Wiley the official guide for GMAT Episode 1 Part 7 ppsx

Arithmetic Operations on rational numbers Th e problem classiﬁed the job applicants into two categories: whether they had more or less than 4 years’ experience, and whether they had a d

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s(1 + r) dollars to spend next year Th is amount is

to be 1

2 the amount he spends this year Th e task

is to ﬁ nd I s , where I and s satisfy the

Arithmetic Negative exponents

Using rules of exponents, m–2 = m–1 ∙ 2 = (m–1)2,

and since m–1 = –1

9.

Th e correct answer is D.

165 Lois has x dollars more than Jim has, and together

they have a total of y dollars Which of the following

represents the number of dollars that Jim has?

(A) y − x

2 (B) y − x

Algebra Simplifying algebraic expressions

Let J be the number of dollars that Jim has

Th en, the amount that Lois has can be expressed

as J + x dollars If Lois and Jim together have a total of y dollars, then:

y = J + ( J + x) total dollars =

Jim’s dollars + Lois’s dollars

Solve this for J to determine the number of dollars

that Jim has:

(A) 180 (B) 170 (C) 156 (D) 150 (E) 105

Arithmetic; Algebra Percents; Applied problems

Let G equal the number of games played by the

team this season Th e given information can be

expressed as (0.80)(100) + 0.50(G − 100) = 0.70G,

that is, 80 percent of the ﬁrst 100 games plus

50 percent of the remaining games equals

70 percent of the total number of games played

Th is equation can be solved for G to determine

the answer to the problem:

(0.80)(100) + 0.50(G − 100) = 0.70G

80 + 0.50G − 50 = 0.70G simplify and distribute

30 = 0.20G simplify and subtract

0.05G from both sides

150 = G multiply by 5

Th e correct answer is D

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167 Of 30 applicants for a job, 14 had at least 4 years’

experience, 18 had degrees, and 3 had less than

4 years’ experience and did not have a degree

How many of the applicants had at least 4 years’

experience and a degree?

Arithmetic Operations on rational numbers

Th e problem classiﬁed the job applicants into two

categories: whether they had more or less than

4 years’ experience, and whether they had a

degree Th e given information can be summarized

in the following table:

Th us, according to the given information,

30 – 14 = 16 applicants had less than 4 years’

experience Th en, of those applicants with

less than 4 years’ experience, it is given that

3 applicants did not have a degree, so 16 – 3 =

13 applicants had less than 4 years’ experience

and had a degree Th erefore, out of the given 18

applicants that had degrees, 13 applicants had less

than 4 years’ experience, so 18 – 13 = 5 applicants

had at least 4 years’ experience with a degree

Th ese results are shown in the following table

Algebra First-degree equations

Work the problem to solve the equation for x.

x + 1 = 2x − 2 multiply through by x

3 = x solve for x by adding 2 to

and subtracting x from

both sides

Th e correct answer is E

169 Last year, for every 100 million vehicles that traveled

on a certain highway, 96 vehicles were involved in accidents If 3 billion vehicles traveled on the highway last year, how many of those vehicles were involved in accidents? (1 billion = 1,000,000,000)

(A) 288 (B) 320

(E) 28,800

According to the given information, 96 out of every 100 million vehicles were in an accident last year Th us, of the 3 billion vehicles on the highway last year, the number of vehicles involved

in accidents was:

Th e correct answer is C

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170 Thirty percent of the members of a swim club have

passed the lifesaving test Among the members

who have not passed the test, 12 have taken the

preparatory course and 30 have not taken the course

How many members are there in the swim club?

