B Neither passage describes a discovery, but rather the world picture line 5 of the medieval mind lines 26–27, that is, medieval theories about the na-ture of the universe.. Although Pas
Trang 110 104 Quadrilateral ABCD is
composed of two identical
triangles, each with an
area of 240 square inches
Area of ΔABC =1⁄2bh
240 =1⁄2(20)(h)
h= 24 inches
To solve for side BC, set up the Pythagorean theorem:
102+ 242= (BC)2
Simplify: 100 + 576 = (BC)2
Combine like terms: 676 = (BC)2
Take the square root: 26 = BC
(Or simply notice that each right triangle is a classic
5-12-13 triangle times 2: 10-24-26.)
The perimeter of quadrilateral ABCD=
26 + 26 + 26 + 26 = 104 (Chapter 10, Lesson 5: Areas and Perimeters)
11 30 If two things are equal, you can substitute
either one for the other Since g(x) = x2− 5,
f( g(x)) = f(x2− 5)
Plug x2− 5 into f(x) and simplify:
f(x) = 7(x2− 5) + 2 Distribute: 7x2− 35 + 2
Plug in 3 for x: 7(3)2− 35 + 2
Simplify: 63 − 35 + 2 = 30
(Chapter 11, Lesson 2: Functions)
12 3 Start by simplifying the expression:
Combine like terms:
This expression is 3 less than x.
(Chapter 8, Lesson 1: Solving Equations)
13 18 Approach this problem logically, but keep the
restrictions in mind If we want the largest possible
value of a and a + b < 20, try a = 19 But that is not a
possibility, because b is a positive integer and so can
be no less than 1, and 19 + 1 is equal to, not less than
20 Therefore, the largest value of a that fits the
re-striction is 18 If a = 18 and b = 1, then ab = (18)(1) =
18, an even number
(Chapter 9, Lesson 3: Numerical Reasoning Problems)
5 15 5
x−
6 9
5
6 5
5
x− − x+ = x− − −x
14 28 Since rectangle U and rectangle V share a side
with integer length, this length must be a common factor of 18 and 21
Similarly, the side that
rectangle U and rectan-gle W share must be a
common factor of 18 and 12 Therefore, the common side between
U and V is 3, and the
common side between
U and W is 6 So U is a 6-by-3 rectangle, V is a 7-by-3 rectangle, and W is a 6-by-2 rectangle, which means rectangle X must have an area of 14 The sum of the
areas of those four rectangles is 18 + 21 + 12 + 14 = 65 The area of the entire rectangle is given as 117 Thus,
the area of rectangles Y and Z together must be 117 −
65 = 52 Set up an equation:
6x + 7x = 52 Combine like terms: 13x= 52 Divide by 13: x= 4
If x = 4, then the area of rectangle Z is 4 × 7 = 28.
(Chapter 10, Lesson 5: Areas and Perimeters)
15 45 Begin by finding the amount the class would
spend on 55 regular-price tickets: 55 × $15 = $825 Then calculate how much 60 discounted tickets cost:
60 × $13 = $780 Then subtract to find the amount saved: $825 − $780 = $45
(Chapter 9, Lesson 3: Numerical Reasoning Problems)
16 22.5
The length of WY ––, as shown above, is 15
Point X is the midpoint of WY –– , so WX = XY = 7.5 YZ
= 2WX = 2(7.5) = 15 So XZ = 22.5.
(Chapter 6, Lesson 2: Analyzing Problems)
Solve for x: 3 ▫
x▫ 3 = 12 ▫ (Chapter 9, Lesson 1: New Symbol or Term Problems)
3 12 3
12 3
108
9 12
2
=( ) ( )
2 3 2
3 2
12
1 12
2
=( )( )
− = = = x
s rs
r s
= −2
U
W
Y
V
X
Z
6
6 6
7
7 7
3 2
x
B
D
C
24
26 26
26 26
24
Trang 218 37.5
Since DC –– ⊥ AB ––
, angle CDB is a right angle, so ΔCDB
is a 45°-45°-90° right triangle Therefore, DB = DC = 5.
