Acing The Gre phần 9 pptx

61. d. It is ironic that in a place where there are so
many ways to describe one food (indicating that
this food is a central part of the culture),
Thomas is hungry. The passage does not men-
tion the language of the reservation, so choice a
is incorrect. The sentence does not show any
measure of how hungry Thomas is, so choice b
is incorrect. The sentence does not describe fry
bread or make it sound in any way appealing, so
choice c is also incorrect. The passage tells us
that it was Thomas’s hunger, not the number of
ways to say fry bread, that provided his inspira-
tion, so choice e is incorrect.
62. c. The author tells us that the new house was in
“the best neighborhood in town,” and the neigh-
borhood’s “prestige outweighed its deadliness”
(lines 5–8). There is no indication that their old
house was falling apart (choice a) or that they
needed more room (choice b). The neighbor-
hood is clearly not great for children (“it was not
a pleasant place to live [especially for chil-
dren]”), so choice d is incorrect. The author tells
us that business was going well for his father—
so well, in fact, that he could pay for the house
in cash—but that does not mean the house was
affordable (choice e). In fact, if it was in the
most prestigious neighborhood, it was probably
expensive.
63. a. The author tells us that his father was “always a
man of habit”—so much so that he forgot he’d
moved and went to his old house, into his old

room, and lay down for a nap, not even noticing
that the furniture was different. This suggests
that he has a difficult time accepting and adjust-
ing to change. There is no evidence that he is a
calculating man (choice b). He may be unhappy
with his life (choice c), which could be why he
chose not to notice things around him, but there
is little to support this in the passage, while
there is much to support choice a. We do not
know if he was proud of the house (choice d).
We do know that he was a man of habit, but we
do not know if any of those habits were bad
(choice e).
64. d. That his father would not realize that someone
else was living in the house—that he would not
notice, for example, different furniture arranged
in a different way—suggests that his father did
not pay any attention to things around him and
just went through the motions of his life by
habit. Being habitual is different from being
stubborn, so choice a is incorrect. The author is
writing about his father and seems to know him
quite well, so choice b is incorrect. We do not
know if the author’s father was inattentive to his
needs (choice c), though if he did not pay atten-
tion to things around him, he likely did not pay
much attention to his children. Still, there is not
enough evidence in this passage to draw this
conclusion. His father may have been very
attached to the old house (choice e), but the

incident doesn’t just show attachment; it shows
a lack of awareness of the world around him.
65. b. The bulk of this excerpt is the story that the
author finds “pathetic,” so the most logical con-
clusion regarding his feelings for his father is
that he lived a sad life. We know that his busi-
ness was going well, but the author does not dis-
cuss his father’s methods or approach to
business, so choice a is incorrect. Choice c is
likewise incorrect; there is no discussion of his
father’s handling of financial affairs. Choice d is
incorrect because there is no evidence that his
father was ever cruel. His father may have been
impressive and strong (choice e), but the domi-
nant theme is his habitual nature and the sad
fact that he did not notice things changing
around him.
– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS–
375

Glossary of Terms: Language
Arts, Reading
alliteration the repetition of sounds, especially at the
beginning of words
antagonist the person, force, or idea working against
the protagonist
antihero a character who is pathetic rather than
tragic, who does not take responsibility for his or her
destructive actions
aside in drama, when a character speaks directly to

the audience or another character concerning the
action on stage, but only the audience or character
addressed in the aside is meant to hear
autobiography the true account of a person’s life
written by that person
ballad a poem that tells a story, usually rhyming abcb
blank verse poetry in which the structure is con-
trolled only by a metrical scheme (also called metered
verse)
characters people created by an author to carry the
action, language, and ideas of a story or play
climax the turning point or high point of action and
tension in the plot
closet drama a play that is meant only to be read,
not performed
comedy humorous literature that has a happy
ending
commentary literature written to explain or illumi-
nate other works of literature or art
complication the series of events that “complicate”
the plot and build up to the climax
conflict a struggle or clash between two people,
forces, or ideas
connotation implied or suggested meaning
context the words and sentences surrounding a
word or phrase that help determine the meaning of
that word or phrase
couplet a pair of rhyming lines in poetry
denotation exact or dictionary meaning
denouement the resolution or conclusion of the

action
dialect language that differs from the standard lan-
guage in grammar, pronunciation, and idioms (natu-
ral speech versus standard English); language used by
a specific group within a culture
dialogue the verbal exchange between two or more
people; conversation
diction the particular choice and use of words
drama literature that is meant to be performed
dramatic irony when a character’s speech or actions
have an unintended meaning known to the audience
but not to the character
elegy a poem that laments the loss of someone or
something
exact rhyme the repetition of exactly identical
stressed sounds at the end of words
exposition in plot, the conveyance of background
information necessary to understand the complica-
tion of the plot
eye rhyme words that look like they should rhyme
because of spelling, but because of pronunciation,
they do not
falling action the events that take place immediately
after the climax in which “loose ends” of the plot are
tied up
feet in poetry, a group of stressed and unstressed
syllables
fiction prose literature about people, places, and
events invented by the author
figurative language comparisons not meant to be

taken literally but used for artistic effect, including
similes, metaphors, and personification
flashback when an earlier event or scene is inserted
into the chronology of the plot
free verse poetry that is free from any restrictions of
meter and rhyme
functional texts literature that is valued mainly for
the information it conveys, not for its beauty of form,
emotional impact, or message about human experience
genre category or kind; in literature, the different
kinds or categories of texts
haiku a short, imagistic poem of three unrhymed
lines of five, seven, and five syllables, respectively
half-rhyme the repetition of the final consonant at
the end of words
hyperbole extreme exaggeration not meant to be
taken literally, but done for effect
iambic pentameter a metrical pattern in poetry in
which each line has ten syllables (five feet) and the
stress falls on every second syllable
imagery the representation of sensory experiences
through language
inference a conclusion based upon reason, fact, or
evidence
– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS–
376
irony see dramatic irony, situational irony,or verbal
irony
literature any written or published text
literary texts literature valued for its beauty of

