manhattan test 2 answers

In fact, for any perfect square, the number of factors will always be odd.. The existence of this “identical pair” will always make the number of factors odd for any perfect square.. Any

Answers:

1 The ratio of buses to cars can be expressed as 2x: 23x We can

write an equation that represents the fact that there are 630 fewer

buses than cars: 2x + 630 = 23x

Solving this equation for x yields the following:

2x + 630 = 23x

630 = 21x

30 = x

There are 23x cars on River Road which equals 23 × 30 = 690 cars

The correct answer is D

2 Simplify this equation by factoring out 49 from the numerator and the denominator as follows:

The correct answer is B

3 Recognize here the basic form (x –y)2, which equals x2 - 2xy +

y2

corresponds here to x, and

So the expression can be simplified to:

Under the radical, recognize the basic form (a + b)(a – b), which equals a2 – b2

The expression can be further simplified to:

The correct answer is C

4 If the square root of p2 is an integer, p is a perfect square Let’s take a look at 36, an example of a perfect square to extrapolate some general rules about the properties of perfect squares

Statement I: 36’s factors can be listed by considering pairs of

factors (1, 36) (2, 18) (3,12) (4, 9) (6, 6) We can see that they are

9 in number In fact, for any perfect square, the number of factors will always be odd This stems from the fact that factors can always

be listed in pairs, as we have done above For perfect squares,

however, one of the pairs of factors will have an identical pair, such

as the (6,6) for 36 The existence of this “identical pair” will always

make the number of factors odd for any perfect square Any number that is not a perfect square will automatically have an even number

of factors Statement I must be true

Statement II: 36 can be expressed as 2 x 2 x 3 x 3, the product of 4 prime numbers

A perfect square will always be able to be expressed as the product

of an even number of prime factors because a perfect square is formed by taking some integer, in this case 6, and squaring it 6 is comprised of one two and one three What happens when we square this number? (2 x 3)2 = 22 x 32 Notice that each prime element of 6

will show up twice in 62 In this way, the prime factors of a perfect

square will always appear in pairs, so there must be an even

number of them Statement II must be true

Statement III: p, the square root of the perfect square p 2 will have

an odd number of factors if p itself is a perfect square as well and

an even number of factors if p is not a perfect square Statement III

is not necessarily true

The correct answer is D, both statements I and II must be true

5 (1) INSUFFICIENT: This gives the definition of the $ function,

however, it gives us no information about p and q

(2) INSUFFICIENT: This statement gives us no information about the $ function

(1) AND (2) SUFFICIENT: We can use the definition of the $ function

given in (1) along with the values of p and q from (2) to solve for the value of p $ q = 2(4)2 - 10 = 6

The correct answer is C

6 At the point where a curve intercepts the x-axis (i.e the x intercept), the y value is equal to 0 If we plug y = 0 in the equation of the curve, we get 0 = (x – p)(x – q) This product would only be zero when x is equal to p or q The question is asking us if (2, 0) is an x-intercept, so it is really asking us if either p or q is

(1) AND (2) SUFFICIENT: Together we have enough information to

see if either p or q is equal to 2 To solve the two simultaneous

equations, we can plug the pvalue from the first equation, p = 8/q, into the second equation, to come up with -2 + 8/q = q

-This simplifies to q2 + 2q – 8 = 0, which can be factored (q + 4)(q – 2) = 0, so q = 2, -4

If q = 2, p = -4 and if q = -4, p =2 Either way either p or q is equal

to 2

The correct answer is C

7 In order to determine the median of a set of integers, we need to find the "middle" value

(1) SUFFICIENT: Statment one tells us that average of the set of

integers from 1 to x inclusive is 11 Since this is a set of consecutive

integers, the "average" term is always the exact middle of the set Thus, in order to have an average of 11, the set must be the

integers from 1 to 21 inclusive The middle or median term is also is

11

(2) SUFFICIENT: Statement two states that the range of the set of

integers from 1 to x inclusive is 20 In order for the range of

integers to be 20, the set must be the integers from 1 to 21

inclusive The median term in this set is 11

The correct answer is D

8

The distance from G to H is 513 - 512

The distance between and two consecutive points is constant, so the

distance from A to G will be 6 times the distance from G to H or

9 If we multiply both sides of the equation by (x + 2), we get 1.5x

+ 3 = 1.8

If we multiply both sides of the equation by 2, we get 3x + 6 = 3.6 Further simplifying, 3x = -2.4, so x = -0.8