Algebra Applied problems

If 30 percent of the club members have passed

the test, then 70 percent have not Among the

members who have not passed the test, 12 have

taken the course and 30 have not, for a total of

12 + 30 = 42 members who have not passed the

test Letting x represent the total number of

members in the swim club, this information can

be expressed as 0.70x = 42, and so x = 60

Th e correct answer is A

171 What is the difference between the sixth and the ﬁ fth

terms of the sequence 2, 4, 7, … whose nth term is

According to the given formula, the sixth term

of the sequence is 6 + 26 – 1 = 6 + 25 and the ﬁ fth

Algebra Second-degree equations

Work the problem by taking the square root of

both sides and solving for x

(C) 0 ≤ x ≤ 1

(D) x ≤ – 1 or x ≥ 1 (E) – 1 ≤ x ≤ 1

Algebra Inequalities

Th e expression 1 – x2 can be factored as

(1 – x)(1 + x) Th e product is positive when both

factors are positive (this happens if 1 ≥ x and

x ≥ –1, or equivalently if –1 ≤ x ≤ 1) or both factors are negative (this happens if 1 ≤ x and

x ≤ 1, which cannot happen), and therefore the solution is –1 ≤ x ≤ 1

Th e correct answer is E.

174 The probability is 1

2 that a certain coin will turn up heads on any given toss If the coin is to be tossed three times, what is the probability that on at least one of the tosses the coin will turn up tails?

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Another way of stating that a coin toss will turn

up tails at least once is to say that it will not turn

up heads every time Th e probability that on at

least one of the tosses the coin will not turn up

heads is 1 minus the probability that the coin

will turn up heads on all three tosses Each toss

is an independent event, and so the probability

of getting heads all three times is 1

2

18

3

⎛

⎝⎜ ⎞⎠⎟ = .

Th us, the probability of not getting heads all

three times (that is, getting tails at least once) is

8

78

− =

175 Of the ﬁnal grades received by the students in a

certain math course, 1

number of students in the course?

Let x be the number of students in the course

20o

15

14

12

420

520

1020

ff the students received grades of A, B, or C

Th is means the 10 remaining grades represent

Investigate each of the functions to determine if

they increase from x = 165 to x = 166.

I Graphically, this represents a line with positive slope Th erefore, the function

increases between any two values of x

A direct computation can also be used:

165

1166

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1662 – 166 > 1652 – 165, and hence

, which shows that decreases from

x = 165 to x = 166.

Th e correct answer is C.

177 A rectangular box is 10 inches wide, 10 inches long,

and 5 inches high What is the greatest possible

(straight-line) distance, in inches, between any two

points on the box?

Geometry Pythagorean theorem

Th e greatest possible distance between any two

points in a rectangular solid is the space diagonal

(AD) of the rectangular solid as shown below

C D

10

10 5

To compute the length of AD, the Pythagorean

theorem must be used twice as follows:

A good way to solve this problem is to create a Venn diagram To determine how many students

to put in each section, begin by putting the given shared-student data in the overlapping sections

Put 0 in the intersection of all three clubs, 10 in the Chess and Drama intersection, 5 in the Chess and Math intersection, and 6 in the Drama and Math intersection, as shown in the Venn diagram below

14

10 0

Drama Chess

MathSubtracting the shared students from the totals in each club that are listed in the table establishes the members who belong only to that club Th rough this process, it can be determined that the Chess club has 25 such members (40 – 10 – 5 = 25), the Drama club has 14 such members (30 – 10 – 6 = 14), and the Math club has 14 such members (25 – 5 – 6 = 14) Putting the number of unshared club members into the Venn diagram and then adding up all the sections of the diagram gives

25 + 14 + 14 + 10 + 5 + 6 = 74 students

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179 The ratio of two quantities is 3 to 4 If each of the

quantities is increased by 5, what is the ratio of these

two new quantities?

given.

Both 3 to 4 and 6 to 8 are examples of two

quantities in the ratio 3 to 4 Increasing both

numbers in each of these examples by 5 gives

8 to 9 and 11 to 13 Since 8

9 ≠

11

13, the ratio of the two new quantities cannot be determined

from the information given

180 If the average (arithmetic mean) of x and y is 60 and

the average (arithmetic mean) of y and z is 80, what is

2 80 since the average of y and z is 80.

