Quantitative - Arithmetic

Question 1

In order to complete a reading assignment on time, T planned to read 90 pages per day. However, she read only 75 pages during the last 6 days before the assignment was to be completed. How many days in all did Terry have to complete the assignment on time?

Accepted Answer

TOTAL NUMBER PAGES = 75n + 690 TOTAL NUMBER DAYS = n+6 The average she requires = 90 75n + 690 / n+6 = 90 n= 10 Total Number of days = 16

Question 2

If n=33^43 + 33^33, what is the units digit of n?

Accepted Answer

units digits of integer n is 3, therefore they are 3, 9, 7, 1, 3, 9, 7 and 1 Thus, the units digits of the powers of n form the sequence in which the digits 3,9,7 and 1 are repeated indefinately in that order. Since 43 = 10*4 + 3, the 43rd number in the sequence is 7 Since 33 = 8*4 + 1, the 33rd number in the sequence is 3 The units digit of 43+33 = 7+3 = 0

Question 3

If d=1/2^3 * 5^7 is expressed as a terminating decimal, how many nonzero digits will d have?

Accepted Answer

It will be helpful to use the fact that a factor that is an integer power of 10 has NO EFFECT on the number of nonzero digits of a terminating decimal. UMFORMEN! d=1/ 23 * 57 = (1/ 2*5)3 * 1/ 54 = 1/103 * 0.24 = 10-3 * 0.0016 2 nonzero digits

Question 4

List t consists of 30 positive decimals, on of which is an integer and the sum of the 30 decimals is S The estimated sum E is defined at follows: each decimal in T whose 10th digit is even is rounded up to the nearest integer and each decimal whose 10th digit is odd is rounded down to the nearest integer. E is the sum of the resulting integers. If 1/3 of the decimals in T have a 10th digit that is even, which of the following is a possible value of E-S?

Accepted Answer

1/3 * 3ß = 10 that have an even 10th digit. Let T represent the estimated sum of all 10 decimals and E represent the sum after rounding The remaining 20 decimals in T have an odd 10th digit.

Question 5

If r is the combination of 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 y?

Accepted Answer

1/r = 1/x + 1/y l MULTIPLY THE FIRST FRACTION BY Y/Y 1/R = y/xy + 1/y l MULTIPLY THE SECOND FRACTION BY X/X 1/R = y/xy + x/xy 1/r = x+y / xy r = xy / x+y

Question 6

if x is replaced by 1/x in (x+1/x-1)^2, the resulting expression is equivalent to what?

Accepted Answer

Substitute 1/x in the expression and simplify: (1/x+1/1/x-1)2 l MULTIPLY NUMERATOR AND DENOMINATOR OUTSIDE THE PARANTHESES BY X x* (1/x+1/1/x -1)2 (1+x / 1-x) 2 l SIMPLIFY FURTHER

Question 7

0.999999999/1.0001 - 0.9999999991 / 1.00003 =

Accepted Answer

1 - (10-8) / 1+ 1*(10-4) - 1 - 9* (10-8) / 1+3*(10-4)

Question 8

DS: What is the sum of a certain pair of consecutive odd integers? 1) at least one of the integers is negative 2) at least one of the integers is positive?

Accepted Answer

1) when at least one of the integers is negative, the sum could be -4 (-1 and -3) and the sum could be 0 (1 and -1). NOT SUFFICIENT 2) is at least one of the integers is positive, the sum could be the sum could be 4 (1 and 3) and the sum could be 0 (1 and -1). NOT SUFFICIENT

Question 9

DS: A clothing store acquired an item at a cost of x dollars and sold the items at y dollars. the store's gross profit from the item was what percent of its cost for the item? 1) y-x=20 2)y/x = 5/4