form, emotional impact, and message(s) about the
human experience
main idea the overall fact, feeling, or thought a writer
wants to convey about his or her subject
melodrama a play that starts off tragic but has a
happy ending
memoir an autobiographical text that focuses on a
limited number of events and explores their impact
metaphor a type of figurative language that com-
pares two things by saying they are equal
meter the number and stress of syllables in a line of
poetry
monologue in drama, a play or part of a play
performed by one character speaking directly to the
audience
narrator in fiction, the character or person who tells
the story
nonfiction prose literature about real people, places,
and events
ode a poem that celebrates a person, place, or thing
omniscient narrator a third-person narrator who
knows and reveals the thoughts and feelings of the
characters
onomatopoeia when the sound of a word echoes its
meaning
paragraph a group of sentences about the same idea
personification figurative language that endows
nonhuman or nonanimal objects with human
characteristics
plot the ordering of events in a story

poetry literature written in verse
point of view the perspective from which something
is told or written
prose literature that is not written in verse or dra-
matic form
protagonist the “hero” or main character of a story,
the one who faces the central conflict
pun a play on the meaning of a word
quatrain in poetry, a stanza of four lines
readability techniques strategies writers use to
make information easier to process, including the use
of headings and lists
rhyme the repetition of an identical or similar
stressed sound(s) at the end of words
rhythm the overall sound or “musical” effect of the
pattern of words and sentences
sarcasm sharp, biting language intended to ridicule
its subject
satire a form of writing that exposes and ridicules its
subject with the hope of bringing about change
setting the time and place in which a story unfolds
simile a type of figurative language that compares two
things using like or as
situational irony the tone that results when there is
incongruity between what is expected to happen and
what actually occurs
soliloquy in drama, a speech made by a character
who reveals his or her thoughts to the audience as if
he or she is alone and thinking aloud
sonnet a poem composed of fourteen lines, usually

in iambic pentameter, with a specific rhyme scheme
speaker in poetry, the voice or narrator of the poem
stage directions in drama, the instructions pro-
vided by the playwright that explain how the action
should be staged, including directions for props, cos-
tumes, lighting, tone, and character movements
stanza a group of lines in a poem, a poetic paragraph
structure the manner in which a work of literature is
organized; its order of arrangement and divisions
style the manner in which a text is written, composed
of word choice, sentence structure, and level of for-
mality and detail
subgenre a category within a larger category
suspense the state of anxiety caused by an unde-
cided or unresolved situation
symbol a person, place, or object invested with spe-
cial meaning to represent something else
theme the overall meaning or idea of a literary work
thesis the main idea of a nonfiction text
thesis statement the sentence(s) that express an
author’s thesis
tone the mood or attitude conveyed by writing or
voice
topic sentence the sentence in a paragraph that
expresses the main idea of that paragraph
tragedy a play that presents a character’s fall due to
a tragic flaw
tragic hero the character in a tragedy who falls from
greatness and accepts responsibility for that fall
tragic flaw the characteristic of a hero in a tragedy

that causes his or her downfall
– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS–
377
tragicomedy a tragic play that includes comic
scenes
understatement a statement that is deliberately
restrained
verbal irony when the intended meaning of a word
or phrase is the opposite of its expressed meaning
voice in nonfiction, the sound of the author speaking
directly to the reader
– GED LITERATURE AND THE ARTS, READING PRACTICE QUESTIONS–
378
379
PART
VI
The GED
Mathematics
Exam
T
his section covers the material you need to know to prepare for the GED Math-
ematics Exam. You will learn how the test is structured so you will know what
to expect on test day.You will also review and practice the fundamental math-
ematics skills you need to do well on the exam.
Before you begin Chapter 40, take a few minutes to do the pretest that follows. The
questions and problems are the same type you will find on the GED. When you are fin-
ished, check the answer key carefully to assess your results. Your pretest score will help you
determine how much preparation you need and in which areas you need the most care-
ful review and practice.


Pretest: GED Mathematics
Directions: Read each of the questions below carefully
and determine the best answer.
To practice the timing of the GED exam, please allow
18 minutes for this pretest. Record your answers on the
answer sheet provided here and the answer grids for
questions 9 and 10.
Note: On the GED, you are not permitted to write in
the test booklet. Make any notes or calculations on a sep-
arate piece of paper.
1. On five successive days, a motorcyclist listed his
mileage as follows: 135, 162, 98, 117, 216.
If his motorcycle averages 14 miles for each
gallon of gas used, how many gallons of gas did
he use during these five days?
a. 42
b. 52
c. 115
d. 147
e. 153
2. Bugsy has a piece of wood 9 feet 8 inches long.
He wishes to cut it into 4 equal lengths. How far
from the edge should he make the first cut?
a. 2.5 ft.
b. 2 ft 5 in.
c. 2.9 ft.
d. 29 ft.
e. 116 in.
Question 3 is based on the following figure.
3. What is the perimeter of the figure?

a. 8a + 5b
b. 9a + 7b
c. 7a + 5b
d. 6a + 6b
e. 8a + 6b
4. Jossie has $5 more than Siobhan, and Siobhan
has $3 less than Michael. If Michael has $30, how
much money does Jossie have?
a. $30
b. $27
c. $32
d. $36
e. Not enough information is given.
3a + b
3a + 2b
2a + b
a + 3b
– THE GED MATHEMATICS EXAM–
380
1. abcde
2. abcde
3. abcde
4. abcde
5. abcde
6. abcde
7. abcde
8. abcde
ANSWER SHEET
Questions 5 and 6 are based on the following graph.
5. The number of persons engaged in Food Service

in the city during this period was
a. 900.
b. 9,000.
c. 14,400.
d. 36,000.
e. 90,000.
6. If the number of persons in trade and finance is
represented by M, then the approximate number
in manufacturing is represented as
a.
ᎏ
M
5
ᎏ
b. M + 3
c. 30M
d.
ᎏ
4
3
M
ᎏ
e. Not enough information is given.
Question 7 is based on the following figure.
7. In the figure AB
៮
៮
៮
| | CD
៮