Th e two equations can be rewritten as x + y = 120 and y + z = 160 Subtracting the ﬁ rst equation from the second equation gives (y + z) – (x + y) =

160 – 120, or z – x = 40.

Th e correct answer is B.

181 If 1

2 of the air in a tank is removed with each stroke

of a vacuum pump, what fraction of the original amount of air has been removed after 4 strokes?

With each stroke’s removal of 1

2 of the tank’s air, the amount of air being removed from the tank on that stroke is equal to the amount of air remaining in the tank after that stroke With the ﬁrst stroke of the pump, 1

2 of the air is removed;

with the second stroke, of the air is removed, leaving 1

4 of the air With the third stroke, of the air is removed, leaving 1

8

of the air, and with the fourth stroke,

of the air is removed Th erefore, with four strokes,

of the air has been removed

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182 If the two-digit integers M and N are positive and have

the same digits, but in reverse order, which of the

following CANNOT be the sum of M and N ?

It is given that M and N have the same digits in

reverse order Let M = 10t + u and N = 10u + t,

where t and u are two digits Th en, M + N =

(10t + u) + (10u + t ) = 11t + 11u = 11(t + u) Th is

means that any sum of the two integers M and

N must also be a multiple of 11 Of the answer

choices, only 181 is not a multiple of 11 and thus

cannot be the sum of M and N

183 Car X and Car Y traveled the same 80-mile route If

Car X took 2 hours and Car Y traveled at an average

speed that was 50 percent faster than the average

speed of Car X, how many hours did it take Car Y to

travel the route?

5

Substituting the given information in the formula

per hour At this speed, Car Y would travel the

80-mile route in 80 hours

60

4

13

= =

184 If the average (arithmetic mean) of the four numbers

K, 2K + 3, 3K – 5, and 5K + 1 is 63, what is the value

of K ?

(A) 11

4 (C) 22 (D) 23

the given information can be expressed in the

following equation and solved for K

185 If p is an even integer and q is an odd integer, which of

the following must be an odd integer?

q

(C) 2p + q (D) 2(p + q)

q

Arithmetic Properties of numbers

Since it is given that p is even and q is odd, use

these properties to test the outcome of each answer choice to determine which one must be odd

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A even

odd = even must be even

B (even)(odd) = even must be even

186 Drum X is 1

2 full of oil and Drum Y, which has twice the capacity of Drum X, is 2

3 full of oil If all of the oil

in Drum X is poured into Drum Y, then Drum Y will be

ﬁlled to what fraction of its capacity?

Let y represent the capacity of Drum Y Since Y

has twice the capacity of Drum X, Drum X has

half the capacity of Drum Y, and thus the

capacity of Drum X can be expressed as 1

2 y.

Since Drum X is half full, the amount of oil in

Drum X is equal to According

to the given information, the initial amount of

Because 3 –1 is within the parentheses, its value

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Th en, 2 (3 –1) = 2 –3, which has the

189 The inside dimensions of a rectangular wooden box

are 6 inches by 8 inches by 10 inches A cylindrical

canister is to be placed inside the box so that it stands

upright when the closed box rests on one of its six

faces Of all such canisters that could be used, what is

the radius, in inches, of the one that has maximum

Th e largest cylinder that can ﬁt in a rectangular

box will have the same height as the box and a

diameter equal to the smaller dimension of the

top of the box By deﬁnition, the diameter of the

canister is twice its radius One possible canister

placement in the box is illustrated below

10 inches

8 inches

However, since the box can rest on any of three

diﬀ erently sized faces, it is necessary to consider

the volume of each possibility Th e formula for calculating volume is volume = π(radius)2(height),

or v = π r 2h; the possible volumes for the canister

are those shown in the following table:

Th e units digit of 134 is 1, since 3 × 3 × 3 × 3 = 81;

the units digit of 172 is 9, since 7 × 7 = 49; and the units digit of 293 is 9, since 9 × 9 × 9 = 729

Th erefore, the units digit of (13)4(17)2(29)3 is 1, since 1 × 9 × 9 = 81

4th Street 3rd Street 2nd Street 1st Street

191 Pat will walk from Intersection X to Intersection Y

along a route that is conﬁned to the square grid of four streets and three avenues shown in the map

above How many routes from X to Y can Pat take

that have the minimum possible length?

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Arithmetic Elementary combinatorics

In order to walk from Intersection X to

Intersection Y by one of the routes of minimum

possible length, Pat must travel only upward or

rightward between the intersections on the map

Let U represent upward movements and R

represent rightward movements It takes 3 upward

and 2 rightward movements to complete the route

Th e following 10 routes are possible:

192 The ratio, by volume, of soap to alcohol to water in

a certain solution is 2:50:100 The solution will be

altered so that the ratio of soap to alcohol is doubled

while the ratio of soap to water is halved If the altered

solution will contain 100 cubic centimeters of alcohol,

how many cubic centimeters of water will it contain?

the amount of water in the altered solution can be found by solving , which gives

193 If 75 percent of a class answered the ﬁrst question

on a certain test correctly, 55 percent answered the second question on the test correctly, and 20 percent answered neither of the questions correctly, what percent answered both correctly?

For questions of this type, it is convenient to draw

a Venn diagram to represent the conditions in the problem For example, the given information can

be depicted:

20%

Q175%

Q255%

In the diagram it can be seen that the 80% of the class answering a question correctly is represented

by the two circles Let x represent the percent of

the class that answered both questions correctly, that is, the shaded region above Since the sum of the circles minus their overlap equals 80% of the class, the information given in the problem can

then be expressed as 75% + 55% – x = 80% Th is

equation can be solved for x as follows:

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194 In the rectangular coordinate system above, the line

y = x is the perpendicular bisector of segment AB (not

shown), and the x-axis is the perpendicular bisector of

segment BC (not shown) If the coordinates of point A

are (2,3), what are the coordinates of point C ?

Since the line y = x is the perpendicular bisector

of AB , B is the reﬂection of A through this line

In any reﬂection through the line y = x, the

x -coordinate and the y-coordinate of a point

become interchanged Th us, if the coordinates of

A are (2,3), the coordinates of B are (3,2)

1 2 3

B

Since the x-axis is the perpendicular bisector of

BC , C is the reﬂection of B through the x-axis In any reﬂection through the x-axis, the x-coordinate remains the same, and the sign of the y-coordinate

–1 –2

Let p be the current price per towel, and let n be

the number of towels that can be bought for $120

Th en the information in the problem can be expressed in the following equations:

(i) pn = 120 (ii) (p + 1)(n – 10) = 120 or equivalently (iii) pn + n – 10p – 10 = 120.

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Th en replace pn in (iii) with 120 to get:

Number of green marbles

Total number

of red and green marbles

First, set up an equation to ﬁnd the total number

of marbles in the three jars as follows:

x + y + y + z + x + z = 80 + 120 + 160

2x + 2y + 2z = 360 combine the like terms

x + y + z = 180 divide both sides by 2

Th en, since it can be seen from the table that the

number of green marbles in Jar R is z, solve for z

to answer the problem To do this most efﬁciently, use the information from the table for Jar P, which

4 minute it rotates 300

4 = 75 revolutions Th e distance the point on the edge of the fan blade rotates in 75 revolutions

is 75 times the circumference of a circle with radius 10 centimeters Th e circumference C of a circle with radius r is C = 2πr Th us the distance the point travels is 75[2π(10)] = 1,500π

Th e correct answer is B.

198 If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ?

(A) Two (B) Three (C) Four (D) Six (E) Eight

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Since p is a prime greater than 2, p must be odd

Th erefore, the possible even divisors of n = 4p are

2, 4, 2p, and 4p Alternatively, choose such a

prime, for example p = 3, and determine the

number of positive even divisors that n = 4p =

12 has

I 72, 73, 74, 75, 76

II 74, 74, 74, 74, 74 III 62, 74, 74, 74, 89

199 The data sets I, II, and III above are ordered from

greatest standard deviation to least standard deviation

in which of the following?