Since AD = 2DB, AD = (5)(2) = 10.
The area of ΔABC =1⁄2(b)(h) =1⁄2(15)(5) = 37.5
(Chapter 10, Lesson 5: Areas and Perimeters)
Section 6
1 B The reflex produces an immediate (or
instan-taneous) response transient = short-lived; stagnant =
not moving; revitalized= filled with new life and energy
2 D Although the cats are emaciated (excessively
thin) and starved for food, summoning energy would
help them to fight hard or aggressively for the scraps.
humanely = with mercy; vigilantly = in a watchful way;
fluently = smoothly; ferociously = fiercely
3 C Jennifer irritated her peers with her
supercil-ious (overly proud) and pretentsupercil-ious (haughty)
re-marks These are characteristic of an arrogant or
showy demeanor reticent = reserved, unwilling to
speak; belligerent = warlike; lofty = pompous;
self-effacing = modest; discomfited = uneasy, uncomfortable
4 A The first part of the sentence indicates that a
sushi master’s work is not easily learned Therefore,
much training and studying are required to become a
master chef Apprenticeship, tutelage, and cultivation
are all good choices for the first word This training
will give someone the autonomy to create his or her
own work apprenticeship = working as a beginner
under the assistance of an instructor; autonomy=
in-dependence; tutelage = instruction; ineptitude = lack of
skill; dormancy = lack of activity; sovereignty =
supreme authority; cultivation= the act of improving;
boorish = rude, lacking manners; quiescent = not active
5 A The journalist had a reputation for breaking
news early, almost as if she were able to see the
fu-ture Later she admitted that she had privileged
sources and did not use prophecy at all prophetic=
able to tell the future; prescience= knowledge of
fu-ture events; premeditated= planned ahead of time;
predilection = preference; dismissive = indifferent;
omniscience = total knowledge; preeminent =
supe-rior; reluctance = resistance; insolvent = bankrupt;
foresight= thinking ahead
6 B Neither passage describes a discovery, but
rather the world picture (line 5) of the medieval mind
(lines 26–27), that is, medieval theories about the na-ture of the universe Although Passage 1 provides a
counterexample to an often-heard charge (lines 14–15),
Passage 2 does not attempt to disprove any assump-tions Neither passage questions the medieval theories presented Instead, the passages merely describe those theories Lastly, neither passage discusses the everyday life in medieval Europe
7 D The passage states that where our universe is thought to be dark, the other one was presumed to
be illuminated, which means that the medieval uni-verse was perceived to be full of light, unlike our modern universe
8 C Pascal is said to be disturbed by the silence of the vast spaces between the stars (lines 10–11), in con-trast to the medieval thinkers who formerly thought that the universe produced the “music of the spheres”
(lines 12–13)
9 E Dante’s theory is described in Passage 1 to counter the charge that medieval thinkers were focused
on man’s sense of self-importance (lines 16–17), but
Hilde-gard’s theory presented in the final sentence of Passage
2 is clearly anthropocentric, or human-centered The world views of both Dante and Hildegard are focused on religion and an ordered hierarchy, but neither ad-dresses scientific methods Lastly, Passage 1 does not discuss the public acceptance of Dante’s theory
10 E This passage is concerned primarily with de-scribing the relationships among the plants, animals, and climate of the Serengeti Therefore, it is describing how a particular ecosystem works Although the passage mentions human intervention tangentially in the last paragraph, where it refers to badly drawn park bound-aries, it is not a central focus of the passage Although it does mention individual plants, the passage as a whole does not focus on them, but instead shows how they play
a role in a larger ecosystem It does not mention natural disasters, and only mentions the distinction between grazers and browsers as a minor point
11 C These sentences suggest that the variety in the diet of grazers increases with the length of the grass When the grass is short, all the animals apparently eat much the same sort of grass, but when it is long, they diversify their diets
12 A Browsers are said to feed on shrubs or the leaves of trees (lines 15–16), as opposed to the grazers, which eat the abundant grass that springs up like a well-mown lawn (lines 9–11).
C
B
45°
45°
5 5
2
Trang 313 B The passage states that unlike all the other
grazers on the plain, (zebras) have teeth in both jaws
(lines 27–28) All the rest (besides the zebras) are
var-ious species of antelope (lines 30–31), which have
toothless upper palates (lines 32–33).
14 E The second paragraph states that where the
grass is all short all the animals apparently eat the
same sort of grass but where the grass is of varied
lengths each animal copes differently with the
avail-able fodder (lines 19–24) This difference is then
de-scribed in the third paragraph, where the grazing
sequence is specified
15 C The rains are said to bring on fresh growth
(line 53), encouraging the grazers to return to old
grazing lands
16 A This paragraph states that if the migrant
herds were confined (lines 63–64), they would so
weaken the grass that it would die out (lines 67–68) So
maintaining the grasslands requires that the animals
not be confined
17 A The thesis of the passage is that men come
greatly to desire that these capricious gifts of Nature
(that is, the natural resources that are hard for some
and easy for others to find , by luck alone) might be
in-tercepted by some agency having the power and the
goodwill to distribute them justly This desire is
So-cialism (lines 41–47).
18 C This primitive cultivator is a person who tries
to stick a spade into the earth and make wheat and
other edible matters spring from it This is a farmer
Although the author uses figurative and metaphorical
language throughout the passage, this particular
phrase is being used literally
19 D The astronomer is said to regard the earth as
simply a ball without ulterior motives (lines 3–4),
while the foolish spendthrift suddenly realizes that
the earth is offering him gold (lines 21–24) Therefore,
the astronomer regards the earth as impersonal,
while the spendthrift regards it as generous
20 B The closed hand represents the tendency of the
Earth to hide its diamonds and good red wheat (line 31).
21 A The author is discussing how capricious
na-ture is in revealing its resources, and suggests that
anyone trying to harvest the earth’s resources must
become a gambler (line 33), and scoff at theorists who
prate (speak inconsequentially) of moral virtues such
as industry and honesty and equality Therefore, the
author is suggesting that these virtues are not as
valu-able as many people claim they are
22 C This fate is the fate of the gambler (line 33),
who is at the whim of mother earth
23 E The author states that the Social Democratic State remains to be tried (lines 51–52).
24 C The author states that our own choice (that is, the choice of his society) is shown by our continual as-piration to possess property (lines 59–60).
Section 7
Combine like terms: 16x= 32
(Chapter 8, Lesson 1: Solving Equations)
2 D
(Chapter 8, Lesson 1: Solving Equations)
3 A 5b− 10 ≥ 15 Add 10: 5b≥ 25 Divide by 5: b≥ 5 (Chapter 8, Lesson 6: Inequalities, Absolute Values, and Plugging In)
4 C First find 30% of 50: (.3)(50) = 15
t% of 60 is 15
Set up the equation:
Cross-multiply: 60t= 1,500
(Chapter 7, Lesson 5: Percents)
5 C First convert 8 hours into minutes:
Cross-multiply: x= 480 minutes Then set up a ratio to answer the question:
Cross-multiply: 40y = 480h
(Chapter 8, Lesson 7: Word Problems) (Chapter 7, Lesson 4: Ratios and Proportions)
6 A First find the product of −1.5 × 1.25: −1.875 Point A is closest to −1.875 on the number line presented
(Chapter 7, Lesson 1: Numbers and Operations)
480minutes 40 cards
minutes cards
60
hours minutes
hour minutes
t
100
15 60
=
1
3 6 8
x
x x
⎛
⎝⎜ ⎞⎠⎟⎛⎝⎜ ⎞⎠⎟( )=
Trang 47 C If the four people must each have a different
positive number of cards, then the least that three
may have is one, two, and three cards This leaves a
maximum of 100 − 6 = 94 for the remaining person
(Chapter 9, Lesson 3: Numerical Reasoning Problems)
8 C A quick plot of the data listed in the table will
point you to answer
choice (C) Since the
miles per gallon are
de-creasing as the age
in-creases, you can
eliminate choice (A)
There is no point where
the data levels out,
which eliminates
an-swer choice (D) Finally,
because it is not
de-creasing at a constant rate, you can eliminate choices
(B) and (E)
(Chapter 11, Lesson 5: Data Analysis)
9 D If the question is at most how many of these
integers could be odd, begin by imagining that ALL of
them are odd The integers may be the same, so
imag-ine that they are all 1: 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7
But 7 is odd, so try 6 odds and 1 even:
1 + 1 + 1 + 1 + 1 + 1 + 2 = 8 Therefore, the most that
could be odd is 6
(Chapter 9, Lesson 3: Numerical Reasoning Problems)
10 D You can write out a quick calendar for
your-self to track the days:
If you do this problem too quickly, you might assume
that since the fourth Wednesday is the 22nd, the fourth
Monday would be the 20th But the first Monday
comes after the first Wednesday, which makes the
fourth Monday the 27th
(Chapter 9, Lesson 3: Numerical Reasoning
Problems)
11 D Be careful with this question Make sure you
understand the chart before choosing an answer The
question asks about teachers who use gas heat There
are 60 + 13 = 73 teachers who use gas heat, and 60 of
these live in a house
(Chapter 11, Lesson 5: Data Analysis)
(Chapter 7, Lesson 3: Fractions)
12 D “Stack” the equations: 5x + 7y = 18
2x − 4y = 6
Add straight down: 7x + 3y = 24
(Chapter 8, Lesson 2: Systems)
13 A
(1,36) (3,31) (5,20) Miles
per gallon
Age (years)
y°
z°
x°
x°
m
Su M T W Th F Sa
1 2 3 4
8 9 10
15 16 17
22 23 24
19 20 21
29 30 31
26 27 28
25
Starting with the angle marked x° in the original
fig-ure, you can mark its vertical angle x° as well Since
line l is parallel to line m, the corresponding angle in the top triangle is also x° Set up an equation for the triangle:
x + z + (180 − y) = 180
Subtract 180: x + z − y = 0
(You might also simply notice that the angle marked
y° is an “exterior” angle to the triangle, so its measure
is equal to the sum of the two “remote interior”
an-gles: y = x + z.)
Subtract z: x = y − z
(Chapter 10, Lesson 1: Lines and Angles) (Chapter 10, Lesson 2: Triangles)
14 B This problem involves rates, so it helps to
re-call the rate equation: d = rt.
Because she travels home along the same route, you can use d for the distance both to and from work
Be-cause she spends a total of 2 hours in the car, if she
spends t hours on the way to work, she will spend
2 − t hours on the way home from work.
Set up rate equations for both legs of the trip:
Set the expressions equal: 40t = 24(2 − t)
Distribute: 40t = 48 − 24t
Divide by 64: t= 75
Plug 0.75 in for t and solve for d: d= 40(.75) = 30
Check by confirming that plugging t = 75 into the other rate equation gives the same distance from home to work
(Chapter 9, Lesson 4: Rate Problems)
15 B The graph of the original function will be shifted up 4 and right 2 Answer choice (B) shows the proper representation of the new graph
(Chapter 11, Lesson 3: Transformations)
Trang 516 A
Line segment BD –– is tangent to the circle at point A, so
angles BAO and DAO are right angles This means
that both ΔDAO and ΔBAO are 30°-60°-90° triangles.
2 B The sentence states that one feature of Joyce’s
work gave way to another, suggesting that the first
missing word is a noun that directly contrasts the ad-jective in the second blank The second word is paired
with arcane, which means secret or little-understood This word must also describe works that feature ne-ologisms (invented words) and obscure literary tricks.
Therefore the first word should mean something like
clarity and the second phrase should include an ad-jective like hard to understand lucidity = clarity;
opaque = very difficult to understand or translate; con-cise = brief and to the point; anachronism = quality of being out of place in time; derivative= copied from others
3 B Churchill was known to choose his strategies arbitrarily (without logical reason), so he was whim-sical or impulsive diligent= working with great effort;
impulsive = acting without thought; vicious = evil, harsh; malevolent= wishing harm, malicious
4 D If the king executed those who acted
irrever-ently (without respect), he must have demanded ut-most respect insolence = brazen rudeness; impudence
= disrespect; truculence = inclination to pick fights; deference = respect; ignominy = humiliation
5 D Since Galileo contradicted church teachings,
he was a heretic ostracized= cut off from society;
hermit = one who seeks solitude; venerated = wor-shipped; demagogue = powerful leader; hallowed = respected as holy; revisionist= one who rethinks or
reshapes a commonly accepted view; denounced=
accused or condemned for being a villain; heretic=
one who holds controversial opinions; reviled=
at-tacked with harsh language; luminary= one who in-spires others
6 A The Senator is known for her iconoclastic
views, which means that she goes against the party line Because of this, she would have a tough time
get-ting traditional party members to support her con-tentious = quarrelsome; orthodox = traditional; litigious = prone to bringing lawsuits; disingenuous = insincere; vituperative= using harsh censure or
con-demnation; dissident = disagreeing; heretical = going against standard beliefs; polemical= pertaining to a highly controversial political or intellectual position
7 D This is an address to the Atlanta Exposition (as the introduction indicates), and the author is
clearly addressing those in the commercial world (lines
26–27) and entreating ex-slaves and Southern whites
to work together for their mutual benefit
O
30°
30°
60° 60°
6
6
3
6 3
Using the 30°-60°-90° reference information at the
be-ginning of this section, you can find the values of the
remaining sides of the triangles
Find the area of ΔBOD:
Plug in values:
The radius of the circle is 6, so the area of the entire
circle can be found using the equation Area = πr2=
π(6)2= 36π
The shaded region of the circle makes up 120°, or 1⁄3of
the circle Therefore, the area of the shaded region is
equal to 1⁄3(πr2) =1⁄3(36π) = 12π
The area of the unshaded region of the triangle can be
found by subtracting the area of the shaded region
from the total area of the triangle: 36 − 12π
(Chapter 10, Lesson 2: Triangles)
(Chapter 10, Lesson 8: Circles)
Section 8
1 E If businesses are having a hard time staying
current, their equipment must be old or outdated
be-cause technology is advancing at a fast rate urgency=
pressing importance; progressive = advancing
for-ward; conventional = standard; torpidity = lethargy;
antiquated = outdated; lassitude = lack of energy;
in-novative= inventive, novel
3
Area=1(12 3 6) ( )=
Area=1(base height) ( )
2
x
x
2x
3
6 3
60°
30°
60°
30°
Trang 68 A The author does not directly address those in
the antislavery movement but does address (B) those
of the white race who look to the incoming of those
of foreign birth for the prosperity of the South
(lines 45–48), (C) those of my race who depend upon
bettering their condition (lines 12–14), (D) those for
whom African Americans have tilled your fields,
cleared your forests (lines 56–57), and (E) those
African Americans who underestimate the
impor-tance of cultivating friendly relations with the
South-ern white man (lines 14–16).
9 C The captain did not heed the first, second, or
third call but heeded the fourth call at last (line 9),
suggesting that he did not believe the responses were
helpful at first
10 C The phrase this chance refers to the man’s
chance (that African Americans can have) in the
com-mercial world (lines 26–27).
11 A In saying that there is as much dignity in tilling
a field as in writing a poem (lines 40–41), the author is
saying that such manual labor is valuable work
12 D These indicate the thoughts of a Negro who
has given earnest thought to the situation of his people
in America (lines 68–70).
13 C In saying that because of incessant
self-questioning (line 83) the best energy of the Negro
people cannot be marshalled to do the bidding of the
race (lines 89–91), the author means that
introspec-tion keeps African Americans from organizing
them-selves to meet the needs of their race
14 D Such people are said to make room for every
rascal and demagogue who chooses to cloak his selfish
deviltry under the veil of race pride (lines 91–93); that
is, they allow themselves to be influenced by selfish
and evil people
15 D That point refers to the point farther than
(which) our Americanism does not go (lines 121–122).
In other words, this is the point up to which African
Americans share much in common with all
Ameri-cans but beyond which they are a unique people
16 C This broader humanity is that which freely
rec-ognizes differences in men, but sternly deprecates
(dis-approves of) inequality in their opportunities of
development (lines 139–141) In other words, its
mem-bers value equal opportunity for all races
17 E Passage 1 focuses on the manual labor that African Americans have performed in tilling fields, clearing forests, building railroads and cities, etc., while the author of Passage 2 emphasizes
contribu-tions like the subtle sense of song that has given Amer-ica its only AmerAmer-ican music, its only AmerAmer-ican fairy tales, its only touch of pathos and humor (lines
130–133)
18 A The black tomorrow in Passage 2 is the influ-ence of African Americans in softening the whiteness
of the Teutonic today (line 129), which suggests a
change in the dominant culture Passage 1, on the other hand, envisions a future in which African
Amer-icans make friends in every manly way of the people of all races by whom we are surrounded (lines 19–20) and
incorporate themselves into the existing dominant
in-dustries of agriculture, mechanics, commerce, (and) domestic service (lines 21–22).
19 E Passage 1 indicates that the dominant culture
can give the African American a man’s chance in the commercial world (lines 26–27) and contains many opportunities (line 44) Passage 2 is more assertive in
suggesting that African Americans have changed and
will continue to change the dominant culture: We are the first fruits of this new nation, the harbinger of that black tomorrow which is yet destined to soften the whiteness of the Teutonic today (lines 126–129).
Section 9
1 C The parallel idiom neither nor requires that the phrase following neither and the phrase fol-lowing nor have the same grammatical form The
only choice that maintains proper idiom and paral-lelism is (C)
(Chapter 15, Lesson 3: Parallelism)
2 E This sentence contains three clauses, each of
which has the same subject, Georgia The underlined
clause, however, is in the passive voice, unlike the other two It should be changed to the active voice like the others
(Chapter 15, Lesson 3: Parallelism)
3 D Since any rehearsal would have been com-pleted before the performance, the participle in the
underlined phrase should be in the perfect form hav-ing rehearsed Choice (B) also uses the nonstandard phrase being that, and choices (C) and (E) create run-on
sentences
(Chapter 15, Lesson 9: Tricky Tenses)
Trang 74 B In the original sentence, the subject outlets
does not agree with the verb has acknowledged In
choice (C), the subject scope disagrees with the verb
have been acknowledged Choice (D) is awkward and
choice (E) uses an illogical verb tense Choice (B)
con-veys the idea clearly and grammatically
(Chapter 15, Lesson 1: Subject Verb Disagreement)
5 E The original verb have had is in the imperative
mood, but should be in the subjunctive mood because
it conveys a hypothetical condition Choice (E) conveys
the mood correctly
(Chapter 15, Lesson 14: The Subjunctive Mood)
6 C The statement made in the main clause is not
against popular opinion, but rather is contrary to it.
Although choice (E) uses the proper modifier, it
illog-ically suggests that an opinion can say something.
(Chapter 15, Lesson 15: Coordinating Ideas)
7 A The original sentence is the most logical and
effective option
8 B The original phrase misuses the semicolon,
because the phrase preceding it is not an independent
clause Similarly, choice (E) uses the conjunction so
to join two clauses, but the first is not independent, so
the sentence is ungrammatical Choices (C) and (D)
use the unidiomatic phrases struggled for finding and
struggled finding Choice (B) avoids these problems,
and is clear and effective
(Chapter 15, Lesson 15: Coordinating Ideas)
9 A The original phrasing is best It provides the
parallel form required by the comparative idiom not
so much by as by , whereas the others violate
parallel form
10 D The original phrasing is unidiomatic The
cor-rect idiom is A appears to be B.
(Chapter 15, Lesson 10: Idiom Errors)
11 C The comparison is logically between the re-sponse to the revised proposal and the rere-sponse to the
original proposal Choice (C) is the only one that makes the correct logical and parallel comparison
(Chapter 15, Lesson 4: Comparison Problems)
12 C The original phrasing breaks the idea into two independent clauses But since it conveys one central idea, it is more effectively phrased with a single inde-pendent clause and a modifying phrase Choice (C) does this effectively, idiomatically, and concisely (Chapter 15, Lesson 15: Coordinating Ideas)
13 E The original phrasing is unnecessarily wordy and does not effectively coordinate the ideas in the sentence Choice (C) has the same problem Choices (B) and (D) create clauses with uncoordinated verbs Only choice (E) conveys the idea concisely and effectively (Chapter 15, Lesson 15: Coordinating Ideas)
14 C The phrase extra superfluous is redundant, and the phrase to edit for eliminating is unidiomatic.
Choice (C) is clear and concise
(Chapter 15, Lesson 12: Other Modifier Problems)
Trang 8Math Smart Cards Math Smart Cards
A right triangle has a leg of length 3 and
a hypotenuse of length 4 What is the
length of the other side?
The average of 3 consecutive even
integers is 80 What is the least of
these integers?
If 5 – 2(x – 3) = 9, then what is the
value of x?
If n is a positive real number, what is the simplest way to express n2× n3?
Stephanie bought a sweater for $42.40, including a 6% sales tax What was the price before tax?
105°
Note: Figure not drawn to scale
In the figure above, what is the value of
a + b?
Trang 9Math Math
Math Math
Formula/Concept:
When you multiply exponentials with the same
base, you add the exponents.
Correct answer: n5 (n2)(n3) =
Add the exponents: (n2+3) = n5
Common mistake: n6
This is the result if you mistakenly multiply the
exponents
Formula/Concept:
To find the price before a 6% tax, divide the
final price by 1.06
Correct answer: $40
$42.40 = (1.06)(x)
Divide by 1.06: $40.00 = x
Common mistake: $39.86
This is the result if you mistakenly deduct 6% of
$42.40 (which is $2.54), from $42.40
Formula/Concept:
(n – 2)180° = the sum of the angles in an
n-sided figure.
Correct answer: 250
The sum of the angles is (5 – 2)(180°) = 540°, so
105° + 93° + 92° + a + b = 540° Therefore, a +
b = 250°.
Common mistake: 70
This is the result if you mistakenly think the sum
is 360° instead of 540°
Formula/Concept:
The Pythagorean Theorem
Correct answer: 32+ x2= 42
Subtract 9: x2= 7 Take square root: x =
Common mistake: 5
Don’t assume it is a 3-4-5 triangle In such a
triangle, the 3-4 sides must both be legs.
7 7
Formula/Concept:
If a set of numbers is “evenly” spaced, the average is the same as the middle number
Correct answer: 78
If the average of consecutive even numbers is
80, then 80 must be the “middle” number in the set, so the numbers are 78, 80, 82
Common mistake: 79
Don’t overlook the fact that the numbers are
even.
Formula/Concept:
Distributing with negative numbers
Correct answer: 1 5 – 2(x – 3) = 9
Distribute: 5 – 2x + 6 = 9
Combine like terms: 11 – 2x = 9
Subtract 11: –2x = –2
Common mistake: –5
This results from improperly distributing the –2
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At the beginning of 1999, stock in ABC
company cost $100 per share It
in-creased by 25% in 1999, dein-creased by
20% in 2000, decreased by 20% in
2001, and increased by 15% in 2002.
What was the price at the end of 2002?
If the average of x, x + 2, and 2x + 8 is
6, what is the value of x?
A set consists of the integers from –12
to n If the sum of the members of that
set is 42, how many integers are in
the set?
Shaquille O’Neal made 4 of his first
12 free throws How many consecutive
shots x must he hit for his free-throw
percentage to reach 60%?
If a triangle has two sides of length 8 and 12, then what is the largest possi-ble integer length of the third side?
If 10 students in a class of 16 have an average score of 82 on a physics test and the remaining students have an average score of 90, what is the average score of the entire class?