៮
៮
, CE
៮
៮
៮
bisects ∠BCD, and
m∠ABC = 112°. Find m∠ECD.
a. 45°
b. 50°
c. 56°
d. 60°
e. Not enough information is given.
8. Mr. DeLandro earns $12 per hour. One week,
Mr. DeLandro worked 42 hours; the following
week, he worked 37 hours. Which of the
following indicates the number of dollars Mr.
DeLandro earned for 2 weeks?
a. 12 × 2 + 37
b. 12 × 42 + 42 × 37
c. 12 × 37 + 42
d. 12 + 42 × 37
e. 12(42 + 37)
9. What is the slope of the line that passes through
points A and B on the coordinate graph below?
Mark your answer in the circles in the grid
below.
1
2
3

4
5
6
7
8
9
•
1
2
3
4
5
6
7
8
9
0
•
/
1
2
3
4
5
6
7
8
9
0
•

/
1
2
3
4
5
6
7
8
9
0
•
/
1
2
3
4
5
6
7
8
9
0
•
y
5
4
3
x
2

1
−5 −4 −3 −2 −1 12345
A (1,3)
B (3,5)
−2
−1
−3
−4
−5
A
B
C
D
E
Personal
Service
12%
Professional
8%
Food
Service
5%
All Others
17%
Trade and
Finance
25%
Manufacturing
33%
There are 180,000

employees total.
– THE GED MATHEMATICS EXAM–
381
10. What is the value of the expression 3(2x − y) +
(3 + x)
2
, when x = 4 and y = 5? Mark your
answer in the circles on the grid below.
Pretest Answers and Explanations
1. b. First, find the total mileage; 135 + 162 + 98 +
117 + 216 = 728 miles. Divide the total mileage
(728) by the number of miles covered for each
gallon of gas used (14) to find the number of
gallons of gas needed; 728 ÷ 14 = 52 gallons.
2. b. 1 ft. = 12 in. 9 ft. 8 in. = 9 × 12 + 8 = 116 in.;
116 ÷ 4 = 29 in. = 2 ft. 5 in.
3. b. To find the perimeter of the figure, find the sum
of the lengths of the four sides: 2a + b + a + 3b
+ 3a + b + 3a + 2b = 9b + 7b.
4. c. Michael has $30. Siobhan has $30 − $3 = $27.
Jossie has $27 + $5 = $32.
5. b. To find 5% of a number, multiply the number
by .05: 180,000 × .05 = 9,000. There are 9,000
food service workers in the city.
6. d. M = number of persons in trade and finance.
Since M = 25% of the total, 4M = total number
of city workers. Number of persons in manufac-
turing = =
ᎏ
4

3
M
ᎏ
.
total number of workers
ᎏᎏᎏ
3
Personal
Service
12%
Professional
8%
Food
Service
5%
All Others
17%
Trade and
Finance
25%
Manufacturing
33%
There are 180,000
employees total.
3a + b
3a + 2b
2a + b
a + 3b
1
2

3
4
5
6
7
8
9
•
1
2
3
4
5
6
7
8
9
0
•
/
1
2
3
4
5
6
7
8
9
0

•
/
1
2
3
4
5
6
7
8
9
0
•
/
1
2
3
4
5
6
7
8
9
0
•
– THE GED MATHEMATICS EXAM–
382
7. c. Since pairs of alternate interior angles of parallel
lines have equal measures, m∠BCD = m∠ABC.
Thus, m∠BCD = 112°.

m∠ECD =
ᎏ
1
2
ᎏ
m∠BCD =
ᎏ
1
2
ᎏ
(112) = 56°
8. e. In two weeks, Mr. Delandro worked a total of
(42 + 37) hours and earned $12 for each hour.
Therefore, the total number of dollars he earned
was 12(42 + 37).
9.1.
The coordinates of point A are (1,3). The
coordinates of point B are (3,5). Use the slope
formula:
ᎏ
x
y
2
2
−
−
y
x
1
1

ᎏ
Substitute and solve:
ᎏ
5
3
−
−
3
1
ᎏ
=
ᎏ
2
2
ᎏ
,or
ᎏ
1
1
ᎏ
= 1
1
2
3
4
5
6
7
8
9

•
1
2
3
4
5
6
7
8
9
0
•
/
1
2
3
4
5
6
7
8
9
0
•
/
1
2
3
4
5

6
7
8
9
0
•
/
2
3
4
5
6
7
8
9
0
•
1
y
5
4
3
x
2
1
−5 −4 −3 −2 −1 12345
A (1,3)
B (3,5)
−2
−1

−3
−4
−5
A
B
C
D
E
– THE GED MATHEMATICS EXAM–
383
10. 58.
3(2x − y) + (3 + x)
2
, x = 4 and y = 5.
3(2 × 4 − 5) + (3 + 4)
2
= 3(8 − 5) + (7)
2
=
3(3) + 49 = 9 + 49 = 58.
Pretest Assessment
How did you do on the math pretest? If you answered
seven or more questions correctly, you have earned the
equivalent of a passing score on the GED Mathematics
Test. But remember that this pretest covers only a frac-
tion of the material you might face on the GED exam. It
is not designed to give you an accurate measure of how
you would do on the actual test. Rather, it is designed to
help you determine where to focus your study efforts.
For success on the GED, review all of the chapters in this

section thoroughly. Focus on the sections that corre-
spond to the pretest questions you answered incorrectly.
1
2
3
4
5
6
7
8
9
•
1
2
3
4
5
6
7
8
9
0
•
/
1
2
3
4
5
6