(A) I, II, III

Th e data set with the least standard deviation will

be the data set with elements most closely

clustered around the mean of the data set, and the

data set with the greatest standard deviation will

be the data set with elements that are spread out

farthest from the mean of the data set

Because set I is symmetric about 74 (73 is 1 less

than 74 and 75 is 1 more than 74; 72 is 2 less

than 74 and 76 is 2 more than 74), the mean of

set I is 74 Because every number in set II is 74,

the mean of set II is 74 Th e mean of set III is

Th e elements of set II do not deviate at all from

74, so set II has the least standard deviation Th e

most that any element of set I diﬀ ers from 74 is 2,

but there are elements of set III that diﬀ er from

74.6 by 12.6 and 14.4 Th erefore, set III has a

greater standard deviation than set I, which has a

greater standard deviation than set II

Th e correct answer is D.

200 Of the 50 researchers in a workgroup, 40 percent will

be assigned to Team A and the remaining 60 percent

to Team B However, 70 percent of the researchers prefer Team A and 30 percent prefer Team B What is the lowest possible number of researchers who will NOT be assigned to the team they prefer?

(A) 15 (B) 17 (C) 20 (D) 25 (E) 30

If all 15 who prefer Team B are assigned to Team B, which is to have 30 researchers, then

15 who prefer Team A will need to be assigned

to Team B Alternatively, since there are only

20 spots on Team A, 35 – 20 = 15 who prefer Team A but will have to go to Team B instead

201 If m is the average (arithmetic mean) of the ﬁ rst

10 positive multiples of 5 and if M is the median

of the ﬁ rst 10 positive multiples of 5, what is the

Arithmetic Statistics

Th e ﬁ rst 10 positive multiples of 5 are 5, 10, 15,

20, 25, 30, 35, 40, 45, and 50 From this, the average (arithmetic mean) of the 10 multiples, that is, number of valuessum of values , can be calculated:

= =

5 10 15 20 25 30 35 40 45 50

10275

10 27 5

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Since there is an even number of multiples, the

median, M, is the average of the middle two

Th is problem can also be solved as follows Since

the values can be grouped in pairs (i.e., 5 and 50,

10 and 45, 15 and 40, etc.), each of which is

symmetric with respect to the median, it follows

that the average and median are equal

203 What is the 25th digit to the right of the decimal point

in the decimal form of 6

11 ?(A) 3

(B) 4 (C) 5 (D) 6 (E) 7

Th e fraction in its decimal form is 6

11 = 0.545454.… Every odd-numbered digit to the right of the decimal point is 5, so the 25th digit must be 5

204 John and Mary were each paid x dollars in advance to

do a certain job together John worked on the job for

10 hours and Mary worked 2 hours less than John If

Mary gave John y dollars of her payment so that they

would have received the same hourly wage, what was

the dollar amount, in terms of y, that John was paid in

advance?

(A) 4y (B) 5y (C) 6y (D) 8y (E) 9y

Let w be the amount of Mary and John’s same

hourly wage To set their hourly pay equal, John,

who worked 10 hours, needs to be paid 10w, and Mary, who worked 8 hours, needs to be paid 8w

Since Mary gave John y dollars, Mary now has

x – y dollars and John now has x + y dollars

Th eir pay can thus be expressed as follows:

x – y = 8w Mary’s pay

x + y = 10w John’s pay Subtract the ﬁrst equation from the second and

solve for w

2y = 2w

y = w

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Substitute y for w in the second equation, and

solve for x, the amount each was paid in advance

205 In the rectangular coordinate system above, if point R

(not shown) lies on the positive y-axis and the area of

triangle ORP is 12, what is the y-coordinate of point R ?

Geometry Simple coordinate geometry; Area

Since O and P of triangle ORP are already drawn

and R has to be on the positive y-axis, the triangle

is a right triangle with its base length the distance

from the origin O (0,0) to P (4,0), which is 4

Since the area of a triangle = (base)(height)

the information about the area and base can be

expressed as follows and solved for the height of

triangle OPR:

12 = (4)(height)

2

12 = 2(height) simplify the right side

6 = height solve for the height

On the y-axis, the x-coordinate is 0 and the

y-coordinate is the distance above the axis that

the point is located In this case, the y-coordinate

is the height of the triangle

(B) 2.0 (C) 2.5 (D) 3.0 (E) 3.5

Understand that Car A ﬁrst has to travel 20 miles

to catch up to Car B and then has to travel an additional 8 miles ahead of Car B, for a total of

28 extra miles to travel relative to Car B It can

be stated that Car A is traveling 58 – 50 = 8 miles per hour faster than Car B Solving the

distance = (rate)(time) formula for time yields distance

207 For the past n days, the average (arithmetic mean)

daily production at a company was 50 units If today’s production of 90 units raises the average to 55 units

per day, what is the value of n ?

(A) 30 (B) 18 (C) 10 (D) 9 (E) 7

Arithmetic; Algebra Statistics; Applied problems; Simultaneous equations

Let x be the total production of the past n days

Using the formula average = sum of values

number of values, the information in the problem can be expressed

in the following two equations:

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n

daily average of 50 units

over the past n days

1

= ++

x

n increased daily average when including today’s

90 units

Solving the ﬁrst equation for x gives x = 50n

Th en substituting 50n for x in the second

equation gives the following that can be solved

for n:

1

= ++

n n

55(n + 1) = 50n + 90 multiply both sides

by (n + 1) 55n + 55 = 50n + 90 distribute the 55

5n = 35 subtract 50n and 55 from

both sides

n = 7 divide both sides by 5

x x

2

(C) x

x

2 2

1 1

− +

1 1

2

x x

Multiply the numerator and denominator inside

the parentheses by x to eliminate the compound

fractions

x x

1111

Since this is not one of the answer choices,

it is necessary to simplify further With the

knowledge that 1 + x = x + 1 and 1 – x = –(x – 1),

it can be stated that

because the negative, when squared, is positive

Geometry Angles; Measures of angles

Refer to the ﬁgure below

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Triangle ABC is a right triangle, and segment

AB is parallel to segment ED since they are

both perpendicular to the same segment (BC )

Th erefore, m∠DEC = m∠BAC = z° = 50° So,

since ∠DEC and ∠AED form a straight line

at E, y + 50 = 180, or y = 130

Th e measure of an exterior angle of a triangle is

the sum of the measures of the nonadjacent

interior angles Th us,

210 In the coordinate system above, which of the following

is the equation of line C ?

Geometry Simple coordinate geometry

Th e line is shown going through the points (0,2) and (3,0) Th e slope of the line can be found with the formula slope =change in

change in

y x

for two points (x1,y1) and (x2,y2) Th us, the slope

of this line equals Using the formula

for a line of y = mx + b, where m is the slope and

b is the y-intercept (in this case, 2), an equation for

this line is y= −2x+

3 2 Since this equation must

be compared to the available answer choices, the following further steps should be taken:

y= −2x+

3y = –2x + 6 multiply both sides by 3

2x + 3y = 6 add 2x to both sides

Th is problem can also be solved as follows From

the graph, when x = 0, y is positive; when y = 0,

x is positive Th is eliminates all but B and C Of these, B is the only line containing (0,2) Still another way is to use (0,2) to eliminate A, C, and

E, and then use (3,0) to eliminate D

Let the one two-digit integer be represented by

10t + s, where s and t are digits, and let the other

integer with the reversed digits be represented

by 10s + t Th e information that the diﬀ erence between the integers is 27 can be expressed in the following equation, which can be solved for the answer

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(10s + t ) − (10t + s) = 27

10s + t − 10t − s = 27 distribute the negative

9s − 9t = 27 combine like terms

s − t = 3 divide both sides by 9

Th us, it is seen that the two digits s and t diﬀ er

212 The circle with center C shown above is tangent to

both axes If the distance from O to C is equal to k,

what is the radius of the circle, in terms of k ?

In a circle, all distances from the circle to the

center are the same and called the radius, r

Since the horizontal distance from C to the y-axis

is also a radius, the base of the triangle drawn will

be r as well Th is creates a right triangle, and so

the Pythagorean theorem (or a 2 + b 2 = c 2) applies

r 2 + r 2 = k 2 substitute values into

Pythagorean theorem;

2r2 = k2 combine like terms

r2 k

22

= divide both sides by 2

combined resistance of these two resistors, then the

reciprocal of r is equal to the sum of the reciprocals of

x and y What is r in terms of x and y ?

Note that two numbers are reciprocals of each other if and only if their product is 1 Th us the

reciprocals of r, x, and y are 1 1 1

r, x, and y ,

respectively So, according to the problem,

r = +x y. To solve this equation for r, begin by

creating a common denominator on the right side

by multiplying the ﬁrst fraction by y

y and the

second fraction by x

x:

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x xy

214 Xavier, Yvonne, and Zelda each try independently to

solve a problem If their individual probabilities for

64 (D) 5

64 (E) 3

64

Arithmetic Probability

Since the individuals’ probabilities are

independent, they can be multiplied to ﬁgure out

the combined probability Th e probability of

Xavier’s success is given as 1

4, and the probability

of Yvonne’s success is given as 1

2 Since the probability of Zelda’s success is given as 5

8, then the probability of her NOT solving the problem

is Th us, the combined probability is

Algebra Second-degree equations

Solve the equation for x Begin by multiplying all the terms by x(x + 1)(x + 4) to eliminate the

denominators

(x + 1)(x + 4) – x(x + 4) = x(x + 1) (x + 4)(x + 1 – x) = x(x + 1) factor the (x + 4) out

front on the left side

(x + 4)(1) = x(x + 1) simplify

x + 4 = x2 + x distribute the x on

the right side

4 = x2 subtract x from both

sides ±2 = x take the square root

of both sidesBoth –2 and 2 are square roots of 4 since (–2)2 = 4 and (2)2 = 4 Th us, x could be –2.

Th is problem can also be solved as follows

then set equal to the right side to get 1

1

14

x x( + )=x+ Next, cross multiply:

(1)(x + 4) = x(x + 1)(1) Th erefore, x + 4 = x2 + x,

or x2 = 4, so x = ± 2.

Trang 20

216 1

2

1 4

1 16

It is clear from the answer choices that all three

factors need to be written with a common

denominator, and they thus become

12

121

4

12

121

2

12

Th e correct answer is B

217 In a certain game, a large container is ﬁlled with red,

yellow, green, and blue beads worth, respectively, 7,

5, 3, and 2 points each A number of beads are then

removed from the container If the product of the point

values of the removed beads is 147,000, how many

red beads were removed?

From this, the red beads represent factors of 7 in the total point value of 147,000 Since 147,000 = 147(1,000), and 1,000 = 10 3, then 147 is all that needs to be factored to determine the factors of 7

Factoring 147 yields 147 = (3)(49) = (3)(72) Th is means there are 2 factors of 7, or 2 red beads

2 (D) 2 (E) 3

Algebra First-degree equations

219 If a, b, and c are consecutive positive integers and

a < b < c, which of the following must be true?

I c – a = 2

II abc is an even integer

III a + b + c

3 is an integer

Tiêu đề	Problem Solving Answer Explanations
Trường học	Wiley
Chuyên ngành	GMAT Preparation
Thể loại	Official Guide for GMAT
Năm xuất bản	2023

Định dạng
Số trang	40
Dung lượng	2,4 MB