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LORD MAY THY WILL BE DONE IN ALL THINGS. 
CHAPTER 6:
Ratio, Proportion,
and Probability
SECTION: 6.1 Ratios and Rates
Objective: TLW find ratios and unit rates.
Standards: M7.A.1.1, M7.A.2.2, M7.A.2.2.1, M7.A.2.2.3,
M7.A.2.2.4, M7.B.1.1
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
Vocabulary:
Ratio:
Equivalent Ratios:
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
WRITING RATIOS
You can write the ratio of two quantities, a and b, where b is not
equal to 0, in three ways…
a
a to b
a : b
b
Each ratio is read “the ratio of a and b.” You should write the ratio
in simplest form, as shown in Example 1 below.
Example 1: Team 7A’s top basketball player shoots 30 free throws, with
19 of them being successful, and 11 of them being misses.
Using this information, write the following ratios (all 3 ways)
a. number of successful baskets to the number of misses
1.
2.
3.
b. number of successful baskets to the number of shots
1.
2.
3.
Example 1 You try:
Compare the number of misses to the number of shots using a ratio.
1.
2.
3.
To compare two ratios, you can write both ratios as fractions or as
decimals. Two ratios are called equivalent ratios when they have the
same value.
Example 2: The ratio comparing the length of a bird’s wings to the
average width of the bird’s wings is the bird’s aspect ratio.
Order the birds in the table from the greatest aspect ratio
to the least.
Bird
White-tailed eagle
European jay
Black-headed gull
Wing Length (cm)
209
47
83
Average wing width (cm)
30
12
8
White-tailed eagle:
European jay:
Black-headed gull:
*****************************************************************
Rates:
Unit Rate:
Example 3:
a. You host a party for 12 people. The food and drinks for the party cost
$66. What is the cost per person?
b. A jet flies 1620 miles in 3 hours. How many miles per hour does the
jet fly?
Example 4: Write the equivalent rate.
a.
b.
c.
d.
e.
f.
Example 5:
a. Find the ratio of the area of the shaded square region to the area
of the unshaded square region.
b. Find the ratio of the area of the shaded square region to the area of the
unshaded square region.
HOMEWORK:
SECTION: 6.2 Writing and Solving Proportions
Objective: TLW write and solve proportions.
Standards: M7.A.2.2, M7.A.2.2.2, M7.A.2.2.5
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
PROPORTIONS
Words: A proportion is an equation that states that two ratios are
equivalent.
Numbers:
Algebra:
Example 1: Mental Math - Use equivalent ratios to solve the proportion.
a.
2
x

7 21
b.
c.
3
x

8 32
d.
x 20

2 10
x
6

48 12
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
Example 2: Use Algebra to solve proportions.
x 2

a.
12 8
x 3

b.
98 7
Examples 1 and 2 You try:
a.
3 x

8 30
b.
5 x

6 9
c.
x 11

15 24
CONSIDER THE FOLLOWING:
Yesterday you rode your bike 18 miles in 2.5 hours. Today you plan to ride
for 3.5 hours. If you ride at the same rate as yesterday, how far will you
ride?
What can you come up with?
Example 3: Each day, an elephant eats 5 pounds of food for every 100
pounds of its body weight. Use this information to write and solve a
proportion to determine how much food an elephant that weighs 9300
pounds eats per day.
How much food would a 12,500 pound elephant eat per day?
HOMEWORK:
SECTION: 6.3 Solving Proportions Using Cross Products
Objective: TLW solve proportions using cross products.
Standards: M7.A.2.2, M7.A.2.2.2, M7.D.2.1, M7.D.2.2
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
Ratios:
Cross Products:
You can use cross products to tell whether two ratios form a proportion.
If the cross products are equal, then the ratios form a proportion.
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
Example 1: Tell whether the ratios form a proportion.
9 6
,
a.
51 34
12 32
,
b.
20 50
Example 1 You Try:
a.
6 3
,
14 7
b.
14 8
,
35 20
c.
6 9
,
11 16
CROSS PRODUCTS PROPERTY
Words: The cross products of a proportion are equal.
2 6

Numbers: Given that
, you know that:
5 15
a c
 , where b  0 and d  0 , then ad = bc.
Algebra: If
b d
d.
15 10
,
24 16
Example 2: Human hair grows about 0.7 centimeters in 2 weeks. How
long does hair take to grow 14 centimeters?
Example 2 You Try: Use the cross products property to solve the
proportion.
a.
18 3