The correct answer is B

10 From the diagram, we see that all 6 of the labeled angles add up

to 360°:

3a + 3b = 360

a + b = 120 (a = 120 – b or b = 120 – a)

(1) SUFFICIENT: We can use the value of a to solve for b (b = 120

- 35 = 85) We can then see that b > a

(2) SUFFICIENT: If a < 60 and b = 120 - a, then b = 120 - less

than 60

Therefore, b must be greater than 60 and consequently greater than

a

The correct answer is D

11 We can determine the sales revenue that the sales associate generated by analyzing her commission earnings for the week

(1) SUFFICIENT: The sales associate earned a total of $1500 in

commission last week We know that on the first $10,000 in sale revenue, the associate earns 8% or $800 in commission This

means that the associate earned $700 in additional commission Since this additional commission is calculated based on a 10%

rate, the sales associate must have generated an additional $7000 worth of sales revenue Thus, we know from statement 1 that the sales associate generated $10,000 + $7000 = $17,000 in sales

revenue last week Statement 1 alone is sufficient

(2) SUFFICIENT: The sales associate was eligible for the 10%

commission rate on $7000 worth of sales Since the 10% rate only kicks in after the first $10,000 in sales, this means that the sales

associate generated $7000 in sales revenue above the $10,000

threshold Thus, we know from statement 2 that the sales associate generated $10,000 + $7000 = $17,000 in sales revenue last week Statement 2 alone is sufficient

The correct answer is D

The correct answer is A

13 We can solve this question as a VIC (Variable in answer choices)

by plugging in values for x, y and z:

If we plug x = 10, y = 20, z = 100 into the answer choices, only

answer choice (A) gives us 88:

If we plug x = 10, y = 20, z = 100 into the answer choices, only

answer choice (A) gives us 88:

The correct answer is A

14 For an overlapping set problem we can use a double-set matrix

to organize our information and solve Because the values here are percents, we can assign a value of 100 to the total number of lights

at Hotel California The information given to us in the question is

shown in the matrix in boldface An x was assigned to the lights

that were “Supposed To Be Off” since the values given in the

problem reference that amount The other values were filled in using the fact that in a double-set matrix the sum of the first two rows equals the third and the sum of the first two columns equals the third

The correct answer is D

15 To determine the value of 10 – x, we must determine the exact value of x To determine the value of x, we must find out what digits

a and b represent Thus, the question can be rephrased: What is a

and what is b?

(1) INSUFFICIENT: This tells us that x rounded to the nearest

hundredth must be 1.44 This means that a, the hundredths

digit, might be either 3 (if the hundredths digit was rounded up to 4) or 4 (if the hundredths digit was rounded down to 4) This

statement alone is NOT sufficient since it does not give us a

definitive value for a and tells us nothing about b

(2) SUFFICIENT: This tells us that x rounded to the nearest

thousandth must be 1.436 This means, that a, the hundredths

digit, is equal to 3 As for b, the thousandths digit, we know that it

is followed by a 5 (the ten-thousandths digit); therefore, if x is

rounded to the nearest thousandth, b must rounded UP Since b is rounded UP to 6, then we know that b must be equal to 5

Statement (2) alone is sufficient because it provides us with

definitive values for both a and b

The correct answer is B

16 It is tempting to view the information in the question as

establishing a pattern as follows:

Green, Yellow, Red, Green, Yellow, Red,

However, consider that the following non-pattern is also possible: Green, Yellow, Red, Green, Green, Green, Green

(1) INSUFFICIENT: This tells us that the 18th tile is Green or Red but this tells us nothing about the 24th tile Statement (1) alone is NOT sufficient

(2) INSUFFICIENT: This tells us that the 19th tile is Yellow or Red but this tells us nothing about the 24th tile Statement (2) alone is NOT sufficient

(1) AND (2) INSUFFICIENT: Together, the statements yield the following possibilities for the 18th and 19th tiles:

GY, GR, RY, or RR

However, only GY adheres to the rules given in the question Thus,

we know that tile 18 is green and tile 19 is yellow However, this does not help us to determine the color of the next tile, much less

tile 24 (the one asked in the question) For example, the next tile

(tile 20) could be green or red Thus, the statements taken together are still not sufficient

The correct answer is E

17 (1) INSUFFICIENT: If we simplify the inequality by adding 3 to

both sides and dividing by 2, we get x < 4 There are an infinite number of x values less than 4

(2) INSUFFICIENT: If we simplify the inequality by dividing both

sides by -4 and switching the direction of the inequality, we get x >

2 There are an infinite number of x values greater than 2

(1) AND (2) SUFFICIENT: If x is an integer and 2 < x < 4, x must

The sum of a set = (the mean of the set) x (the number of terms in the set)

There are 9 terms in the set: 20th - 12th + 1 = 8 + 1 = 9

The mean of the set = (the first term + the last term) divided by 2: (84 + 140)/2 = 112

The sum of this set = 112 x 9 = 1008

Alternatively, one could list all nine terms in this set (84, 91, 98 140) and add them

When adding a number of terms, try to combine terms in a way that makes the addition easier

(i.e 98 + 112 = 210, 119 + 91 = 210, etc)

The correct answer is C

19 Begin by counting the number of relationships that exist among the 7 individuals whom we will call A, B, C, D, E, F, and G

First consider the relationships of individual A: AB, AC, AD, AE, AF,

AG = 6 total Then consider the relationships of individual B without counting the relationship AB that was already counted before: BC,

BD, BE, BF, BG = 5 total Continuing this pattern, we can see that C will add an additional 4 relationships, D will add an additional 3

relationships, E will add an additional 2 relationships, and F will add

1 additional relationship Thus, there are a total of 6 + 5 + 4 + 3 +

2 + 1 = 21 total relationships between the 7 individuals

We are told that 4 people have exactly 1 friend This would account for 2 "friendship" relationships (e.g AB and CD) We are also told that 3 people have exactly 2 friends This would account for

another 3 "friendship" relationships (e.g EF, EG, and FG) Thus, there are 5 total "friendship" relationships in the group

The probability that any 2 individuals in the group are friends is 5/21 The probability that any 2 individuals in the group are not friends = 1 – 5/21 = 16/21 The correct answer is E

20 Since BE CD, triangle ABE is similar to triangle ACD (parallel

lines imply two sets of equal angles) We can use this relationship

to set up a ratio of the respective sides of the two triangles:

So AD = 8

We can find the area of the trapezoid by finding the area of triangle

CAD and subtracting the area of triangle ABE

Triangle CAD is a right triangle since it has side lengths of 6, 8 and

10, which means that triangle BAE is also a right triangle (they

share the same right angle)

Area of trapezoid = area of triangle CAD – area of triangle BAE

= (1/2)bh – (1/2)bh

= 0.5(6)(8) – 0.5(3)(4)

= 24 – 6

= 18

The correct answer is B

21 (1) INSUFFICIENT: If we subtract 3x from both sides and factor out an x, we get:

x(x + 3)(x – 1) = 0, so x = -3, 0, or 1

(2) INSUFFICIENT: This can be factored as (x – 5)(x + 3) = 0, so

x = -3 or 5

(1) AND (2) SUFFICIENT: With the two statements together we

know x must equal -3

The correct answer is C

22 m/n will be an integer if m is divisible by n For m to be

divisible by n, the elements of n's prime box (i.e the prime factors that make up n) must also appear in m's prime box

(1) INSUFFICIENT: If 2m is divisible by n, the elements of n's prime box are in 2m's prime box However, since 2m contains a 2 in its prime box because of the coefficient 2, m alone may not have all of the elements of n's prime box For example, if 2m = 6 and n = 2, 2m is divisible by n but m is not