Accepted Answer

Determine the value of y-x/x as a percent! 1) when y-x, there still is no inofrmation to determine the value of x. NOT SUFFICIENT 2) there is enough information to determine the value of x any y. y=5/4 x and so 1/4 x, assume that y=5 an x=4 => so 5-4/4 = 1/4 SUFFICIENT

Question 10

the sum of 4 different off integers is 64. what is the value of the greatest of those integers? 1) integers are consecutive odd numbers 2) the greatest is 6 more than the least

Accepted Answer

Determine the greatest of four odd integers whose sum is 64 1) consecutive odd integers: z is the greatest of those integers => z+ (x-2) + (x-4) + (x-6) = 64, from which z can be determined SUFFICIENT 2) w, x, y, z => with a 6 in between z and w. z-w = 6 z-x = 4 z-y = 2 RESOLVE AFTER Z and determine z => SUFFICIENT

Question 11

A discounted price of a coast and the discount price of a sweater. Which of the two articles was discounted by the greater dollar amount? 1.) percentage discount of the coat was greater than that of the sweeter 2) the price before the discount of the coat was 10$ bigger than that of the sweater

Accepted Answer

1.) percentage discount: NOT SUFFICIENT 2.) price of articles before the discount in dollar NOT SUFFICIENT

Question 12

If the positive integer n is added to each of the integers 69, 94, 121, what is the value of n? 1) 69+n, 94+n are squares of 2 consecutive integers 2) 94+n and 121+n are the squares of 2 consecutive integers

Accepted Answer

Determine the value of the positive integers n 1) when 69+n, 94+n are squares of 2 consecutive integers, it follows that x and (x+1) represent the consecutive integers => (x+1)2-x2= 2x+1 SUFFICIENT 2) same as above! SUFFICIENT

Question 13

Of the four numbers represented on the number line above, is r closest to zero? number line: q,r,s,t (1) q = -s (2) -t < q

Accepted Answer

1) since q=-s one of q and s is positive and the other is negative. Since s is to the right of q and s is positive, q is negative. is q ≠ 0, then either (i) q<0<r<s<t or ii) q<r<0<s<t ... in each case, r is closest to 0 SUFFICIENT 2) If -t<g, then -t is to the left of q. If t = 5, s=4, r=3 and q=-2, then -5<-2, so (2) is satisfied. In this case, q is clostst to 0. On the other hand, if t=5, s=4, r=-1 and q=-2, then -5>-2, so (2) is satisfies but r is clostest to 0. NOT SUFFICIENT

Question 14

What is the ratio of the average (arithmetic mean) height of students in class X to the average height of students in class Y? 1) the average height of students in class X is 120 cm 2) the average height of students in class X and class Y combined is 126 cm

Accepted Answer

1) given that the average height of students in class X is 120 cm, but given no information about the average height of the students in class Y, the desired ratio cannot be determined NOT SUFFICIENT2) Given that the average height of students in class X and Y combined is 126 cm, the ratio of the averaes of the individual classes cannot be determined. (e.g., if class X consists of 10 students and class Y consists of 20 students, the ratio cannot be determined. NOT SUFFICIENT E

Question 15

Determine the value of integer k, where 56

Accepted Answer

1) it is given that the remainder is 1 when k is divided by 2 which implies that k is odd. Therefore, the value of k can be 57,59,61,63,65 NOT SUFFICIENT 2.) it is given that the remainder is 0 when k+1 is divided by 3, which implies that k+1 is divisible by 3. therefore 56+1<k+1<66+1 => 57<k+1<67 the value of k+1 can be 60,63, or 66, so the value of k can be 59,62,65 NOT SUFFICIENT

Question 16

If x is a positive integer, then x is prime? 1) 3x+1 is prime 2) 5x+1 is prime

Accepted Answer

Determine if the positive integer x is prime 1) This indicates that 3x+1 is prime. if 3x+1 is prime, then x=2 then 3x+1=7 is prime and so is x. However, if x=6, then 3x+1=19 is prime but 6 is not prime NOT SUFFICIENT 2.) This indicates that 5x+1 is prime. if 5x+1 is prime, then x=2 then 5x+1=11 is prime and so is x. However, if x=6, then 5x+1=31 is prime but 6 is not prime NOT SUFFICIENT

Question 17

When are all the numbers in a certain list of 15 numbers equal?

Accepted Answer

the sum of ANY 3 numbers in the list ---> EFFICIENT the SUM of all the numbers --> NOT EFFICIENT

Question 18

A company mase a profit last year from selling product X and had total expenses for product X of 100000+0.05R, where R is the total revenue for selling product X. Determine whether the company sold more than 21000 units of product X last year. 1) Total R from sale of product X was greater than 110000 2) The unit price for each Product was 5

Accepted Answer

1) Not sufficient, as R was more than 110000. It is porrisble that R> 110000 and more than 21000 units were sold, and also that R > 110000 and less than 21000 units were sold as we do not know the unit price. NOT SUFFICIENT 2) The unit price is known, this statement is sufficient.

Question 19

What is the median number of employees assigned to a project at company Z? 1) 25% of employees have 4 or more employees assigned to a project. 2) 35% of employees have 2 or fewer employees assigned to each project.

Accepted Answer

1) Even though 25 % of project have 4 or more employees, there is no information of the number of employees per project. For example, if there were a total of 100 projects, then the median could be 2 (75 projects that have excactly 2 employees each and 25 projects that have exactly 4 employees each) or the median could be 3 (75 projects that have excactly 3 emploees each and 25 projects that have excactly 4 employees each NOT SUFFICIENT 2) of 35% of the projects have 2 or fewer employees, there is no information of the number of employees per project. For example, if there were a total of 100 projects, then the median could be 3 (65 projects that have excactly 3 employees each and 35 projects that have exactly 2 employees each) or the median could be 4 (35 projects that have excactly 2 employees each and 65 projects that have excactly 4 employees each) NOT SUFFICIENT Taken together, we can say that 100 - (25+35) = 40 projects have excactly 3 employees. when the numbers of employeed per project are listed from least to greatest, 35% of the number are 2 or less, and 75% are 3 or less, and hence the median is 3!

Question 20

A school administrator will assign each student in a group of n students in one of m classrooms. IF 3

Accepted Answer

1) given that 3n is divisible by m, then n is divisible by m is m=9 and n=27 => 3<m<13<n, 3n=81 and m=9, so 3n is divisible by m and n is not sivisible by m if m=9 and n=30 (note that 3<m<13<n, 3m=90 and m=9 NOT SUFFICIENT 2) given that 13 n is divisible by m, then 13n = qm or n/m=q/13 for some integer q. Since 13 is a prime number tht divides qm because 13n = qm and 13 does not divide m, it follows that 13 divides q. Therefore, q/13 is an integer and since n/m = q/13, then n/m is an integer SUFFICIENT B

Question 21

If x,y, and z are three-digit positive integers and if x=y+z, is there a hundreds digit of x equal to the sum of the hundreds digits of y and z? 1) then tens digit of x is equal to the sum of the tens digits of y and z 2) the units digit of x is equal to the sum of the units digits of y and z

Accepted Answer

Letting x= 100a+10b+c y= 100p+10q+r and z=100t+10u+v, where a,b,c,p,q,r,t,u and v are digits, determine if a=p+t 1) it is given that b=q+u (which omplies that c+v>9), then in the addition process a ten would need to be carried over to then tens columns and b would be q+u+1. Since b is a digit, 0>b>9. Thence, 010>90 SUFFICIENT 2) NOT SUFFICIENT

Question 22

DS: For each integer c and y, min (x;y) and max (x;y) denote the minimum and the maximum of x and y, respectively. E.g. min (5,2)=2 and max (5,2) =5. For the integer w, what is the value of min (10;w)? 1)Given than w=max(20,z) for some integer z 2) Given that w=max(10,w)

Accepted Answer

If w>10, then (10,w) and if w<10, then (10,w)=w. Therefore, the value of min (10,w) can be determined if the value of w can be determined. !! 1) Given that w>max(20;z), then w>20. Hence w>10 and so min(10,w)=10 SUFFICIENT 2) Given that w=max (10,w), then w>10 and so min (10,w) = 10 SUFFICIENT

Question 23

DS Instructions

Accepted Answer

Assessment Instructions:Data SufficiencyEach data sufficiency problem consists of a question and two statements, labeled (1) and (2), which contain certain data. Using these data and your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), decide whether the data given are sufficient for answering the question and then indicate one of the following answer choices: A Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. B Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. C BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. D EACH statement ALONE is sufficient. E Statements (1) and (2) TOGETHER are not sufficient. Note: In data sufficiency problems that ask for the value of a quantity, the data given in the statements are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Question 24

A certain bookshelf has 2 shelves of books. On the upper self, the book with the greatest number of pages has 400 pages. On the lower shelf, the book with the least number of pages hat 475 pages. What is the median number of pages for all of the books on the 2 shelves? 1) There are 25 books on the upper shelf 2) There are 24 books on the lower shelf

Accepted Answer

1) The information says nothing about the number of book on the lower shelf. If there a fewer than 2 books on the lower shelf, then the median number of pages will be the number of pages in one of the books on the uper sheld. or the average number of pages in the two books on the upper sheld. Hence, the median will be at most 400. If there are 25 books on the lower shelf, then the median number of pages will be the number of pages in one of the books on the lower shelf or the average numbr of pages in the two books on the lower shelf. Hence, the median will be at least 475. NOT SUFFICIENT 2) An analysis very similar in 1) shows the information given is NOT sufficient to determine the median. NOT SUFFICIENT Given 1) and 2), it follows that there is a total of 49 books. therefore, the median will be the 25th book when the books are orderen by number of pages. Since the 25th book in this ordering is the book on the upper shelf with the greatest number of pages, the median is 400. Therefore 1) and 2) TOGETHER ARE SUFFICIENT C

Question 25

TWO intertwined equilateral perimeters of triangles

Accepted Answer

the perimeter of the figure equals the sum of the bigger perimeters - the small perimeters within them

Question 26

Set S has a range of x, set T has a subset of S. If all the numbers in set T are also in set S, is x greater than y? 1) Set s contains 7 numbers 2) Set T contains 6 numbers

Accepted Answer

Set S has a range of x, set T has a subset of S. Determine if x is greater than y. 1 ) It is given that S contains exactly 7 numbers but nothing additional is known about T. Thus, if S=(1,2,3,4,5,6) and T=(1,2,3,4,5,6,), then x=7 - 1 = 6, y=6-1=5 and x is greater than y. On the other hand, if S= (1,2,3,4,5,6,7) amd T= (1,2,3,4,5,6,7), then x= 7-1 = 6, y=7.1=6 and x is not greater than y. NOT SUFFICIENT 2) It is given that T contains exaclty 6 numbers, but nothin additional is known about T. Since the same examples given in 1) can also be used in 2), it cannot be determined if x is greater than y. THE CORRECT ANSWER IS E. WE CANNOT SAY

Question 27

The annual rent collected by a corporation from a certain building was x percent more in 1998 than in 1997 and y percent less in 1999 than in 1998. Was the annual rent collected by the corporation from the building more in 1999 than in 1997? 1) x>y 2) xy/100 < x-y

Accepted Answer

Let A be the annual rent collected in 1997. Then the annual rent collected in 1998 is (1+x/100)*A and the annual rent collected in 1999 is (1+x/100)(1-y/100)A. Determine if (1+x/100)(1-y/100)A > A; or equivalently, if (1+x/100)(1-y/100)> 1. 1) Given that x>y, (1+x/100)(1-y/100)> 1.is possible by choosing x=100 and y=10. since (1+100/100)(1- 10/100)> 2*0.9 = 1.8 and (1+x/100)(1-y/100)<1 is possible by choosing x=100 and y=90, since (1+ 100/100)(1- 90/100)> 2*0.1= 0.2 NOT SUFFICIENT 2) The inequality xy/100 1, which can be justified by the following steps, where each step's inequality is equivalent to the previous step's inequality. SUFFICIENT

Question 28

If Q is an odd number and the median of Q consecutive integers is 120, what is the largest of these integers?

Accepted Answer

_ _ _ _ _ _ _ _ Q = 7, Median = place 4 = 120 Largest number = (Q-1) / 2 + 120

Question 29

When a positie integer x is divided by positive integer y, the remainder is 9. If x/y = 96.12, what is the value of y?

Accepted Answer

1) x/y = q + 9/y 2) x/y = 96.12 = 96 + 0.12 EQUATE THE TWO EQUATIONS q + 9/y = 96 + 0.12 q= 96 and 9/y = 0.12 9= 0.12 * y Y = 75

Question 30

If x and y are positive numbers such that x+y = 1, which of the following could be the value of 100x + 200y ? - 80 - 140 - 199

Accepted Answer

Since x+y = 1, then y = 1-x and 100x + 200y can be expressed as 100x + 200(1-x) = 200 - 100x TEST EACH VALUE: Since x+y = 1, then y = 1-x and 100x+200y can be expressed as 100x + 200 (1-x) = 200-100x 1) 200-100x = 80, then x= 1.2 and y = 1. CANNOT BE A VALUE 2) 200-100x= 140, then x= 200-140/100 = 0.6 and y=1-0.6=0.4, so 140 can be a value of 100x+200y 3.) 199 can be a value of 100x+200y!

Question 31

0.0036 * 2.8 / 0.04 * 0.1 * 0.003 = what?

Accepted Answer

0.0036 * 2.8 ⁄ 0.04 * 0.1 * 0.003 (36*10-4)(28*10-1) ⁄ (4*10-2)* 10-1*3*10-3 36*28 / 4*3 * 10 -4-1-(-2-1-3) = 36*28 / 4*3 * 10 -5-(-6) = 36*28 / 4*3 * 101 = 840

Question 32

On a scale that measures the attest of a certain phenomenon, a reading n+1 corresponds to an intensity that is 10 times the intensity corresponding to a reading of n. On the scale, the intensity corresponding to a reading of 8 is how many times as great as the intensity corresponding to a reading of 3?

Accepted Answer

8 can be obtained from 3 by assing 1 five times. The intensity reading is greater by a factor 10*10*10*10*10 = 105

Question 33

If n is an integer greater than 6, which of the following must be divisible by 3? 1) n*(n+1)(n-4) 2) n*(n+2)(n-1) 3) n*(n+3)(n-5) 4) n*(n+4)(n-2) 5) n*(n+1)(n-4) 6) n*(n+5)(n-6)

Accepted Answer

1) n*(n+1)(n-4) = 7*8*3 DIVISIBLE BY 32) n*(n+2)(n-1) = 7*9*6 DIVISIBLE BY 33) n*(n+3)(n-5) = 7*10*2 NOT DIVISIBLE by 34) n*(n+4)(n-2) = 7*11*5 NOT DIVISIBLE by 35) n*(n+5)(n-6) = 7*12*1 DIVISIBLE by 3 THE EASIEST WAY IT TO CHOOSE AN INTEGER GREATER THAN 6, SUCH AS 7 AND ELIMINATE THE ANSWER CHOICES IN WHICH THE VALUE IS NOT DIVISIBLE BY 3. SECOND ROUND: Choose another integer greater 6, such as 8 and further test the choices 1) n*(n+1)(n-4) = 8*9*4 DIVISIBLE BY 32) n*(n+2)(n-1) = 8*19*7 NOT DIVISIBLE BY 3 ANSWER A!