7
8
9
0
•
/
1
2
3
4
6
7
8
9
0
•
/
1
2
3
4
5
6
7
9
0
•
58
– THE GED MATHEMATICS EXAM–
384


What to Expect on the GED Mathematics Exam
The GED Mathematics Exam measures your understanding of the mathematical knowledge needed in everyday
life. The questions are based on information presented in words, diagrams, charts, graphs, and pictures. In addi-
tion to testing your math skills, you will also be asked to demonstrate your problem-solving skills. Examples of
some of the skills needed for the mathematical portion of the GED are:
■
understanding the question
■
organizing data and identifying important information
■
selecting problem-solving strategies
■
knowing when to use appropriate mathematical operations
■
setting up problems and estimating
■
computing the exact, correct answer
■
reflecting on the problem to ensure the answer you choose is reasonable
This section will give you lots of practice in the basic math skills that you use every day as well as crucial
problem-solving strategies.
CHAPTER
About the GED
Mathematics
Exam
IN THIS chapter, you will learn all about the GED Mathematics
Exam, including the number and type of questions, the topics and
skills that will be tested, guidelines for the use of calculators, and
recent changes in the test.

40
385
The GED Mathematics Test is given in two separate
sections. The first section permits the use of a calculator;
the second does not. The time limit for the GED is 90
minutes, meaning that you have 45 minutes to complete
each section. The sections are timed separately but
weighted equally. This means that you must complete
both sections in one testing session to receive a passing
grade. If only one section is completed, the entire test
must be retaken.
The test contains 40 multiple-choice questions and
ten gridded-response questions for a total of 50 ques-
tions overall. Multiple-choice questions give you several
answers to choose from and gridded-response questions
ask you to come up with the answer yourself. Each
multiple-choice question has five answer choices, a
through e. Gridded response questions use a standard
grid or a coordinate plane grid. (The guidelines for
entering a gridded-response question will be covered
later in this section.)
Test Topics
The math section of the GED tests you on the following
subjects:
■
measurement and geometry
■
algebra, functions, and patterns
■
number operations and number sense

■
data analysis, statistics, and probability
Each of these subjects is detailed in this section along
with tips and strategies for solving them. In addition, 100
practice problems and their solutions are given at the end
of the subject lessons.
Using Calculators
The GED Mathematics Test is given in two separate
booklets, Part I and Part II. The use of calculators is per-
mitted on Part I only. You will not be allowed to use your
own. The testing facility will provide a calculator for you.
The calculator that will be used is the Casio fx-260. It is
important for you to become familiar with this calcula-
tor as well as how to use it. Use a calculator only when it
will save you time or improve your accuracy.
Formula Page
A page with a list of common formulas is provided with
all test forms. You are allowed to use this page when you
are taking the test. It is necessary for you to become
familiar with the formula page and to understand when
and how to use each formula. An example of the formula
page is on page 388 of this book.
Gridded-Response and Set-Up
Questions
There are ten non-multiple-choice questions in the math
portion of the GED. These questions require you to find
an answer and to fill in circles on a grid or on a coordi-
nate axis.
STANDARD GRID-IN QUESTIONS
When you are given a question with a grid like the one

below, keep these guidelines in mind:
■
First, write your answer in the blank boxes at the
top of the grid. This will help keep you organized
as you “grid in” the bubbles and ensure that you
fill them out correctly.
■
You can start in any column, but leave enough
columns for your whole answer.
■
You do not have to use all of the columns. If your
answer only takes up two or three columns, leave
the others blank.
■
You can write your answer by using either frac-
tions or decimals. For example, if your answer
is
ᎏ
1
4
ᎏ
, you can enter it either as a fraction or as a
decimal, .25.
The slash “/” is used to signify the fraction bar of the
fraction. The numerator should be bubbled to the left of
the fraction bar and the denominator should be bubbled
in to the right. See the example on the next page.
– ABOUT THE GED MATHEMATICS EXAM–
386
■

When your answer is a mixed number, it must be
represented on the standard grid in the form of
an improper fraction. For example, for the
answer 1
ᎏ
1
4
ᎏ
, grid in
ᎏ
5
4
ᎏ
.
■
When you are asked to plot a point on a coordi-
nate grid like the one below, simply fill in the
bubble where the point should appear.
SET
-UP QUESTIONS
These questions measure your ability to recognize the
correct procedure for solving a problem. They ask you to
choose an expression that represents how to “set up” the
problem rather than asking you to choose the correct
solution. About 25 percent of the questions on the GED
Mathematics Test are set-up questions.
Example: Samantha makes $24,000 per year at a new
job. Which expression below shows how much
she earns per month?
a. $24,000 + 12

b. $24,000 − 12
c. $24,000 × 12
d. $24,000 ÷ 12
e. 12 ÷ $24,000
Answer: d. You know that there are 12 months in a
year. To find Samantha’s monthly income, you
would divide the total ($24,000) by the number
of months (12). Option e is incorrect because it
means 12 is divided by $24,000.
Graphics
Many questions on the GED Mathematics Test use
diagrams, pie charts, graphs, tables, and other visual
stimuli as references. Sometimes, more than one of these
questions will be grouped under a single graphic. Do not
let this confuse you. Learn to recognize question sets by
reading both the questions and the directions carefully.
What’s New for the GED?
The structure of the GED Mathematics Test, revised in
2002, ensures that no more than two questions should
include “not enough information is given” as a correct
answer choice. Given this fact, it is important for you to
pay attention to how many times you select this answer
choice. If you find yourself selecting the “not enough
information is given” for the third time, be sure to check
the other questions for which you have selected this
choice because one of them must be incorrect.
The current GED has an increased focus on “math in
everyday life.” This is emphasized by allowing the use of
a calculator on Part I as well as by an increased empha-
sis on data analysis and statistics. As a result, gridded-

response questions and item sets are more common. The
number of item sets varies.
1
2
−3
4
−5
−6
0
1
−2
3
4
5
6
−1
2 3
−4
5
−6
−1
−2−3−4
−5
6
1
2
3
4
5
6

7
8
9
•
1
2
3
4
5
6
7
8
9
0
•
/
1
2
3
4
55
6
7
8
9
0
/
1
3
4

6
7
88
9
0
•
/
1
2
3
4
6
7
9
0
•
2
.
5
1
2
3
4
5
6
7
8
9
•
1

2
3
4
5
6
7
8
9
0
•
/
2
3
4
5
6
7
8
9
0
•
/
1
2
3
4
5
6
7
8

9
0
•
1
2
3
5
6
7
8
9
0
•
– ABOUT THE GED MATHEMATICS EXAM–
387
Area of a:
square Area = side
2
rectangle Area = length ϫ width
parallelogram Area = base ϫ height
triangle Area =
ᎏ
1
2
ᎏ
ϫ base ϫ height
trapezoid Area =
ᎏ
1
2

ᎏ
ϫ (base
1
+ base
2
) ϫ height
circle Area = π ϫ radius
2
; π is approximately equal to 3.14
Perimeter of a:
square Perimeter = 4 ϫ side
rectangle Perimeter = 2 ϫ length + 2 ϫ width
triangle Perimeter = side
1
+ side
2
+ side
3
Circumference of a circle Circumference = π ϫ diameter; π is approximately equal to 3.14
Volume of a:
cube Volume = edge
3
rectangular solid Volume = length ϫ width ϫ height
square pyramid Volume =
ᎏ
1
3
ᎏ
ϫ (base edge)
2

ϫ height
cylinder π ϫ radius
2
ϫ height π is approximately equal to 3.14
cone Volume =
ᎏ
1
3
ᎏ
ϫ π ϫ radius
2
ϫ height; π is approximately equal to 3.14
Coordinate Geometry distance between points = ͙(x
2
– x
ෆ
1
)
2
+ (y
ෆ
2
– y
1
)
ෆ
2
ෆ
; (x
1

,y
1
) and (x
2
,y
2
) are two points
in a plane
slope of a line =
ᎏ
y
x
2
2
–
–
y
x
1
1
ᎏ
; (x
1
,y
1
) and (x
2
,y
2
) are two points on the line

Pythagorean Relationship a
2
+ b
2
= c
2
; a and b are legs and c is the hypotenuse of a right triangle
Measures of mean =
ᎏ
x
1
+ x
2
+
n
+x
n
ᎏ
, where the x's are the values for which a mean is desired,
Central Tendency and n is the total number of values for x.
median = the middle value of an odd number of ordered scores, and halfway
between the two middle values of an even number of ordered scores.
Simple Interest interest = principal ϫ rate ϫ time
Distance distance = rate ϫ time
Total Cost total cost = (number of units) ϫ (price per unit)
Adapted from official GED materials.
388
Formulas
T
HE USE OF measurement enables you to form a connection between mathematics and the real world.

To measure any object, assign a unit of measure. For instance, when a fish is caught, it is often weighed
in ounces and its length measured in inches. This lesson will help you become more familiar with the
types, conversions, and units of measurement.
Also required for the GED Mathematics Test is knowledge of fundamental, practical geometry. Geometry is the
study of shapes and the relationships among them. A comprehensive review of geometry vocabulary and con-
cepts, after this measurement lesson, will strengthen your grasp on geometry.
CHAPTER
Measurement
and Geometry
THE GED Mathematics Test emphasizes real-life applications of
math concepts, and this is especially true of questions about meas-
urement and geometry. This chapter will review the basics of meas-
urement systems used in the United States and other countries,
performing mathematical operations with units of measurement, and
the process of converting between different units. It will also review
geometry concepts you’ll need to know for the exam, such as prop-
erties of angles, lines, polygons, triangles, and circles, as well as the
formulas for area, volume, and perimeter.
41
389

Types of Measurements
The types of measurements used most frequently in the
United States are listed below:
Units of Length
12 inches (in.) = 1 foot (ft.)
3 feet = 36 inches = 1 yard (yd.)
5,280 feet = 1,760 yards = 1 mile (mi.)
Units of Volume
8 ounces* (oz.) = 1 cup (c.)

2 cups = 16 ounces = 1 pint (pt.)
2 pints = 4 cups = 32 ounces = 1 quart (qt.)
4 quarts = 8 pints = 16 cups = 128 ounces = 1 gallon
(gal.)
Units of Weight
16 ounces* (oz.) = 1 pound (lb.)
2,000 pounds = 1 ton (T.)
Units of Time
60 seconds (sec.) = 1 minute (min.)
60 minutes = 1 hour (hr.)
24 hours = 1 day
7 days = 1 week
52 weeks = 1 year (yr.)
12 months = 1 year
365 days = 1 year
*Notice that ounces are used to measure both the volume and
weight.

Converting Units
When performing mathematical operations, it is neces-
sary to convert units of measure to simplify a problem.
Units of measure are converted by using either multipli-
cation or division:
■
To change a larger unit to a smaller unit, simply
multiply the specific number of larger units by
the number of smaller units that makes up one of
the larger units.
For example, to find the number of inches in 5
feet, simply multiply 5, the number of larger units,

by 12, the number of inches in one foot:
5 feet = how many inches?
5 feet × 12 inches (the number of inches in a single
foot) = 60 inches
Therefore, there are 60 inches in 5 feet.
Try another:
Change 3.5 tons to pounds.
3.5 tons = how many pounds?
3.5 tons × 2,000 pounds (the number of pounds in
a single ton) = 6,500 pounds
Therefore, there are 6,500 pounds in 3.5 tons.
■
To change a smaller unit to a larger unit, simply
divide the specific number of smaller units by the
number of smaller units in only one of the larger
units.
For example, to find the number of pints in 64
ounces, simply divide 64, the smaller unit, by 16,
the number of ounces in one pint.
= 4 pints
Therefore, 64 ounces are equal to four pints.
Here is one more:
Change 24 ounces to pounds.
= 2 pounds
Therefore, 32 ounces are equal to two pounds.

Basic Operations with
Measurement
It will be necessary for you to review how to add, sub-
tract, multiply, and divide with measurement. The

mathematical rules needed for each of these operations
with measurement follow.
Addition with Measurements
To add measurements, follow these two steps:
1. Add like units.
2. Simplify the answer.
32 ounces
ᎏᎏ
16 ounces
64 ounces
ᎏᎏ
16 ounces
specific number of the smaller unit
ᎏᎏᎏᎏᎏ
the number of smaller units in one larger unit
– MEASUREMENT AND GEOMETRY–
390
Example: Add 4 pounds 5 ounces to 20 ounces.
4 lb. 5 oz. Be sure to add ounces to ounces.
+
20 oz.
4 lb. 25 oz. Because 25 ounces is more than 16
ounces (1 pound), simplify by
dividing by 16. Then add the 1
pound to the 4 pounds.

4 lb. + 25 oz.

1 lb.
4 lb. + 16ͤ25

ෆ
−16
9 oz.
4 pounds 25 ounces =
4 pounds + 1 pound 9 ounces =
5 pounds 9 ounces
Subtraction with Measurements
1. Subtract like units.
2. Regroup units when necessary.
3. Write the answer in simplest form.
For example, to subtract 6 pounds 2 ounces
from 9 pounds 10 ounces,
9 lb. 10 oz. Subtract ounces from ounces.
−
6 lb. 2 oz. Then, subtract pounds from pounds.
3 lb. 8 oz.
Sometimes, it is necessary to regroup units when
subtracting.
Example: Subtract 3 yards 2 feet from 5 yards 1
foot.
5
4
΋
yd. 1
4
΋
ft.
− 3 yd. 2 ft.
1 yd. 2 ft.
From 5 yards, regroup 1 yard to 3 feet. Add 3

feet to 1 foot. Then subtract feet from feet and
yards from yards.
Multiplication with Measurements
1. Multiply like units.
2. Simplify the answer.
Example: Multiply 5 feet 7 inches by 3.
5 ft. 7 in. Multiply 7 inches by 3, then multiply 5
× 3 feet by 3. Keep the units separate.
15 ft. 21 in. Since 12 inches = 1 foot, simplify 21
inches.
15 ft. 21 in. = 15 ft. + 1 ft. + 9 inches =
16 feet 9 inches
Example: Multiply 9 feet by 4 yards.
First, change yards to feet by multiplying the
number of feet in a yard (3) by the number of
yards in this problem (4).
3 feet in a yard × 4 yards = 12 feet
Then, multiply 9 feet by 12 feet =
108 square feet.
(Note: feet × feet = square feet)
Division with Measurements
1. Divide into the larger units first.
2. Convert the remainder to the smaller unit.
3. Add the converted remainder to the existing
smaller unit if any.
4. Then, divide into smaller units.
5. Write the answer in simplest form.
Example:
Divide 5 quarts 4 ounces by 4.
1 qt. R1 First, divide 5 ounces

1. 4ͤ5
ෆ
ෆ
by 4, for a result of 1
−4 quart and a reminder
1 of one.
2. R1 = 32 oz. Convert the remainder
to the smaller unit
(ounces).
3. 32 oz. + 4 oz. = 36 oz. Add the converted
remainder to the
existing smaller unit.
4. 9 oz. Now divide the smaller
4ͤ36
ෆ
units by 4.
5. 1 qt. 9 oz.
– MEASUREMENT AND GEOMETRY–
391

Metric Measurements
The metric system is an international system of meas-
urement also called the decimal system. Converting units
in the metric system is much easier than converting
units in the English system of measurement. However,
making conversions between the two systems is much
more difficult. Luckily, the GED test will provide you
with the appropriate conversion factor when needed.
The basic units of the metric system are the meter,
gram, and liter. Here is a general idea of how the two sys-

tems compare:
METRIC S
YSTEM ENGLISH SYSTEM
1 meter A meter is a little more than a
yard; it is equal to about 39 inches.
1 gram A gram is a very small unit of
weight; there are about 30 grams
in one ounce.
1 liter A liter is a little more than a quart.
Prefixes are attached to the basic metric units listed
above to indicate the amount of each unit.
For example, the prefix deci means one-tenth (
ᎏ
1
1
0
ᎏ
);
therefore, one decigram is one-tenth of a gram, and one
decimeter is one-tenth of a meter. The following six pre-
fixes can be used with every metric unit:
Kilo Hecto Deka Deci Centi Milli
(k) (h) (dk) (d) (c) (m)
1,000 100 10
ᎏ
1
1
0
ᎏ
ᎏ

1
1
00
ᎏ
ᎏ
1,0
1
00
ᎏ
Examples:
■
1 hectometer = 1 hm = 100 meters
■
1 millimeter = 1 mm =
ᎏ
1,0
1
00
ᎏ
meter =
.001 meter
■
1 dekagram = 1 dkg = 10 grams
■
1 centiliter = 1 cL* =
ᎏ
1
1
00
ᎏ

liter = .01 liter
■
1 kilogram = 1 kg = 1,000 grams
■
1 deciliter = 1 dL* =
ᎏ
1
1
0
ᎏ
liter = .1 liter
*Notice that liter is abbreviated with a capital letter—“L.”
The chart shown here illustrates some common rela-
tionships used in the metric system:
Length Weight Volume
1 km = 1,000 m 1 kg = 1,000 g 1 kL = 1,000 L
1 m = .001 km 1 g = .001 kg 1 L = .001 kL
1 m = 100 cm 1 g = 100 cg 1 L = 100 cL
1 cm = .01 m 1 cg = .01 g 1 cL = .01 L
1 m = 1,000 mm 1 g = 1,000 mg 1 L = 1,000 mL
1mm = .001 m 1 mg = .001 g 1 mL = .001 L
Conversions within the Metric
System
An easy way to do conversions with the metric system is
to move the decimal point to either the right or the left
because the conversion factor is always ten or a power of
ten. As you learned previously, when you change from a
large unit to a smaller unit, you multiply, and when you
change from a small unit to a larger unit, you divide.
Making Easy Conversions within