(2) SUFFICIENT: If m is divisible by 2n, m's prime box contains a 2 and the elements of n's prime box Therefore m must be divisible

by n

The correct answer is B

23 Begin by assigning variables to the unknown quantities:

The question asks for the number of writers who are NOT

left-handed which is the same as asking for the number of right-left-handed writers (34) The correct answer is C

numerator ab2 is even

If ab2 were odd, the quotient would never be divisible by 2,

regardless of what c is To prove this try to divide an odd number

by any integer to come up with an even number; you can't If ab2

is even, either a is even or b is even

(I) TRUE: Since a or b is even, the product ab must be even

(II) NOT NECESSARILY: For the quotient to be positive, a and c

must have the same sign since b2 is definitely positive We know

nothing about the sign of b The product of ab could be negative or

positive

(III) NOT NECESSARILY: For the quotient to be even, ab2 must be

even but c could be even or odd An even number divided by an

odd number could be even (ex: 18/3), as could an even number

divided by an even number (ex: 16/4)

The correct answer is A

25 The solution to a problem such as this often looks less appealing

than some of the incorrect answers Thus, it is important to methodically analyze each answer choice

B: Any fraction between 0 and 1 multiplied by itself will decrease in

value Thus (2/3) multiplied by itself will yield a result that is less

D (0.9)2 × (0.9)2 = (0.81) ×(0.81) This is approximately 0.65,

which is less than 2/3

Then, 27 × 27 is clearly greater than 2/3

The correct answer is E

We can use this information to find the area of the circular base

Because the probability of the stone landing outside the triangle

is 3/4 , the triangle must comprise 1/4 of the area of the circular base

The height of an equilateral triangle splits the triangle into two 60-90 triangles (Each 30-60-90 triangle has sides in the ratio of 1: : 2) Because of this, the area for an equilateral triangle can be expressed in terms of one side If we call the side of the equilateral

30-triangle, s, the height must be (s ) / 2 (using the 30-60-90 relationships)

The area of a triangle = 1/2 × base × height, so the area of an equilateral triangle can be expressed as: 1/2 × s × (s ) / 2

0.08

0.003 =

80

3 This is approximately 27

E:

Here the triangle has an area of , so:

= 1/2 × s × (s ) / 2

s = 2

The correct answer is E

27 We can rephrase the question by opening up the absolute value

sign There are two scenarios for the inequality |n| < 4

If n > 0, the question becomes “Is n < 4?”

If n < 0, the question becomes: “Is n > -4?”

We can also combine the questions: “Is -4 < n < 4?” ( n is not

equal to 0)

(1) SUFFICIENT: The solution to this inequality is n > 4 (if n > 0) or

n < -4 (if n < 0) This provides us with enough information to

guarantee that n is definitely NOT between -4 and 4 Remember

that an absolute no is sufficient!

(2) INSUFFICIENT: We can multiply both sides of the inequality by

|n| since it is definitely positive To solve the inequality |n| × n < 1, let’s plug values If we start with negative values, we see that n can

be any negative value since |n| × n will always be negative and

therefore less than 1 This is already enough to show that the

statement is insufficient because n may not be between -4 and 4

The correct answer is A

28 The question asks about the sign of d

(1) INSUFFICIENT: When two numbers sum to a negative value, we have two possibilities:

Possibility A: Both values are negative (e.g., e = -4 and d = -8) Possibility B: One value is negative and the other is positive.(e.g., e

= -15 and d = 3)

(2) INSUFFICIENT: When the difference of two numbers produces

a negative value, we have three possibilities:

Possibility A: Both values are negative (e.g., e = -20 and d = -3) Possibility B: One value is negative and the other is positive (e.g., e

= -20 and d = 3)

Possibility C: Both values are positive (e.g., e = 20 and d = 30) (1) AND (2) SUFFICIENT: When d is ADDED to e, the result (-12) is greater than when d is SUBTRACTED from e This is only possible if

d is a positive value If d were a negative value than adding d to a

number would produce a smaller value than subtracting d from that

number (since a double negative produces a positive) You can test

numbers to see that d must be positive and so we can definitively

answer the question using both statements

29 (1) INSUFFICIENT: If we test values here we find two sets of possible x and y values that yield conflicting answers to the

question

(2) INSUFFICIENT: If we test values here we find two sets of

possible x and y values that yield conflicting answers to the

(1) AND (2) SUFFICIENT: Let’s start with statement 1 and add the

constraints of statement 2 From statement 1, we see that x has to

be positive since we are taking the square root of x There is no point in testing negative values for y since a positive value for x against a negative y will always yield a yes to the question Lastly,

we should consider x values between 0 and 1 and greater than 1

because proper fractions behave different than integers with regard

to exponents When we try to come up with x and y values that fit both conditions, we must adjust the two variables so that x is

always greater than y

x x3 y Is x > y?