42 t
b.
16 10

p 45
c.
.4 18

6
z
HOMEWORK:
SUMMARY: Methods for Solving a Proportion
To solve the proportion
Equivalent ratios:
Algebra:
Cross products:
5
x
 , use one of the following:
12 36
EXTRA EXTRA…Basic Geometry Concepts!
Objective: TLW review basic geometry concepts
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
POINTS, LINES, AND PLANES
WORD
NOTATION
DIAGRAM
Point
Line
Plane
EXAMPLE: Use the diagram to name three points, two lines, and a plane.
SEGMENTS, RAYS, AND ANGLES
WORD
Line segment, or
segment length of a
line segment
Ray
NOTATION
DIAGRAM
Angle
Measure of an angle
EXAMPLE: Use the diagram to name two segments and their lengths, two
rays, and an angle and its measure.
TRIANGLES, QUADRILATERALS, AND CONGRUENT PARTS
WORD
NOTATION
DIAGRAM
Triangle
A quadrilateral (made
of 4 line segments that
intersect only at their
endpoints)
Congruent Segments:
Congruent Angles:
EXAMPLE: Identify the angles, sides, congruent angles, and congruent
sides of the triangle.
YOU TRY:
1. Name three points.
2. Name two lines.
3. Name two planes.
4. Name two rays.
5. Name a segment.
6. Name an angle and give its measure.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
7. Name the quadrilateral.
8. Name the sides.
9. Name the angles.
10. Identify the congruent angles and congruent sides.
SECTION: 6.4 Similar and Congruent Figures
Objective: TLW identify similar and congruent figures.
Standards: M7.A.2.2, M7.A.2.2.3, M7.C.1.2, M7.C.1.2.1
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
Similar figures:
Corresponding parts:
D
A
E
U
X
F
V
Y
B
W
Z
C
PROPERTIES OF SIMILAR FIGURES
C
ABC ~ DEF
4
1. Corresponding angles of similar
figures are congruent.
3
B
5
F
A
8
2. The ratios of the lengths of
corresponding sides of similar
figures are equal.
D
6
E
10
Example 1: Given LMN ~ PQR , name the corresponding angles and the
corresponding sides.
M
N
L
P
R
Q
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Example 2: Given ABCD ~ JKLM, find the ratio of the lengths of
corresponding sides of ABCD to JKLM.
A
8ft
4ft
D
8ft
M
6ft
B
2ft
C
J
12 ft
12 ft
L
3ft
K
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Example 2: A soccer field is a rectangle that is 70 yards long and 40
yards wide. The penalty area of the soccer field is a rectangle that is 35
yards long and 14 yards wide. Is the penalty area similar to the field?
Congruent Figures:
Two figures are congruent if
_______________________________________________________
_______________________________________________________.
If two figures are congruent, then
_______________________________________________________
_______________________________________________________.
Congruent figures are also congruent.
L
J
Q
K
P
R
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Example 3: Given ABCD  WXYZ , find the indicated measure.
B
a. WZ
b. mW
A
105
12m
80
D
C
X
Y
16 m
W
Z
PUZZLER: If you double the length of the sides of a square, does the perimeter
double or quadruple?
HOMEWORK:
SECTION: 6.5 Similarity and Measurement
Objective: TLW find unknown side lengths of similar figures.
Standards: M7.A.2.2, M7.A.2.2.6, M7.C.1.2.1, M7.C.1.2.2
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
Example 1: Given ABCD ~ EFGH, find EH.
Example 1 You try:
1. Given STU ~ DEF , find
2. Given JKLM ~ PQRS, Find PQ.
Example 2: A man who is 6 feet tall is standing near a saguaro cactus.
The length of the man’s shadow is 2 feet. The cactus casts a
shadow 5 feet long. The cactus and the man are
perpendicular to the ground. The sun’s rays strike the cactus
and the man at the same angle, forming two similar triangles.
How tall is the cactus?
A cactus is 5 feet tall and casts a shadow that is 1.5 feet long. How tall is
a nearby cactus that casts a shadow that is 8 feet long?
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Example 3: Given ABC ~ DEC , find BE.
HOMEWORK:
SECTION: 6.6 Scale Drawings
Objective: TLW use proportions with scale drawings.
Standards: M7.A.2.2, M7.A.2.2.1, M7.A.2.2.6,
M7.B.2.2
+ = + = + = + = + = + = + = + = + = + = + = + = + =
Scale drawing:
Scale model:
Scale:
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
Example 1: A map uses a scale of 1 inch – 5 miles. Two towns on the map
are 4.5 inches apart. How far apart are the actual towns?
Example 2: Mike’s map shows a bicycle route that is 112 miles long. On
his map, this distance is 14 inches. What is the scale of the
map?
Example 3: A model of a sailboat has a scale of 1:20. The actual sailboat
is 32 feet long. How long is the model?
PUZZLER: Our solar system has a radius of about 5,900,000,000 kilometers. What scale
could you use to make a model that would fit in a school gymnasium?
HOMEWORK:
SECTION: 6.7 Probability and Odds
Objective: TLW find probabilities.
Standards: M7.A.2.2, M7.A.2.2.1, M7.E.3.1, M7.E.3.1.1
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
You are conducting an experiment by rolling a six-sided die.
 The possible results of an experiment are OUTCOMES.
The possible results are ___, ___, ___, ___, ___, ___.
 An EVENT is an outcome or a collection of outcomes.
For example: _____________________________________.
 Once you specify an event, the outcomes for that event are
called FAVORABLE OUTCOMES.
For example: _____________________________________.
 The PROBABILITY that an event occurs is a measure of the
likelihood that the event will occur.
Probability of an Event
The probability of an event when all the outcomes are equally likely is:
P(event) = Number of favorable outcomes
Number of possible outcomes
Example 1: Suppose you roll a number cube. What is the probability that
you roll an even number?
P(rolling an even number) = Number of favorable outcomes =
Number of possible outcomes
What is the probability that you roll a number greater than 1?
What is the probability that you roll either a 4 or a 5?
Theoretical probability –
Experimental probability –
Experimental Probability
The experimental probability of an event is:
P(event) = Number of successes
Number of trials
Example 2: A miniature golf course offers a free game to golfers who
make a hole-in-one on the last hole. Last week, 44 out of 256
golfers made a hole-in-one on the last hole. Find the
experimental probability that a golfer makes a hole-in-one on
the last hole.
Example 2 You try:
You interviewed 45 randomly chosen students for the newspaper. Of the
students you interviewed, 15 play sports. Find the experimental
probability that the next randomly chosen student will play sports.
Interpreting Probabilities:
Probabilities can range from 0 to 1. The closer the probability of an event
is to 1, the more likely the event will occur.
0
1
Example 3: Today, you attempted 50 free throws and made 32 of them.
Use experimental probability to predict how many free throws you will
make tomorrow if you attempt 75 free throws.
-if you attempt 150 free throws?
When all outcomes are equally likely, the ratio of the number of
favorable outcomes to the number of unfavorable outcomes is called
____________________ of an event.
The ratio of the number of unfavorable outcomes to the number of
favorable outcomes is called _________________ an event.
Odds in favor = Number of favorable outcomes
Number of unfavorable outcomes
Odds against = Number of unfavorable outcomes
Number of favorable outcomes
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Example 4: Suppose you randomly choose a number between 1 and 20.
a. What are the odds in favor of choosing a prime number?
b. What are the odds against choosing a prime number?
HOMEWORK:
SECTION: 6.8 The Counting Principle
Objective: TLW use the counting principle to find probabilities.
Standards: M7.E.3.1, M7.E.3.1.1
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
Example 1: You are buying new eyeglasses and must choose the frame
material and shape. The frame material can be plastic or
metal. The frame shape can be rectangular, oval, cat’s eye,
or round. How many different frames are possible?
Make a tree diagram to count the number of possible choices.
The Counting 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 n .
The counting principle can be extended to three or more events.
Example 2: You roll a blue and a red number cube. Use the counting
principle to find the number of different outcomes that are
possible.
Example 2 You try:
How many different outcomes are possible when you flip a coin and roll a
number cube?
Example 3: A combination lock has 40 numbers on its dial. To open the
lock, you must turn the dial right to the first number, left to the second
number, then right to the third number. You randomly choose three
numbers on the lock. What is the probability that you choose the correct
combination?
HOMEWORK:
SECTION: 11.9 Independent and Dependent Events
Objective: TLW find the probability that event A and event B occur.
Standards: M7.A.2.2, M7.A.2.2.2, M7.A.2.2.5
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
Vocabulary:
Independent events:
Dependent events:
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
Example 1: Tell whether the events are independent or dependent.
a. You toss a coin and it shows heads. You toss the coin again and it
shows tails.
b. You randomly draw a name from a hat. Then, without putting the first
name back, you randomly draw a second name.
c. You roll a 5 on a number cube, then you roll a 6.
d. You randomly draw a marble from a bag. Then you put it back in the
bag and randomly draw another marble from the bag.
+ = + = + = + = + = + = + = + = + = + = + = + = + = + = + = + = +
PROBABILITY of INDEPENDENT EVENTS
Words:
Algebra:
Example 2:
A computer randomly generates 4-digit passwords. Each digit can be used
more than once. What is the probability that the first two digits in your
password are both 1?
Example 2 You Try:
What is the probability of rolling a 5 on each of three different
number cubes?
PROBABILITY of DEPENDENT EVENTS
Words:
Algebra:
Example 3:
A jar of jelly beans contains 50 red jelly beans, 45 yellow jelly beans, and 30
green jelly beans. Find the probability that both the first and second jelly
bean drawn from the jar is red assuming that the first jelly bean was eaten.
Example 3 You Try:
HOMEWORK:
A group of students consists of 6 girls and 7 boys. Two students
are chosen at random one at a time. What is the probability that
both students who are selected are girls?