the Metric System
When you multiply by a power of ten, you move the dec-
imal point to the right. When you divide by a power of
ten, you move the decimal point to the left.
To change from a large unit to a smaller unit, move
the decimal point to the right.
kilo hecto deka UNIT deci centi milli
To change from a small unit to a larger unit, move the
decimal point to the left.
Example:
Change 520 grams to kilograms.
Step 1: Be aware that changing meters to kilome-
ters is going from small units to larger units, and
thus, you will move the decimal point three places
to the left.
Step 2: Beginning at the UNIT (for grams), you
need to move three prefixes to the left.
یی ی
k h dk unit d c m
– MEASUREMENT AND GEOMETRY–
392
Step 3: Move the decimal point from the
end of 520 to the left three places. 520.

Place the decimal point before the 5. .520
Your answer is 520 grams = .520 kilograms.
Example:
You are packing your bicycle for a trip from
New York City to Detroit. The rack on the back
of your bike can hold 20 kilograms. If you

exceed that limit, you must buy stabilizers for
the rack that cost $2.80 each. Each stabilizer can
hold an additional kilogram. If you want to pack
23,000 grams of supplies, how much money will
you have to spend on the stabilizers?
Step 1: First, change 23,000 grams to kilograms.
یی ی
kg hg dkg g dg cg mg
Step 2: Move the decimal point three places to the
left.
23,000 g = 23.000 kg = 23 kg
Step 3: Subtract to find the amount over the limit.
23 kg − 20 kg = 3 kg
Step 4: Because each stabilizer holds one kilogram
and your supplies exceed the weight limit of the
rack by three kilograms, you must purchase three
stabilizers from the bike store.
Step 5: Each stabilizer costs $2.80, so multiply
$2.80 by 3: $2.80 × 3 = $8.40.

Geometry
As previously defined, geometry is the study of shapes
and the relationships among them. Basic concepts in
geometry will be detailed and applied in this section. The
study of geometry always begins with a look at basic
vocabulary and concepts. Therefore, here is a list of def-
initions of important terms:
area—the space inside a two-dimensional figure
bisect—cut in two equal parts
circumference—the distance around a circle

diameter—a line segment that goes directly through
the center of a circle—the longest line you can
draw in a circle
equidistant—exactly in the middle of
hypotenuse—the longest leg of a right triangle,
always opposite the right angle
line—an infinite collection of points in a straight
path
point—a location in space
parallel—lines in the same plane that will never
intersect
perimeter—the distance around a figure
perpendicular—two lines that intersect to form 90-
degree angles
quadrilateral—any four-sided closed figure
radius—a line from the center of a circle to a point
on the circle (half of the diameter)
volume—the space inside a three-dimensional
figure
– MEASUREMENT AND GEOMETRY–
393

Angles
An angle is formed by an endpoint, or vertex, and two
rays.
Naming Angles
There are three ways to name an angle.
1. An angle can be named by the vertex when no
other angles share the same vertex: ∠A.
2. An angle can be represented by a number written

across from the vertex: ∠1.
3. When more than one angle has the same vertex,
three letters are used, with the vertex always
being the middle letter: –1 can be written as
∠BAD or as ∠DAB; –2 can be written as ∠DAC
or as ∠CAD.
Classifying Angles
Angles can be classified into the following categories:
acute, right, obtuse, and straight.
■
An acute angle is an angle that measures less than
90 degrees.
■
A right angle is an angle that measures exactly 90
degrees. A right angle is represented by a square
at the vertex.
■
An obtuse angle is an angle that measures more
than 90 degrees, but less than 180 degrees.
■
A straight angle is an angle that measures 180
degrees. Thus, its sides form a straight line.
Straight Angle
180°
Obtuse Angle
Right
Angle
A
cute
Angle

1
2
A
C
D
B
Endpoint (or Vertex)
ray
ray
– MEASUREMENT AND GEOMETRY–
394
COMPLEMENTARY
ANGLES
Two angles are complementary if the sum of their meas-
ures is equal to 90 degrees.
SUPPLEMENTARY ANGLES
Two angles are supplementary if the sum of their meas-
ures is equal to 180 degrees.
ADJACENT ANGLES
Adjacent angles have the same vertex, share a side, and do
not overlap.
The sum of the measures of all adjacent angles around
the same vertex is equal to 360 degrees.
Angles of Intersecting Lines
When two lines intersect, two sets of nonadjacent angles
called vertical angles are formed. Vertical angles have
equal measures and are supplementary to adjacent
angles.
■
m∠1 = m∠3 and m∠2 = m∠4

■
m∠1 + m∠2 = 180 and m∠2 + m∠3 = 180
■
m∠3 + m∠4 = 180 and m∠1 + m∠4 = 180
Bisecting Angles and Line
Segments
Both angles and lines are said to be bisected when
divided into two parts with equal measures.
Example
Line segment AB is bisected at point C.
According to the figure, ∠A is bisected by ray AC.
35°
35°
1
2
3
4
1
2
3
4
∠1 + ∠2 + ∠3 + ∠4 = 360°
1
2
∠
1 and ∠2 are adjacent.
Adjacent
Angles
1
2

∠1 + ∠2 = 180°
Supplementary
Angles
1
2
∠1 + ∠2 = 90°
Complementar
y

Angles
– MEASUREMENT AND GEOMETRY–
395
Angles Formed by Parallel Lines
When two parallel lines are intersected by a third line,
vertical angles are formed.
■
Of these vertical angles, four will be equal and
acute, four will be equal and obtuse, or all four
will be right angles.
■
Any combination of an acute and an obtuse angle
will be supplementary.
In the above figure:
■
∠b, ∠c, ∠f, and ∠g are all acute and equal.
■
∠a, ∠d, ∠e, and ∠h are all obtuse and equal.
■
Also, any acute angle added to any obtuse angle
will be supplementary.

Examples
m∠b + m∠d = 180°
m∠c + m∠e = 180°
m∠f + m∠h = 180°
m∠g + m∠a = 180°
Example
In the figure below, if m || n and a || b, what is
the value of x?
Solution:
Because both sets of lines are parallel, you know
that x can be added to x + 10 to equal 180. The
equation is thus, x + x + 10 = 180.
Example
Solve for x:
2x + 10 = 180
−10 −10
ᎏ
2
2
x
ᎏ
=
ᎏ
17
2
0
ᎏ
x = 85
Therefore, m∠x = 85° and the obtuse angle is
equal to 180 − 85 = 95°.

Angles of a Triangle
The measures of the three angles in a triangle always
equal 180 degrees.
EXTERIOR ANGLES
An exterior angle can be formed by extending a side from
any of the three vertices of a triangle. Here are some rules
for working with exterior angles:
■
An exterior angle and interior angle that share the
same vertex are supplementary.
+ = 180° and = +
+ + = 180°
x °
(x + 10)°
b
m
n
a
– MEASUREMENT AND GEOMETRY–
396
■
An exterior angle is equal to the sum of the
nonadjacent interior angles.
■
The sum of the exterior angles of a triangle
equals 360 degrees.

Triangles
Classifying Triangles
It is possible to classify triangles into three categories

based on the number of equal sides:
Scalene Isosceles Equilateral
(no equal sides) (two equal sides) (all sides equal)
It is also possible to classify triangles into three cate-
gories based on the measure of the greatest angle:
Acute Right Obtuse
greatest angle greatest angle greatest angle
is acute is 90° is obtuse
Angle-Side Relationships
Knowing the angle-side relationships in isosceles, equi-
lateral, and right triangles will be useful in taking the
GED exam.
■
In isosceles triangles, equal angles are opposite
equal sides.
■
In equilateral triangles, all sides are equal and all
angles are equal.
Equilateral
60°
60°60°
55
5
AB
ba
C
m∠a = m∠b
Isosceles
50°
60°

70°
150°
Obtuse
Right
Acute
Scalene
Isoceles
Equilateral
– MEASUREMENT AND GEOMETRY–
397
■
In a right triangle, the side opposite the right
angle is called the hypotenuse. This will be the
longest side of the right triangle.
Pythagorean Theorem
The Pythagorean theorem is an important tool for work-
ing with right triangles. It states: a
2
+ b
2
= c
2
,where a and
b represent the legs and c represents the hypotenuse.
This theorem allows you to find the length of any side
as along as you know the measure of the other two.
a
2
+ b
2

= c
2
1
2
+ 2
2
= c
2
1 + 4 = c
2
5 = c
2
͙5
ෆ
= c
45-45-90 Right Triangles
A right triangle with two angles each measuring 45
degrees is called an isosceles right triangle. In an isosceles
right triangle:
■
The length of the hypotenuse is ͙2
ෆ
multiplied by
the length of one of the legs of the triangle.
■
The length of each leg is
ᎏ
͙
2
2

ෆ
ᎏ
multiplied by the
length of the hypotenuse.
x = y = ×
ᎏ
1
1
0
ᎏ
= 10 = 5͙2
ෆ
30-60-90 Triangles
In a right triangle with the other angles measuring 30
and 60 degrees:
■
The leg opposite the 30-degree angle is half the
length of the hypotenuse. (And, therefore, the
hypotenuse is two times the length of the leg
opposite the 30-degree angle.)
■
The leg opposite the 60-degree angle is ͙3
ෆ
times
the length of the other leg.
60°
30°
2s
s
s

√
¯¯¯
3
͙2
ෆ
ᎏ
2
͙2
ෆ
ᎏ
2
10
x
y
45°
45°
2
1
c
Hypotenuse
Right
– MEASUREMENT AND GEOMETRY–
398
Example
x = 2 × 7 = 14 and y = 7͙3
ෆ
Comparing Triangles
Triangles are said to be congruent (indicated by the sym-
bol Х) when they have exactly the same size and shape.
Two triangles are congruent if their corresponding parts

(their angles and sides) are congruent. Sometimes, it is
easy to tell if two triangles are congruent by looking.
However, in geometry, you must be able to prove that the
triangles are congruent.
If two triangles are congruent, one of the three crite-
ria listed below must be satisfied.
Side-Side-Side (SSS) The side measures for both
triangles are the same.
Side-Angle-Side (SAS) The sides and the angle
between them are the same.
Angle-Side-Angle (ASA) Two angles and the side
between them are the same.
Example: Are triangles ᭝ABC and ᭝BCD
congruent?
Given: ∠ABD is congruent to ∠CBD and ∠ADB
is congruent to ∠CDB. Both triangles share
side BD.
Step 1: Mark the given congruencies on the
drawing.
Step 2: Determine whether this is enough
information to prove the triangles are
congruent.
Yes, two angles and the side between them are
equal. Using the ASA rule, you can determine
that triangle ABD is congruent to triangle CBD.

Polygons and Parallelograms
A polygon is a closed figure with three or more sides.
Terms Related to Polygons
■

Ve rt ices are corner points, also called endpoints,
of a polygon. The vertices in the above polygon
are: A, B, C, D, E, and F.
■
A diagonal of a polygon is a line segment between
two nonadjacent vertices. The two diagonals in
the polygon above are line segments BF and AE.
■
A regular polygon has sides and angles that are all
equal.
■
An equiangular polygon has angles that are all
equal.
Angles of a Quadrilateral
A quadrilateral is a four-sided polygon. Since a quadri-
lateral can be divided by a diagonal into two triangles,
FE
D
C
B
A
A
B
C
D
A
B
C
D
60°

30°
x
7
y
– MEASUREMENT AND GEOMETRY–
399