1/4 1/2 1/64 1/128 YES

Logically it also makes sense that if the cube and the square root of

a number are both greater than another number than the number itself must be greater than that other number

The correct answer is C

30 First we must find the total number of 5 member teams, with our without John and Peter We can solve this using an anagram model in which each of the 9 players (A – I) is assigned either a Y (for being chosen) or an N (for not being chosen):

Player A B C D E F G H I Chosen ? Y Y Y Y Y N N N N

It is the various arrangements of Y’s and N’s above that would yield all of the different combinations, so we can find the number of

possible teams here by considering how many anagrams of

YYYYYNNNN exist:

(because there are 9! ways to order 9 objects)

(because the 5Y's and 4N's are identical)

So there are 126 possible teams of 5 Since the question asks for the probability of choosing a team that includes John and Peter, we need to determine how many of the 126 include John and Peter If

we reserve two of the 5 spots on a team for John and Peter, there will be 3 spots left, which must be filled by 3 of the remaining 7

players (remember John and Peter were already selected)

Therefore the number of teams including John and Peter will be equal to the number of 3-player teams that can be formed from a 7-player pool We can approach the problem as we did above:

9!

5! 4! =

9 × 8 × 7 × 6 × 5

5 × 4 × 3 × 2 × 1 = (3 × 7 × 6) = 126

Player A B C D E F G Chosen Y Y Y N N N N

The number of possible YYYNNNN anagrams is:

Since 35 of the total possible 126 teams include John and Peter, the probability of selecting a team with both John and Peter is 35/126 or 5/18

The correct answer is D

31 We can factor the equation in the question :

Alternatively we could solve this question as a VIC (Variable in

answer choice) by plugging a value for x

If x = 2, the original equation becomes y2 – 16 = y – 4 or (y + 4)(y – 4) = y – 4

Since we are told that y does not equal 0, we can divide both sides

by (y – 4) to get y + 4 = 1

This means that y = -3 when x = 2

Unfortunately two answer choices yield a y value of -3 when x = 2

(both B and C) In this case we would have to repeat the process

y + 4 = x

2 , so y =

x – 8

2 Finally,

for a different x value x = 4, y = -2 only works for answer choice

B The correct answer is B

32 We can solve this problem as a VIC (Variable In answer Choice)

and plug in values for the variable x Let’s say x = 6 (Note that there is a logical restriction here in terms of the value of x Lindsay

has to have a rate of less than less than 1 room per hour if she needs Joseph’s help to finish in an hour)

1/2 + 1/J J / 2J + 2 / 2J (J + 2) / 2J

If the two of them finish the room in one hour, using the formula of

rt = w, we can solve for J

rt = w and t = 1 (hour), w = 1 (job)

((J + 2) / 2J )(1) = 1 J + 2 = 2J J = 2

That means that Joseph’s rate is 1/2, the same as Lindsay’s The question though asks us what fraction of the room Joseph would complete in 20 minutes, or 1/3 of an hour

rt = w

(1/2)(1/3) = w

w = 1/6

Now we must look at the answer choices to see which one is equal

to 1/6 when we plug in x = 6 Only C works: (6 – 3) / 18 = 1/6

The correct answer is C

33 We can rewrite the information in the question as an equation representing the T, the total dollar value of the sale:

L + M + S = T

L = the dollar amount received by the partner with the largest share

M = the dollar amount received by the partner with the middle

(second largest) share

S = the dollar amount received by the partner with the smallest share

We are also told in the question that L = (5/8)T Thus we can

rewrite the equation as follows:

(5/8)T + M + S = T

Since the question asks us the value of S, we can simplify the

equation again as follows:

(2) SUFFICIENT: The second statement tells us that M = (1/2)L =

$1 million Additionally, since we know from the question that L = (5/8)T, then M must be equal to 1/2 of 5/8(T) or 5/16(T) We can therefore solve for T as follows: