Sat - MC Grawhill part 71 pptx

Fifty students study two of the three languages, so let’s say that 50 students study both Spanish and Latin.. This means that zero students study both Spanish and French, zero students s

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8 A You are told that: 12v = 3w

Multiply by 2: 8v = 2w

The question asks for the value of: 2w − 8v

Substitute for 2w: 8v − 8v = 0

Alternatively, you can try ﬁnding values for v and w

that work, like 1 and 4, and plug them into 2w − 8v

and into the choices and ﬁnd the match

(Chapter 8, Lesson 1: Solving Equations)

9 C 2x + 1 > 5

Subtract 1: 2x > 4

Divide by 2: x > 2

Interpret the absolute value: x > 2 OR x < −2

You are told that x is negative, so x < −2 is the answer

(Chapter 8, Lesson 6: Inequalities, Absolute Values,

and Plugging In)

Substitute −2 for x: −(−2)2− 8(−2) − 5

Square −2: −(4) − 8(−2) − 5

Simplify: −4 + 16 − 5 = 7

When evaluating −x2, don’t forget to square the value

before taking its opposite!

11 D

Cross-multiply: 15 ≤ 2m

Divide by 2: 7.5 ≤ m

Since m is greater than or equal to 7.5, (D) is the answer.

and Plugging In)

12 B First ﬁnd the price after the 6% sales tax:

$60.00 × 06 = $3.60 tax

$60.00 + $3.60 = $63.60 price with tax

(A simpler way is just to multiply 60 by 1.06.)

Now ﬁnd how much change Theo received:

$70.00 − $63.60 = $6.40 change

(Chapter 7, Lesson 5: Percents)

13 A Write an equation for the ﬁrst sentence

n − m = r Because none of the answer choices contain m, solve

for m in terms of r and n: n − m = r

Subtract r: n − r = m

Now write an expression for what the question asks for:

s + 2m Substitute for m: s + 2(n − r)

Distribute: s + 2n − 2r

Alternatively, you can substitute numbers for n, m,

and r, making sure they “work,” and get a numerical

answer to the question

5 2 3

m≤

14 D Two points on line l are (0, 0) and (10, y).

Find the slope of the line:

Cross-multiply: 5y= 30 Divide by 5: y= 6

Since y= 6, the height of the triangle is 6 Find the area:

A=1⁄2(base)(height)

Substitute 48 for A: 48=1⁄2(base)(6) Simplify: 48 = 3(base) Divide by 3: 16 = base = x Now ﬁnd x + y = 16 + 6 = 22.

(Chapter 10, Lesson 4: Coordinate Geometry)

15 A Ellen travels the ﬁrst 15 miles at 30 miles per hour Find out how much time that takes:

d = (rate)(time)

Plug in known values: 15 = 30t

Divide by 30: 1⁄2hour = t The rest of the trip, which is (y− 15) miles long, she travels at an average speed of 40 miles per hour:

d = (rate)(time) Plug in known values: (y − 15) = 40t

Divide by 40:

Add the two values together to ﬁnd the total time:

(Chapter 9, Lesson 4: Rate Problems)

16 B Set up the relationship in equation form:

Plug in what you’re given:

Divide by 16: 1⁄2= k

Write the new equation:

Plug in new values:

(Chapter 11, Lesson 4: Variation)

y= ( ) ( ) = =

1

28 4

4 16

1 4 2

y m n

= ( ) ( )

1 2 2

8 16

12

= ( ) ( )

k

n

= 2

1 2

15 40 + −y

y

t

−15= 40

= −

− =

−

2 1

0

10 0 10

3 5

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17 D a + b = s

a − b = t

Add straight down: 2a = s + t

Divide by 2:

a + b = s

a − b = t

Subtract straight down: 2b = s − t

Divide by 2:

Find the

product:

(Chapter 8, Lesson 2: Systems)

The answer is in terms of y alone, so ﬁnd m and n in

Take the 4th root: y1/4= m

y = n3 Take the cube root: y1/3= n

Find the product mn: mn = (y1/4)(y1/3) = y1/3 + 1/4

Add exponents: mn = y7/12

(Chapter 11, Lesson 6: Negative and Fractional

Exponents)

19 A This question deals with similar triangles:

Set up ratio:

Cross-multiply: 6x= 48

Divide by 6: x= 8

Area of big triangle =1⁄2(base)(height) =1⁄2(12)(6) = 36

Area of small triangle =1⁄2(base)(height) =1⁄2(8)(4) = 16

Shaded area = area of big triangle − area of small

triangle = 36 − 16 = 20

(Chapter 10, Lesson 6: Similar Figures)

(Chapter 10, Lesson 5: Areas and Perimeters)

20 A Set up a Venn diagram to visualize the

information

6

12

4

=

x

( )( )=a b ⎛s t+ s t s t

⎝⎜

⎞

⎠⎟

−

⎛

⎝⎜

⎞

⎠⎟=

−

⎛

⎝⎜

⎞

⎠⎟

2 2

b= −s t 2

a= +s t 2

Notice that 1⁄3the number of sedans must equal 1⁄5the number of convertibles Say the number of

convert-ible sedans is x If this is 1⁄3the number of sedans, then

there must be 3x sedans in total, and 3x − x = 2x of these are not convertibles Similarly, if x is 1⁄5the

num-ber of convertibles, then there must be 5x convertibles altogether, and 5x − x = 4x of these are not sedans So

now your diagram can look like this:

So there must be a total of 2x + x + 4x = 7x cars at the

dealership The only choice that is a multiple of 7 is (A): 28

(Chapter 9, Lesson 5: Counting Problems)

Section 4

1 E

Perimeter of a square = 4s

36 = 4s

Divide by 4: 9 = s

Area of a square = (s)2 Area = (9)2= 81

2 C

Cross-multiply: b = 10a Try positive integer values of a to see how many work:

There are nine integer pairs that satisfy the equation (Chapter 9, Lesson 3: Numerical Reasoning Problems)

3 E The ten bathrooms cost $20 each to clean:

Total cost = $20 × 10 = $200

To clean each bathroom twice would cost:

$200 × 2 = $400 There are 30 ofﬁces, and they cost $15 each to clean:

Total cost = $15 × 30 = $450

To clean each ofﬁce once and each bathroom twice will cost: $400 + $450 = $850

(Chapter 11, Lesson 5: Data Analysis)

a

b= 1 10

both

2

3s

1

3s 1

5c

4

5c

sedans convertibles

x

sedans convertibles

both

s

6 12

4

x

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4 A Remember the “difference of squares”

factor-ing formula: a2− b2= (a − b)(a + b)

Substitute: 10 = (2)(a + b)

Divide by 2: 5 = a + b

(Chapter 8, Lesson 5: Factoring)

5 A

To ﬁnd the value of f(14), ﬁnd all the factors of 14:

1, 2, 7, 14 There are two prime factors, 2 and 7

2 + 7 = 9

f(14) = 9

To ﬁnd the value of f(6), ﬁnd all the factors of 6:

1, 2, 3, 6 There are two prime factors, 2 and 3

2 + 3 = 5

f(6) = 5

f(14) − f(6) = 9 − 5 = 4

(Chapter 11, Lesson 2: Functions)

6 D First write an equation to ﬁnd the average

Multiply by 4: a + b + c + d = 80

If you want a to be as large as possible, make b, c, and

d as small as possible You are told that they are all

different positive integers: a + b + c + d = 80

Let b = 1, c = 2, d = 3: a + 1 + 2 + 3 = 80

Combine like terms: a+ 6 = 80

(Chapter 9, Lesson 2: Mean/Median/Mode Problems)

7 B Let the radius of circle A = a and the radius of

circle B = b It is given that a = 2b The circumference

of a circle can be found with the equation C = 2πr.

The sum of their circumferences is 36π:

36π = 2πa + 2πb

Divide by π: 36 = 2a + 2b

Substitute for a: 36 = 2(2b) + 2b

Simplify: 36 = 4b + 2b

Combine like terms: 36 = 6b

Divide by 6: 6 = b

Solve for a: a = 2(b) = 2(6) = 12

a b c+ + + =d

8 C This is a visualization problem The six possi-ble planes are illustrated below Notice that the six faces of the cube “don’t count,” because each of those contains four edges of the cube

(Chapter 10, Lesson 7: Volumes and 3-D Geometry)

9 16 Set up an equation: 2x− 10 = 22

Divide by 2: x = 16

10 36

There are 180° on one side of a line:

2y + y + y + y = 180°

Combine like terms: 5y= 180°

Divide by 5: y= 36°

(Chapter 10, Lesson 1: Lines and Angles)

2y°

y° y°

y°

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15 25 First calculate how many grams of sucrose

there are in 200 grams of a 10% mixture

(200 grams)(.10) = 20 grams of sucrose

Since you will be adding x grams of sucrose, the total

weight of sucrose will be 20 + x grams, and the total

weight of the mixture will be 200 + x grams Since the

fraction that will be sucrose is 20%,

Cross-multiply: (20 + x)(100) = 20(200 + x)

Distribute: 2,000 + 100x = 4,000 + 20x

Subtract 2,000: 100x = 2,000 + 20x Subtract 20x: 80x= 2,000

(Chapter 7, Lesson 5: Percents) (Chapter 7, Lesson 4: Ratios and Proportions)

16 24 First calculate how long the race took.

distance = rate × time

16 = (8)(time)

Divide by 8: 2 hours = time = 120 minutes

Next, ﬁnd the new rate that is 25% faster:

new rate= (8)(1.25) = 10 mph Calculate how long the new race would take:

distance = rate × time

16 = (10)(time)

Divide by 10: 1.6 hours = time = 96 minutes

So she can improve her time by (120 − 96) = 24 minutes (Chapter 9, Lesson 4: Rate Problems)

20 200

20 100

+ + =

x x

11 5 Think simple: What’s the simplest way to turn

8x + 4y into 2x + y? Just divide by 4!

8x + 4y = 20

Divide by 4: 2x + y = 5

(Chapter 6, Lesson 4: Simplifying Problems)

12 12 Just substitute x = 3 and y = 5 into the

equa-tion and solve for m:

3m− 15 = 21

13 15 Ratios such as 4:5 can also be written as 4x:5x.

So the number of men m is 4x and the number of

women w is 5x.

Plug those values into the equation w = m + 3

5x = 4x + 3 Subtract 4x: x= 3

Plug 3 in to 5x: w = 5x = 5(3) = 15

(Chapter 7, Lesson 4: Ratios and Proportions)

14 8 or 12 y = ⎟ 2x − b⎟

Plug in (5, 2): 2=⎟ 2(5) − b⎟

Simplify: 2 =⎟ 10 − b⎟

(10 − b) = 2 or (10 − b) = −2

Subtract 10: −b = −8 or −b = −12

Multiply by −1: b = 8 or b = 12

and Plugging In)

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17 52

Break a shape like

this into

recogniz-able four-sided

ﬁgures and

trian-gles that are easier

to deal with

The area of the

rectangle on the

left is 7 × 4 = 28

The area of the rectangle on the right is 5 × 4 = 20

The sum of those two areas is 28 + 20 = 48 The

area remaining for the triangle is the difference

78 − 48 = 30 Set up an equation for the area of a

triangle to solve for x:

Area =1⁄2(base)(height)

30 =1⁄2(5)(height) Divide by 1⁄2: 60 = 5(height)

Divide by 5: 12 = height

To ﬁnd the hypotenuse of

the right triangle, set up

the Pythagorean theorem

and solve:

52+ 122= c2

25 + 144 = c2

169 = c2

c= 13

(Or just notice that it’s a 5-12-13 triangle!)

To ﬁnd the perimeter of the ﬁgure, add up all of the

sides:

13 + 12 + 4 + 5 + 7 + 4 + 7 = 52

(Chapter 10, Lesson 3: The Pythagorean Theorem)

18 225 Set up a three-circle Venn diagram to

visual-ize this information

Fifty students study two of the three languages, so let’s say that 50 students study both Spanish and Latin (It

doesn’t matter which two languages those 50 students

take; the result turns out the same.) This means that zero students study both Spanish and French, zero students study both French and Latin, and zero students study all three languages

There are 120 Spanish students in all There are there-fore 120 − 50 = 70 students who study Spanish alone There are 80 French students in all, all of whom study just French, and there are 75 total Latin students in-cluding 75 − 50 = 25 students who study only Latin This means that there are 70 + 50 + 80 + 25 = 225 sophomores at Hillside High School

(Chapter 9, Lesson 5: Counting Problems)

Section 5

1 C The clients were forced to seek more reliable

investment advice, so the manager must have

man-aged their funds badly ineptitude= lack of skill

2 E Vartan is Armenian; he was born in Iran and educated in Lebanon and is now president of the

American Brown University He has a lot of worldly experience perpetual = lasting forever; authoritative = showing authority; cosmopolitan= worldly

3 D They didn’t consider it in great detail, so the

reading must have been without great care verbatim=

word for word; meandering = wandering; tormented = feeling anguish or pain; cursory= quick and without

care; substantial= of substance, quite large

Latin

25 0

0

4

7

28

5

5 20

4 7

7

4 28

5

5 20

12

4 13

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4 A If the pathogens (infectious agents) spread

more quickly in close quarters, the crowding would be

a problem This would cause the disease to spread.

squalor = horrible or dirty conditions; circulation =

moving of something around from place to place;

poverty = state of being poor; deterioration = wearing

down; congestion = crowdedness; proximity =

close-ness; resilience= ability to recover from a challenge

5 E The purpose of research is to ﬁnd answers to

questions of interest Therefore, the research

endeav-ors (attempts) to determine or understand the

mecha-nisms by which our brains do things If the data must

be turned into coherent and understandable

informa-tion, it must not have been coherent to begin with, but

rather just a big rush of information enhance= make

better; attenuate = reduce in amount; dearth = scarcity,

lack; elucidate = make clear; deluge = huge ﬂood

6 D The fruits mentioned in line 10 refer to the

means of acquiring food and shelter, because they are

described as the fruits for maintaining human life.

7 B The question is whether one can get quick

re-turns of interest (make money) from the capital of

knowledge and learning (from one’s education) (lines

13–15)

8 A The pointing of dogs is mentioned as an

in-stinctive tendency to the performance of an action

(lines 1–2)

9 E Inherited tendencies tend to show themselves

in the behavior of an organism The paragraph

men-tions the calf and the caterpillar as examples of

or-ganisms with instincts that show themselves in later

behavior

10 D The ﬁnal paragraph begins with The best life is

the one in which the creative impulses play the largest part

and the possessive impulses the smallest (lines 56–58).

11 D Lines 22–26 say that the food and clothing of

one man is not the food and clothing of another; if the

supply is insufﬁcient, what one man has is obtained at

the expense of some other man Therefore, food and

clothing exist in ﬁnite amounts and can be used up

12 E This section of the passage discusses matters

such as good-will (line 38), science (line 31), and

paint-ing pictures or writpaint-ing poems (lines 35–36) as thpaint-ings

that are not denied to someone else when one person

possesses them

13 E This sentence discusses the possessive

im-pulses (line 49) as distinct from the creative imim-pulses

discussed in the next sentence The impulse of

prop-erty in lines 51–52 is the desire to possess propprop-erty.

14 C This statement echoes the point made in lines

71–72 that spiritual possessions cannot be taken in this

way, that is, by force.

15 D Lines 58–59 say This is no new discovery and

go on to cite the Gospel as a prior source expressing the same opinions as Russell’s

16 B The author’s main point is that creativity is of higher value than possessiveness The invention mentioned in answer choice (B) was created to make money for its inventor (a possessive and materialis-tic motive) but has the side effect of beneﬁtting all of humankind

17 A The passage discusses the perspective one

Native American has on the appearance of the new

superstition (line 44) It discusses how some villagers

have taken to the new religion and also mentions one fellow tribe member’s attempting to convert the main character

18 E In saying that men of the same color are like the

ivory keys of one instrument where each represents all the rest, yet varies from them in pitch and quality of voice (lines 4–7), the author is saying that people of

the same race possess important differences

19 D The author describes the preacher as

mouth[ing] most strangely the jangling phrases of a bigoted creed (lines 11–12), indicating that she

consid-ers him to be an intolerant pconsid-erson She describes

her-self as having compassion (line 7) and respect (line 10),

but does not attribute these qualities to the preacher

20 B Lines 13–14 say that our tribe is one large

fam-ily, where every person is related to all the others.

21 C Both the preacher and the author’s mother

have become followers of the new superstition (line 44).

22 C In saying that a pugilist commented upon a

re-cent article of mine, grossly perverting the spirit of my pen (lines 66–68), the author is saying that the pugilist

distorted the author’s words in a grotesque way

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23 E The author characterizes herself as a wee child

toddling in a wonder world (lines 72–73), indicating

that she is in awe of the world around her Although

one might expect her to be vengeful in response to the

pugilist (line 66) who grossly pervert[ed] the spirit of

[her] pen (line 68), there is no indication in the

para-graph that she is vengeful

24 A The author says in lines 68–72 that still I would

not forget that the pale-faced missionary and the

aborig-ine are both God’s creatures, though small indeed in

their own conceptions of Inﬁnite Love In other

words, the author respects the missionary but believes

he is small-minded

Section 6

1 D The verb must agree with the plural subject

claims Choice (D) is most concise and correct.

(Chapter 15, Lesson 1: Subject-Verb Disagreement)

2 A The original sentence is best

3 B The participial phrase opening the sentence

modiﬁes Sartre himself, not his writing This being

the case, the phrase dangles

(Chapter 15, Lesson 7: Dangling and Misplaced

Participles)

4 C Choice (C) best follows the law of parallelism

(Chapter 15, Lesson 3: Parallelism)

6 B Choice (B) is the most concise, logical, and

complete

(Chapter 12, Lesson 9: Write Concisely)

7 C The original phrasing contains an incomplete

thought Choice (C) is by far the most concise and direct

(Chapter 15, Lesson 15: Coordinating Ideas)

8 E The participle having spread modiﬁes the

dis-ease, not the doctors.

(Chapter 15, Lesson 7: Dangling and Misplaced

Participles)

9 C The original phrasing contains an incomplete

thought Choice (C) is by far the most concise and direct

10 D The participle singing modiﬁes Anita, not her

hoarseness Furthermore, the participle is in the

wrong form; it should be in the perfect form having

sung, because only the previous singing could have

contributed to her hoarseness

(Chapter 15, Lesson 7: Dangling and Misplaced Participles)

(Chapter 15, Lesson 9: Tricky Tenses)

12 A The word quick is an adjective and can thus

modify only a noun But since it modiﬁes the verb

turned, the adverb quickly is needed here.

(Chapter 15, Lesson 12: Other Modiﬁer Problems)

13 B This sentence violates the law of parallelism

If she is known for her initiative, she should also be known for devoting her own time.

(Chapter 15, Lesson 3: Parallelism)

14 C Since the Medieval era is long past, its

begin-ning is “completed” or, in grammar terms, “perfect.”

So this phrase should be the “perfect” form of the

in-ﬁnitive: to have begun.

15 B The word neither is almost always part of the phrase neither of or neither A nor B So choice (B) should read nor even.

(Chapter 15, Lesson 10: Idiom Errors)

16 D The word less is used to compare only

quanti-ties that can’t be counted If the quantiquanti-ties are

count-able, as accidents are, the word should be fewer.

(Chapter 15, Lesson 4: Comparison Problems)

17 B To convey the proper sequence of events, the

perfect tense is required: had spent.

18 A The subject of the verb has is the plural noun

newspapers (The sentence is “inverted,” because the

subject follows the verb.) The proper form of the verb,

then, is have.

(Chapter 15, Lesson 1: Subject-Verb Disagreement) (Chapter 15, Lesson 2: Trimming Sentences)

19 B The original sentence has a “comma splice” that incorrectly joins two sentences with only a

comma A better phrasing is dream that led.

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20 C The subject of the verb is the singular noun

movement, so the proper verb form is has led.

(Chapter 15, Lesson 2: Trimming Sentences)

21 E The sentence is correct as written

22 D This is a prepositional phrase, so the

pro-noun is the object of the preposition and should be

in the objective case The correct phrasing is for

Maria and me.

(Chapter 15, Lesson 6: Pronoun Case)

23 A The word successive means consecutive, so it

does not make sense in this context The right word is

successful.

(Chapter 15, Lesson 11: Diction Errors)

25 C The word underneath means that it is

physi-cally below something else It should be changed to

under.

(Chapter 15, Lesson 10: Idiom Errors)

27 B The subject of the verb were is arrogance,

which is singular It should instead be was.

28 B The sentence mentions there are numerous

strains of the bacteria, which means that more should

instead be most.

(Chapter 15, Lesson 4: Comparison Problems)

29 C The subject company is singular Therefore,

they should instead be it.

(Chapter 15, Lesson 5: Pronoun-Antecedent

Disagreement)

30 D Choice (D) is most consistent, logical, and

concise

31 A Choice (A) is most logical

(Chapter 12, Lesson 7: Write Logically)

32 B The ﬁrst paragraph ends with the description

of an idea The second paragraph begins with an illustration of how students experience this idea in their daily lives and then goes on to explain how it can

help them get through their brain freezes Choice (B)

is the best introduction to the paragraph, because it explains that a student using the phenomenon can improve his or her studies

33 C The sentence begins using the pronoun you,

so that usage should be maintained throughout the sentence Choice (D) is incorrect because a person has only one brain

(Chapter 15, Lesson 5: Pronoun-Antecedent Disagreement)

34 E Sentence 11 concludes a discussion of Isaac Asimov’s “eureka” experience The additional sen-tence expands upon that idea, relating it back to the lives of students

35 C Choice (C) is the most concise and logical revision

(Chapter 12, Lesson 7: Write Logically) (Chapter 12, Lesson 9: Write Concisely)

Section 7

1 B Set up a ratio to solve this problem:

Cross-multiply: 4x= 200 Divide by 4: x= 50 cents (Chapter 7, Lesson 4: Ratios and Proportions)

2 C Solve for b: 2b= 8

b= 3 Plug in 3: 3b= 33= 27 (Chapter 8, Lesson 3: Working with Exponentials)

3 A The sum of a, b, and 18 is 6 greater than the sum of a, b, and 12 Since there are three terms in the group, it follows that the average of a, b, and 18 would

be 6 ÷ 3 = 2 greater than the average of a, b, and 12.

(Chapter 9, Lesson 2: Mean/Median/Mode Problems)

4apples 10

20 cents

apples

x cents

=

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4 B If you have the patience, you can write out a

quick calendar for yourself to track the days:

Or you can use the simple fact that successive

Tues-days (like any other Tues-days) are always 7 Tues-days apart

Therefore, if the 1st of the month is a Tuesday, so are

the 8th, the 15th, the 22nd, and the 29th Therefore, the

30th is a Wednesday and the 31st is a Thursday

(Chapter 9, Lesson 3: Numerical Reasoning Problems)

5 A From the given information: m = 8n

0 < m + n < 50 Substitute for m: 0 < 8n + n < 50

Combine like terms: 0 < 9n < 50

Divide by 9: 0 < n < 55⁄9

Since n must be an integer, n can be 1, 2, 3, 4, or 5.

and Plugging In)

6 D First ﬁnd the value of y: y% of 50 is 32.

Simplify:

Cross-multiply: 50y= 3,200

Divide by 50: y= 64

What is 200% of 64?

Interpret: 2.00 × 64 = 128

(Chapter 7, Lesson 5: Percents)

Plug in 16 for x: g(16) = 16 + 161/2

Take square root of 16: g(16) = 16 + 4

Combine like terms: g(16) = 20

8 C The slope of the line is −3⁄4, so use the slope

equation and the coordinates of point A (0, 12) to ﬁnd

the coordinates of point B (x, 0):

Cross-multiply: 4(−12) = −3(x)

Simplify: −48 = −3x

Divide by −3: 16 = x

The base of the triangle is 16, and its height is 12

Area =1⁄2(base)(height) Substitute: Area =1⁄2(16)(12)

Simplify: Area = 96

(Chapter 10, Lesson 4: Coordinate Geometry)

= −

− = −− = − = −

0 12 0

12 3 4

y

100×50=32

9 A Find the sum of each repetition of the pattern:

−1 + 1 + 2 = 2 Next, determine how many times the pattern repeats in the first 25 terms: 25 ÷ 3 = 8 with a remainder of 1

Multiply the sum of the pattern by 8 to obtain the sum

of the ﬁrst 24 terms: 2 × 8 = 16 The 25th term is −1, which makes the sum 16 + −1 = 15 (Chapter 11, Lesson 1: Sequences)

10 D The ratio of white marbles to blue marbles is

4 to b The probability of randomly selecting a white

marble from the jar is 1⁄4 This means that one out of every four marbles in the jar is white and three out of every four marbles are blue If there are four white marbles, then there are 4 × 3 = 12 blue marbles (Chapter 7, Lesson 4: Ratios and Proportions)

11 B Area =1⁄2(base)(height)

Substitute: 10 =1⁄2(base)(height) Divide by 1⁄2: 20 = (base)(height) The base and the height are both integers Find all the

“factor pairs” of 20: 1, 20; 2, 10; and 4, 5 Plug each pair into the Pythagorean theorem to ﬁnd the least possible length of the hypotenuse:

a2+ b2= c2

42+ 52= c2 Combine like terms: 41 = c2 Take square root:

a2+ b2= c2

is the shortest possible hypotenuse

(Chapter 10, Lesson 5: Areas and Perimeters) (Chapter 10, Lesson 3: The Pythagorean Theorem)

12 B −1 < y < 0 This means that y is a negative decimal fraction.

Answer choices (A), (C), and (E) will all be negative num-bers Answer choices (B) and (D) are positive numnum-bers When you raise a simple fraction to a positive number

larger than 1, it gets smaller y4< y2, which makes (B) the

greatest value Pick a value like y= −1⁄2and see

41

401= c

104= c

41= c

Su M T W Th F Sa

16 17 18

23 24 25

20 21 22

30 31

27 28 29

26

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13 E Any statement of the form “If A is true, then B

is true” is logically equivalent to “If B is not true, then

A is not true.” Try this with some common-sense

examples of such statements For instance, saying “If

I am under 16 years old, then I am not allowed to drive”

is the same as saying “If I am allowed to drive, then I

must not be under 16 years old.” The statement in (E)

is logically equivalent to the original

(Chapter 6, Lesson 7: Thinking Logically)

14 E If each bus contained only the minimum

number of students, the buses would accommodate

6 × 30 = 180 students But since you have 200 students

to accommodate, you have 20 more students to place

To maximize the number of 40-student buses, place

10 more students in two of the buses Therefore, a

maximum of two buses can have 40 students

15 D The volume of a cylinder is equal to πr2h Let’s

say that the radius of cylinder A is a and the radius of

cylinder B is b Since the height of cylinder B is twice

the height of cylinder A, if the height of cylinder A is

h, then the height of cylinder B is 2h The volume of

A is twice that of B: πa2h = 2πb2(2h)

Simplify: πa2h = 4πb2h

Divide by π: a2h = 4b2h

Take the square root of both sides: a = 2b

Divide by b:

(Chapter 10, Lesson 7: Volumes and 3-D Geometry)

16 C The key is to ﬁnd a pattern among the many

possible solutions Pick some values for x to see if you

can see a pattern For instance, if x= 3, then the

gar-den looks like this:

In this case w = 8 But if x = 4, the garden looks like this:

a

b= 2 1

And here, w= 12 You might notice that the value of

w has increased by 4 Does this pattern continue?

Let’s try x= 5 to check:

Sure enough, w= 16, and it seems that the pattern

continues and w is always a multiple of 4 Only choice

(C), 40, is a multiple of 4, so that must be the correct answer

(Chapter 6, Lesson 3: Finding Patterns)

Section 8

1 B A reputable scientist is well known and well respected Saying the evidence is - at best indicates

that there is not much evidence at all It must be

ﬂimsy Reputable scientists would not likely admit that

a phenomenon exists if the evidence is weak meager=

scanty, deﬁcient; regret = feel bad about an action,

wish it hadn’t happened; paltry= lacking worth

2 D The concept that the Earth is round is now

ac-cepted as an inarguable truth It can be inferred that it

was at some point a fact that was thought to be wrong

incontrovertible = cannot be questioned; melliﬂuous = smooth ﬂowing; dubious= doubtful

3 B A profound break of a political party or religion

into factions is a schism (The Latin word schisma=

split.) unanimity = full agreement; schism = division into factions; caucus = meeting of party members;

someone; prognostication= prediction

4 C As the father of the American public school

system, Horace Mann would pressure or push the Massachusetts legislature to institute a system for

en-suring or guaranteeing universal access to eduction petitioned = requested, lobbied for; vouchsaﬁng =

con-ceding, granting

5 A Since the light from most stars takes millions

of years to reach us, it is plausible to imagine that by the time we see the light the star might actually no longer be there This would make the present

exis-tence of these stars questionable debatable =

dis-putable; methodical = systematic; indecorous = not proper; imperious= acting as if one is superior to

an-other; profuse= abundant

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