair of corresponding sides. Then substitute
the lengths of the sides and simplify.
AD
4
8
5
10
QT
A
9.6
8
D
B
8
6.4 C
12
R
10
T
10
8
S
Answer: The ratio of the lengths of the corresponding sides is 4 .
Chapter 6 • Pre-Algebra Notetaking Guide
5
Copyright © Holt McDougal. All rights reserved.
Checkpoint
2. Given FGHJ S KLMN, find the
ratio of the lengths of the
corresponding sides of FGHJ
to KLMN.
Example 3
F
8
K
12
J
G
6 L
9
16
12
N
20
M
15
H
Finding Measures of Congruent Figures
Given DEFG c KLMN, find the indicated
measure.
a. KL
b. aL
111
8 ft
12 ft
D
Solution
Because the quadrilaterals are congruent,
the corresponding angles are congruent
and the corresponding sides are congruent.
&c
a. KL
. So, KL b. aL c
. So, maL F
E
93
77
G
N
K
L
M
Checkpoint
3. Given TABC c TLMK, find maL.
A
M
58
C
74
B
Copyright © Holt McDougal. All rights reserved.
L
K
Chapter 6 • Pre-Algebra Notetaking Guide
125
Checkpoint
2. Given FGHJ S KLMN, find the
ratio of the lengths of the
corresponding sides of FGHJ
to KLMN.
F
8
K
12
J
G
6 L
9
16
12
N
20
M
15
H
4
3
Example 3
Finding Measures of Congruent Figures
Given DEFG c KLMN, find the indicated
measure.
a. KL
b. aL
111
8 ft
12 ft
D
Solution
Because the quadrilaterals are congruent,
the corresponding angles are congruent
and the corresponding sides are congruent.
F
E
93
77
G
N
&
& . So, KL DE 8 ft
& c DE
a. KL
b. aL c aE . So, maL maE 111
K
L
M
Checkpoint
3. Given TABC c TLMK, find maL.
A
M
58
C
74
B
L
K
58
Copyright © Holt McDougal. All rights reserved.
Chapter 6 • Pre-Algebra Notetaking Guide
125
6.5
Similarity and Measurement
Goal: Find unknown side lengths of similar figures.
Finding an Unknown Side Length in Similar Figures
Example 1
Given RSTV S WXYZ, find VR.
R
S
Solution
Use the ratios of the lengths of
corresponding sides to write a
proportion involving the unknown
length, VR.
XY
ST
X
10 in.
15 in.
T
Y
V
Z
Write proportion involving VR.
p
W
14 in.
x
Substitute.
p
x
Cross products property
Multiply.
Divide each side by
Answer: The length of VR
&* is
.
inches.
Checkpoint
1. Given TPQR S TV TS, find TS.
P
15 in.
R
12 in.
x
T
S
10 in.
126
Chapter 6 • Pre-Algebra Notetaking Guide
V
Copyright © Holt McDougal. All rights reserved.
6.5
Similarity and Measurement
Goal: Find unknown side lengths of similar figures.
Finding an Unknown Side Length in Similar Figures
Example 1
Given RSTV S WXYZ, find VR.
R
S
Solution
Use the ratios of the lengths of
corresponding sides to write a
proportion involving the unknown
length, VR.
XY
ZW
VR
ST
10
15
14
x
10 p x 14 p 15
10x 210
x 21
W
14 in.
x
X
10 in.
15 in.
T
Y
V
Z
Write proportion involving VR.
Substitute.
Cross products property
Multiply.
Divide each side by 10 .
Answer: The length of VR
&* is 21 inches.
Checkpoint
1. Given TPQR S TV TS, find TS.
P
15 in.
R
12 in.
x
T
S
10 in.
V
8 in.
126
Chapter 6 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Using Indirect Measurement
Example 2
Height At a certain time of day, a person who
is 6 feet tall casts a 3-foot shadow. At the same
time, a tree casts an 11-foot shadow. The triangles
formed are similar. Find the height of the tree.
h
6 ft
Solution
3 ft
Write and solve a proportion to find the
height h of the tree.
11 ft
Height of tree
Height of person
p
Substitute values.
p
Cross products property
Multiply.
h
Divide each side by
Answer: The tree has a height of
.
feet.
Using Algebra and Similar Triangles
Example 3
Given T ABC S TDEC, find BE.
C
E
30 in.
To find BE, write and solve a proportion.
AB
DE
B
x
16 in.
24 in.
Write proportion.
D
A
Use fact that BC Substitute.
Cross products property
Multiply.
Subtract
x
Divide each side by
Answer: The length of BE
&* is
Copyright © Holt McDougal. All rights reserved.
.
from each side.
.
inches.
Chapter 6 • Pre-Algebra Notetaking Guide
127
Example 2
Using Indirect Measurement
Height At a certain time of day, a person who
is 6 feet tall casts a 3-foot shadow. At the same
time, a tree casts an 11-foot shadow. The triangles
formed are similar. Find the height of the tree.
h
6 ft
Solution
3 ft
Write and solve a proportion to find the
height h of the tree.
11 ft
Length of tree’s shadow
Height of tree
Length of person’s shadow
Height of person
11
h
3
6
Substitute values.
3 p h 6 p 11
3h 66
Cross products property
Multiply.
h 22
Divide each side by 3 .
Answer: The tree has a height of 22 feet.
Example 3
Using Algebra and Similar Triangles
Given T ABC S TDEC, find BE.
C
E
30 in.
To find BE, write and solve a proportion.
AB
BC
DE
EC
x
16 in.
24 in.
Write proportion.
D
A
AB
BE EC
DE
EC
Use fact that BC BE EC .
24
x 30
16
30
Substitute.
24 p 30 16(x 30)
Cross products property
720 16x 480
Multiply.
240 16x
Subtract 480 from each side.
15 x
B
Divide each side by 16 .
Answer: The length of BE
&* is 15 inches.
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Chapter 6 • Pre-Algebra Notetaking Guide
127
6.6
Scale Drawings
Goal: Use proportions with scale drawings.
Vocabulary
Scale
drawing:
Scale
model:
Scale:
Using a Scale Drawing
Example 1
On a map, the distance between two cities is 3 inches. What is the
actual distance (in miles) between the two cities if the map’s scale
is 1 in. : 125 mi?
Solution
Let x represent the actual distance (in miles) between the two
cities. The ratio of the map distance between the two cities to the
actual distance x is equal to the scale of the map. Write and solve
a proportion using this relationship.
Map distance
Actual distance
Cross products property
x
Answer: The actual distance is
Multiply.
.
Checkpoint
1. On a map, the distance between two cities is 4 inches. What is the
actual distance (in miles) between the two cities if the map’s scale
is 1 in. : 80 mi?
128
Chapter 6 • Pre-Algebra Notetaking Guide
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6.6
Scale Drawings
Goal: Use proportions with scale drawings.
Vocabulary
A scale drawing is a two-dimensional drawing that is
Scale
drawing: similar to the object it represents.
A scale model is a three-dimensional model that is
Scale
model: similar to the object it represents.
Scale:
The scale of a scale drawing or scale model gives the
relationship between the drawing or model’s dimensions
and the actual dimensions.
Example 1
Using a Scale Drawing
On a map, the distance between two cities is 3 inches. What is the
actual distance (in miles) between the two cities if the map’s scale
is 1 in. : 125 mi?
Solution
Let x represent the actual distance (in miles) between the two
cities. The ratio of the map distance between the two cities to the
actual distance x is equal to the scale of the map. Write and solve
a proportion using this relationship.
3 in.
1 in.
x mi
125 mi
1x 125 p 3
x 375
Map distance
Actual distance
Cross products property
Multiply.
Answer: The actual distance is 375 miles .
Checkpoint
1. On a map, the distance between two cities is 4 inches. What is the
actual distance (in miles) between the two cities if the map’s scale
is 1 in. : 80 mi?
320 miles
128
Chapter 6 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Example 2
Finding the Scale of a Drawing
Architecture In a scale drawing, a wall is 2 inches long. The
actual wall is 12 feet long. Find the scale of the drawing.
Solution
Write a ratio using corresponding side lengths of the scale drawing
and the actual wall. Then simplify the ratio so that the numerator
is
.
2 in.
12 ft
Length of wall in scale drawing
Length of actual wall
2 in.
12 ft
Simplify.
Answer: The drawing’s scale is
The scale of a scale
drawing or scale
model can be written
without units if the
measurements have
the same unit. For
example, the scale
1 cm : 2 m can be
written without units
as follows.
1 cm : 2 m
Example 3
Solution
Write a proportion using the scale.
Scale
with
units
1 cm
200 cm
Scale
without
units
Finding a Dimension of a Scale Model
A model of the Sears Tower in Chicago has a scale of 1 : 103. The
height of the Sears Tower’s observation deck is about 412 meters.
Find the height of the observation deck of the model.
1 cm
2m
1 : 200
.
Dimension of model
Dimension of Sears Tower
Cross products property
x
Divide each side by
.
Answer: The height of the model’s observation deck is
.
Checkpoint
2. The height of one antenna on the Sears Tower is about 521.1
meters. Find the height of the antenna on the model to the
nearest tenth of a meter.
Copyright © Holt McDougal. All rights reserved.
Chapter 6 • Pre-Algebra Notetaking Guide
129
Example 2
Finding the Scale of a Drawing
Architecture In a scale drawing, a wall is 2 inches long. The
actual wall is 12 feet long. Find the scale of the drawing.
Solution
Write a ratio using corresponding side lengths of the scale drawing
and the actual wall. Then simplify the ratio so that the numerator
is 1 .
2 in.
12 ft
Length of wall in scale drawing
Length of actual wall
2 in.
1 in.
6 ft
12 ft
Simplify.
Answer: The drawing’s scale is 1 in. : 6 ft .
The scale of a scale
drawing or scale
model can be written
without units if the
measurements have
the same unit. For
example, the scale
1 cm : 2 m can be
written without units
as follows.
1 cm : 2 m
Example 3
A model of the Sears Tower in Chicago has a scale of 1 : 103. The
height of the Sears Tower’s observation deck is about 412 meters.
Find the height of the observation deck of the model.
Solution
Write a proportion using the scale.
Scale
with
units
1 cm
2m
1
x
103
412
Dimension of model
Dimension of Sears Tower
412 103x
Cross products property
4 x
1 cm
200 cm
1 : 200
Scale
without
units
Finding a Dimension of a Scale Model
Divide each side by 103 .
Answer: The height of the model’s observation deck is 4 meters .
Checkpoint
2. The height of one antenna on the Sears Tower is about 521.1
meters. Find the height of the antenna on the model to the
nearest tenth of a meter.
5.1 meters
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Chapter 6 • Pre-Algebra Notetaking Guide
129
6.7
Probability and Odds
Goal: Find probabilities.
Vocabulary
Outcomes:
Event:
Favorable
outcomes:
Probability:
Theoretical
probability:
Experimental
probability:
Odds in
favor:
Odds
against:
130
Chapter 6 • Pre-Algebra Notetaking Guide
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6.7
Probability and Odds
Goal: Find probabilities.
Vocabulary
Outcomes: The possible results of an experiment are outcomes.
Event: An event is an outcome or a collection of outcomes.
Favorable The outcomes for a specified event are called
outcomes: favorable outcomes.
The probability that an event occurs is a measure
Probability: of the likelihood that the event will occur.
Theoretical A theoretical probability is based on knowing all of
probability: the equally likely outcomes of an experiment.
A probability that is based on repeated trials of an
Experimental experiment is called an experimental probability.
probability:
Each trial in which the event occurs is a success.
The ratio of the number of favorable outcomes to the
Odds in number of unfavorable outcomes is called the odds in
favor:
favor of an event.
The ratio of the number of unfavorable outcomes to
Odds
against: the number of favorable outcomes is called the odds
against an event.
130
Chapter 6 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Probability of an Event
The probability of an event when all outcomes are equally likely is:
Number of favorable outcomes
Number of possible outcomes
P(event) Example 1
Finding a Probability
Suppose you roll a number cube. What is the probability that you
roll an odd number?
Solution
Rolls of
There are
P
(
are odd, so there are
favorable outcomes.
possible outcomes.
)
Checkpoint
1. Suppose you roll a number cube. What is the probability that you
roll a number less than 5?
2. Suppose you roll a number cube. What is the probability that you
roll a number that is a multiple of 3?
Copyright © Holt McDougal. All rights reserved.
Chapter 6 • Pre-Algebra Notetaking Guide
131
Probability of an Event
The probability of an event when all outcomes are equally likely is:
Number of favorable outcomes
Number of possible outcomes
P(event) Example 1
Finding a Probability
Suppose you roll a number cube. What is the probability that you
roll an odd number?
Solution
Rolls of 1, 3, and 5 are odd, so there are 3 favorable outcomes.
There are 6 possible outcomes.
(
P rolling an odd number
)
Number of favorable outcomes
Number of possible outcomes
3
6
1
2
Checkpoint
1. Suppose you roll a number cube. What is the probability that you
roll a number less than 5?
2
3
2. Suppose you roll a number cube. What is the probability that you
roll a number that is a multiple of 3?
1
3
Copyright © Holt McDougal. All rights reserved.
Chapter 6 • Pre-Algebra Notetaking Guide
131
Experimental Probability
The experimental probability of an event is:
Number of successes
Number of trials
P(event) Example 2
Finding Experimental Probability
You plant 32 seeds of a certain flower and 18 of them sprout.
Find the experimental probability that the next flower seed
planted will sprout.
Solution
Number of successes
Number of trials
P(flower seed will sprout) Simplify.
Answer: The experimental probability that the next flower seed will
sprout is
, or
.
Example 3
Finding the Odds
Suppose you randomly choose a number between 1 and 16.
a. What are the odds in favor of choosing a prime number?
b. What are the odds against choosing a prime number?
Solution
a. There are
16 favorable outcomes
(
) and
unfavorable outcomes.
Number of favorable outcomes
Number of unfavorable outcomes
Odds in favor The odds are
, or
to
, that you choose a prime number.
b. The odds against choose a prime number are
132
Chapter 6 • Pre-Algebra Notetaking Guide
, or
to
.
Copyright © Holt McDougal. All rights reserved.
Experimental Probability
The experimental probability of an event is:
Number of successes
Number of trials
P(event) Finding Experimental Probability
Example 2
You plant 32 seeds of a certain flower and 18 of them sprout.
Find the experimental probability that the next flower seed
planted will sprout.
Solution
Number of successes
Number of trials
18
P(flower seed will sprout) 3
2
9
1
6
Simplify.
Answer: The experimental probability that the next flower seed will
9
sprout is 1
6 , or 0.5625 .
Example 3
Finding the Odds
Suppose you randomly choose a number between 1 and 16.
a. What are the odds in favor of choosing a prime number?
b. What are the odds against choosing a prime number?
Solution
a. There are 6 favorable outcomes
(
2, 3, 5, 7, 11, and 13
) and
16 6 10 unfavorable outcomes.
Number of favorable outcomes
6
3
Odds in favor 1
5
0
Number of unfavorable outcomes
3
The odds are 5 , or 3 to 5 , that you choose a prime number.
5
b. The odds against choose a prime number are 3 , or 5 to 3 .
132
Chapter 6 • Pre-Algebra Notetaking Guide
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Focus On
Probability
Fair Games
Use after Lesson 6.7
Goal: Determine whether a game is fair.
Vocabulary
Fair game
Example 1
Determining Whether a Game is Fair
In a game, two players roll a number cube. For each round, player 1
scores a point if the number cube shows a multiple of 2. Player 2
scores a point if the number cube shows a multiple of 3. The player
who scores 8 points first wins. Determine whether the game is fair.
Solution
For player 1, rolls of
are multiples of 2, so there are
favorable outcomes out of
possible outcomes.
P(rolling a multiple of 2)
For player 2, rolls of
are
favorable outcomes out of
(
P
Because the probabilities
, so there are
possible outcomes.
)
the same, the players
equally likely to win the game. Therefore, the game
Copyright © Holt McDougal. All rights reserved.
fair.
Chapter 6 • Pre-Algebra Notetaking Guide
133
Focus On
Probability
Fair Games
Use after Lesson 6.7
Goal: Determine whether a game is fair.
Vocabulary
Fair game A game in which all players are equally likely to win
Example 1
Determining Whether a Game is Fair
In a game, two players roll a number cube. For each round, player 1
scores a point if the number cube shows a multiple of 2. Player 2
scores a point if the number cube shows a multiple of 3. The player
who scores 8 points first wins. Determine whether the game is fair.
Solution
For player 1, rolls of 2, 4, and 6 are multiples of 2, so there are
3 favorable outcomes out of 6 possible outcomes.
P(rolling a multiple of 2)
3
6
1
2
For player 2, rolls of 3 and 6 are multiples of 3 , so there are
2 favorable outcomes out of 6 possible outcomes.
(
P rolling a multiple of 3
)
2
6
1
3
Because the probabilities are not the same, the players are not
equally likely to win the game. Therefore, the game is not fair.
Copyright © Holt McDougal. All rights reserved.
Chapter 6 • Pre-Algebra Notetaking Guide
133
Example 2
Determining Whether a Game is Fair
A bag contains 5 cards, numbered 1–5. In a game, two players
each draw a card from the bag. The player who draws the lesser
number wins. Determine whether the game is fair.
Solution
Number that player
draws
5
4
Number of ways
that player wins
0
1
Number of ways
that player loses
4
3
The total number of ways of winning
equal to the total
number of ways of losing. Each player has a
winning, so the game
Example 3
% chance of
fair.
Determining Whether a Game is Fair
In a game, two players spin the spinner shown once. If the sum of
the spins is even, player 1 wins. If the sum is odd, player 2 wins.
Determine whether the game is fair.
Solution
Player 1
Player 2
Winner
1
1
Player 1
1
2
3
From the list, you can see that there are
1 wins and
outcomes in which player
outcomes in which player 2 wins. Because there are
outcomes in which player 1 can win, the game
134
Chapter 6 • Pre-Algebra Notetaking Guide
fair.
Copyright © Holt McDougal. All rights reserved.
Example 2
Determining Whether a Game is Fair
A bag contains 5 cards, numbered 1–5. In a game, two players
each draw a card from the bag. The player who draws the lesser
number wins. Determine whether the game is fair.
Solution
Number that player
draws
5
4
Number of ways
that player wins
0
1
Number of ways
that player loses
4
3
The total number of ways of winning
2
1
2
3
4
2
1
0
3
is
equal to the total
number of ways of losing. Each player has a 50 % chance of
winning, so the game
Example 3
is
fair.
Determining Whether a Game is Fair
In a game, two players spin the spinner shown once. If the sum of
the spins is even, player 1 wins. If the sum is odd, player 2 wins.
Determine whether the game is fair.
Solution
Player 1
Player 2
Winner
1
1
Player 1
1
2
Player 2
1
3
Player 1
2
1
Player 2
2
2
Player 1
2
3
Player 2
3
1
Player 1
3
2
Player 2
3
3
Player 1
1
2
3
From the list, you can see that there are 5 outcomes in which player
1 wins and 4 outcomes in which player 2 wins. Because there are
more outcomes in which player 1 can win, the game is not fair.
134
Chapter 6 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
6.8
The Multiplication Principle
Goal: Use the multiplication principle to find probabilities.
Vocabulary
Tree
diagram:
Multiplication
principle:
Example 1
Making a Tree Diagram
At a picnic lunch, you can choose one sandwich and one salad.
The choices include the following: veggie burger, hamburger,
tuna sandwich, garden salad, and fruit salad. How many different
picnic lunches are possible?
Solution
To count the number of possible picnic lunches, you can make a
tree diagram.
List the
sandwiches.
Answer:
Copyright © Holt McDougal. All rights reserved.
List the salads for
each sandwich.
List the possibilities
for each picnic lunch.
different picnic lunches are possible.
Chapter 6 • Pre-Algebra Notetaking Guide
135
6.8
The Multiplication Principle
Goal: Use the multiplication principle to find probabilities.
Vocabulary
One way to count the number of possibilities is to use
Tree
diagram: a tree diagram. A tree diagram uses branching to list
choices.
Multiplication The multiplication principle uses multiplication to find
the number of ways two or more events can occur.
principle:
Example 1
Making a Tree Diagram
At a picnic lunch, you can choose one sandwich and one salad.
The choices include the following: veggie burger, hamburger,
tuna sandwich, garden salad, and fruit salad. How many different
picnic lunches are possible?
Solution
To count the number of possible picnic lunches, you can make a
tree diagram.
List the
sandwiches.
List the salads for
each sandwich.
List the possibilities
for each picnic lunch.
garden salad
veggie burger and
garden salad
fruit salad
veggie burger and
fruit salad
veggie burger
garden salad
hamburger and
garden salad
fruit salad
hamburger and
fruit salad
hamburger
garden salad
tuna sandwich and
garden salad
fruit salad
tuna sandwich and
fruit salad
tuna sandwich
Answer: Six different picnic lunches are possible.
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Chapter 6 • Pre-Algebra Notetaking Guide
135
Checkpoint
1. Suppose for each picnic lunch in Example 1 you also get to choose
one fruit. The choices include the following: apple, banana, orange,
and grapefruit. Copy the tree diagram in Example 1 and add the
new choices. How many possible picnic lunch choices are there?
The Multiplication Principle
If one event can occur in m ways, and for each of these ways a
second event can occur in n ways, then the number of ways that
the two events can occur together is m p n.
The multiplication principle can be extended to three or more
events.
Example 2
The set of all possible
outcomes for an
experiment is
sometimes called the
sample space.
Use the Multiplication Principle
You roll a number cube and randomly draw a marble from a bag.
There is one marble for each of the following colors: red, blue, green,
and yellow. Use the multiplication principle to find the number
of different outcomes that are possible.
Number of outcomes Number of outcomes
Total number of
for the number cube
for the marble
possible outcomes
p
Answer: There are
136
Chapter 6 • Pre-Algebra Notetaking Guide
different possible outcomes.
Copyright © Holt McDougal. All rights reserved.
Checkpoint
1. Suppose for each picnic lunch in Example 1 you also get to choose
one fruit. The choices include the following: apple, banana, orange,
and grapefruit. Copy the tree diagram in Example 1 and add the
new choices. How many possible picnic lunch choices are there?
24
The Multiplication Principle
If one event can occur in m ways, and for each of these ways a
second event can occur in n ways, then the number of ways that
the two events can occur together is m p n.
The multiplication principle can be extended to three or more
events.
Example 2
The set of all possible
outcomes for an
experiment is
sometimes called the
sample space.
Use the Multiplication Principle
You roll a number cube and randomly draw a marble from a bag.
There is one marble for each of the following colors: red, blue, green,
and yellow. Use the multiplication principle to find the number
of different outcomes that are possible.
Number of outcomes Number of outcomes
Total number of
for the number cube
for the marble
possible outcomes
6
p
4
24
Answer: There are 24 different possible outcomes.
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Checkpoint
2. You roll a number cube, randomly draw a marble from a bag, and
flip a coin. There is one marble for each of the following colors:
red, blue, and yellow. Use the multiplication principle to find the
number of different outcomes that are possible.
Finding a Probability
Example 3
Car Security The access code for a car’s security system consists
of 4 digits. You randomly enter 4 digits. What is the probability that
you choose the correct code?
Solution
First find the number of different codes.
Use the multiplication principle.
Then find the probability that you choose the correct code.
P(correct code) Answer: The probability that you choose the correct code
is
.
Checkpoint
3. Your computer password has 2 lowercase letters followed by
6 digits. Your friend randomly chooses 2 lowercase letters
and 6 digits. Use a calculator to find the probability that your
friend chooses your password.
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Chapter 6 • Pre-Algebra Notetaking Guide
137
Checkpoint
2. You roll a number cube, randomly draw a marble from a bag, and
flip a coin. There is one marble for each of the following colors:
red, blue, and yellow. Use the multiplication principle to find the
number of different outcomes that are possible.
36
Example 3
Finding a Probability
Car Security The access code for a car’s security system consists
of 4 digits. You randomly enter 4 digits. What is the probability that
you choose the correct code?
Solution
First find the number of different codes.
10 p 10 p 10 p 10 10,000
Use the multiplication principle.
Then find the probability that you choose the correct code.
1
P(correct code) 10,000
Answer: The probability that you choose the correct code
1
.
is 10,000
Checkpoint
3. Your computer password has 2 lowercase letters followed by
6 digits. Your friend randomly chooses 2 lowercase letters
and 6 digits. Use a calculator to find the probability that your
friend chooses your password.
1
676,000,000
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Chapter 6 • Pre-Algebra Notetaking Guide
137
Focus On
Probability
Use after Lesson 6.8
The Addition Principle
Goal: Use the addition principle.
Vocabulary
Addition
principle:
The Addition Principle
Suppose two events have no outcomes in common. If one of the
events can occur in m ways and the other event can occur in n
ways, then either of the events can occur in m + n ways.
The addition principle can be extended to three or more events.
Example 1
Using the Addition Principle
Trivia Board Game You are playing a trivia board game that
involves rolling two number cubes, one red and one white. You use
the sum of the numbers rolled to determine the next position of
your marker on the trivia board. For one of your turns, you want the
sum of the numbers to be either 4 or 6 in order to move to a
desirable position. How many ways can this happen?
Solution
There are
ways you can get a sum of 4.
There are
Answer:
138
Chapter 6 • Pre-Algebra Notetaking Guide
ways you can get a sum of 6.
ways
Copyright © Holt McDougal. All rights reserved.
Focus On
Probability
Use after Lesson 6.8
The Addition Principle
Goal: Use the addition principle.
Vocabulary
Addition The addition principle uses addition to find the number
principle: of possible ways any of two or more events can occur.
The Addition Principle
Suppose two events have no outcomes in common. If one of the
events can occur in m ways and the other event can occur in n
ways, then either of the events can occur in m + n ways.
The addition principle can be extended to three or more events.
Example 1
Using the Addition Principle
Trivia Board Game You are playing a trivia board game that
involves rolling two number cubes, one red and one white. You use
the sum of the numbers rolled to determine the next position of
your marker on the trivia board. For one of your turns, you want the
sum of the numbers to be either 4 or 6 in order to move to a
desirable position. How many ways can this happen?
Solution
There are three ways you can get a sum of 4.
1. Red cube shows a 1, and white cube shows a 3.
2. Red cube shows a 3, and white cube shows a 1.
3. Both number cubes show a 2.
There are five ways you can get a sum of 6.
1. Red cube shows a 1, and white cube shows a 5.
2. Red cube shows a 5, and white cube shows a 1.
3. Red cube shows a 2, and white cube shows a 4.
4. Red cube shows a 4, and white cube shows a 2.
5. Both number cubes show a 3.
Answer: 3 5 8 ways
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Example 2
Using the Addition Principle
Trivia Board Game You are playing the same game as in Example 1.
On your next turn, you want the sum of the number cubes to be
less than 6 in order to move your marker to the center of the
board. In how many ways can this happen?
Solution
In order to move your marker to the center of the board, you need
the sum to be 2, 3,
, or
. There is
way to get 2. There
, and
are
ways to get 3,
ways to get
ways to
get
.
Answer:
ways
Checkpoint Determine the number of ways that the specified event can
occur when two number cubes are rolled.
1. Getting a sum of 2 or 8
Example 3
2. Getting a sum that is even
Finding a Probability
Trivia Board Game You are playing the same game as in Example 1.
What is the probability that you get a sum less than 6?
Solution
The number of possible outcomes when rolling two number cubes
can be found using the
:
·
.
The number of favorable outcomes was found in Example 2 using
the
:
.
(
)
P
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Chapter 6 • Pre-Algebra Notetaking Guide
139
Using the Addition Principle
Example 2
Trivia Board Game You are playing the same game as in Example 1.
On your next turn, you want the sum of the number cubes to be
less than 6 in order to move your marker to the center of the
board. In how many ways can this happen?
Solution
In order to move your marker to the center of the board, you need
the sum to be 2, 3, 4 , or 5 . There is one way to get 2. There
are two ways to get 3, three ways to get 4 , and four ways to
get 5 .
Answer: 1 2 3 4 10 ways
Checkpoint Determine the number of ways that the specified event can
occur when two number cubes are rolled.
1. Getting a sum of 2 or 8
2. Getting a sum that is even
6
18
Finding a Probability
Example 3
Trivia Board Game You are playing the same game as in Example 1.
What is the probability that you get a sum less than 6?
Solution
The
can
The
the
number of possible outcomes when rolling two number cubes
be found using the multiplication principle : 6 · 6 36 .
number of favorable outcomes was found in Example 2 using
addition principle : 1 2 3 4 10 .
(
P sum less than 6
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10
36
)
number of favorable outcomes
number of possible outcomes
5
18
Chapter 6 • Pre-Algebra Notetaking Guide
139
6
Words to Review
Give an example of the vocabulary word.
140
Ratio
Equivalent ratios
Rate
Unit Rate
Inverse variation, Constant
of variation
Direct variation, Constant
of variation
Proportion
Cross products
Similar figures
Corresponding parts
Chapter 6 • Pre-Algebra Notetaking Guide
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6
Words to Review
Give an example of the vocabulary word.
Ratio
Equivalent ratios
3 : 4 and 9 : 12 are equivalent
ratios.
3
4
3 to 4, 3 : 4, or Rate
Unit Rate
80 miles in 2 hours or
80 mi
2h
Inverse variation, Constant
of variation
Cost (dollars per foot), x 1
Amount (feet), y
40 miles per hour or 40 mi
2h
Direct variation, Constant
of variation
2 3
Amount (feet),x
1
2
3
12 6 4
Cost (dollars), y
4
8
12
Constant of variation: 12
Constant of variation: 4
Proportion
Cross products
2
x
5
15
The cross products of
2
6
are 2 p 15 5 p 6.
5
15
Similar figures
Corresponding parts
B
A
F
E
B
A
F
E
D
C H
ABCD S EFGH
D
G
C H
G
ABCD S EFGH
Corresponding angles:
aA and aE, aB and aF,
aC and aG, aD and aH
Corresponding sides:
AB
&* and EF
&*, BC
&* and FG
&*,
CD
&* and GH
&*, AD
&* and EH
&*
140
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Congruent figures
Scale drawing
Scale model
Scale
Outcomes
Event
Favorable outcomes
Probability
Theoretical probability
Experimental probability
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Chapter 6 • Pre-Algebra Notetaking Guide
141
Congruent figures
B
A two-dimensional drawing
that is similar to the object it
represents.
F
A
D
Scale drawing
E
C
H
G
ABCD c EFGH
Scale model
Scale
A three-dimensional model
that is similar to the object it
represents.
Outcomes
A ratio that gives the
relationship between the
dimensions of a scale
drawing or a scale model
and the actual dimensions
of the object.
Event
The possible outcomes from
flipping a coin are flipping
heads and flipping tails.
Favorable outcomes
The favorable outcomes for
rolling a number cube and
getting an even number are
2, 4, and 6.
Theoretical probability
The theoretical probability of
rolling a number cube and
1
6
getting a 5 is .
Rolling a number cube and
getting an even number
Probability
If you roll a number cube, the
probability of getting an even
1
2
number is .
Experimental probability
You plant 12 seeds of a certain
flower and 8 of them sprout.
The experimental probability
that the next flower seed
2
3
planted will sprout is .
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Chapter 6 • Pre-Algebra Notetaking Guide
141
Odds in favor
Odds against
Fair game
Tree diagram
Multiplication principle
Addition principle
Review your notes and Chapter 6 by using the Chapter Review on
pages 334–337 of your textbook.
142
Chapter 6 • Pre-Algebra Notetaking Guide
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Odds in favor
You randomly choose a
number between 1 and 12.
The odds in favor of choosing
a prime number are 5 to 7.
Fair game
You and a friend flip a coin
to see who gets the last slice
of pizza.
Odds against
You randomly choose a
number between 1 and 12.
The odds against choosing a
prime number are 7 to 5.
Tree diagram
garden salad
veggie burger
fruit salad
garden salad
hamburger
fruit salad
Multiplication principle
You have 5 pairs of pants
and 3 shirts. You can make
15 different shirt-and-pants
outfits.
Addition principle
You roll two number cubes.
There are 2 ways you can get
a sum of 3. There are 3 ways
you can get a sum of 10.
There are 5 ways you can get
a sum of 3 or 10.
Review your notes and Chapter 6 by using the Chapter Review on
pages 334–337 of your textbook.
142
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