Question 34

If there are fewer than 8 zeros between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t? 1) 3 2) 5 3) 9

Accepted Answer

The decimal expansion of (1/1000)4 = (10-3)-4 = 10-12 has 11 zeros to the right of the decimal point followed by a single digit that is 1. Therefore, in the the decimal expansion of (t/1000)4 = t4•10-12 Normally, there would be 12-1 zeros = 11 zeros between the decimal point and the integer. The question asks for fewers than 8, so there would need to be a number with 4 digits 34 = 81 => 2 digits 54 = 625 => 3 digits 94 = 6561 => 4 digits None of these numbers fulfills this constraint!

Question 35

If n = 33^43 + 43^33, what is the units digit of n?

Accepted Answer

TAKE THE UNITS DIGIT OF EXPONENT = 3FIND THE SEQUENCE -> If the units digit of an integer n is 3, then : n^1,n^2,n^3,n^4 are 3,9,7,1,…..,3,9,7,1. Then the powers of n form the sequence in which the digits 3,9,7 and 1 repeat indefinitely in that order. 3.) FIND THE PLACE IN THE SEQUENCE 43 = 10*4 + 3, the 43rd number in the sequence is 7 33^43 is 7 33= 8*4 = 32+1, so the 33rd number in the sequence is 3 7+3 = 10, so the units digit must be 0

Question 36

If n= 20!+17, then n is divisible by which of the following? i) 15, ii) 17 or iii)19?

Accepted Answer

1. ) Because 20! is the product of all integers from 1 to 20, it follows that 20! is divisible by each integer from 1 through 20. In particular, 20! is divisible by each of the integers 15,17 and 19. 2.) Since 20! and 17 are each divisible by 17, their Sum will be divisible by 17 CHECK FOR EACH VALUE IF IT IS DIVISIBLE

Question 37

If p is the product of the integers from 1-30, what is the greatest integer for which 3^k is a factor of p?

Accepted Answer

for which 3k is a factor of p P=product from 1-30 1. MULTIPLES OF § 3 =1*3 6 = 2*3 9 = 3*3 12 = 3*4 15 = 3*5 18 = 2*3*3 21 = 3*7 27 = 3*3*3 30 = 2*3*5 2. COUNT THE 3's : 14 mal die 3... 3. k=14

Question 38

How many of the integers that satisfy the inequality (x+2)(x+3) / (x-2) > 0 are less than 5?

Accepted Answer

TEST IT OUT -3 up to 4 (x+2)(x+3) / (x-2) >= 0 x + 3 => - - - 0 + + + + + + + + + + x + 2 => - - - - 0 + + + + + + + + + x - 2 => - - - - - - - - - - - - - - - 0 + + -3 -2 -1 0 1 2 3 4 For the equation to be positive, we need 2 "-" and 1 "+" or three "+" and 0 "-"

Question 39

List T consists of 30 positive decimals, none of which is an integer and the sum of the 30 decimals is S. The estimated Sum of the 30 decimals is defined as follows: Each decimal in T whose tenth digit is even is rounded up to the nearest integer and each decimal in T whose tenth digit is off is rounded down to the nearest integer. E is the sum of the resulting integers. If 1/3 of the decimals in T have a tenth digit that is even, which of the following is a possible value of E-S?

Accepted Answer

xx

Question 40

if (x) denotes the least integer greater than or equal to x, is (x) = 0? 1) -1< x < 1 2) x

Accepted Answer

least integer greater than or equal to x = -1 x could be 0.5 => NOT SUFFICIENT 2) x<0 => x could be-0.5 => NOT SUFFICIENT Taken 1) and 2 ) together, the result is SUFFICIENT C

Question 41

if 2^x+y = 4^8, what is the value of y? 1) x^2 = 81 2) x-y = 2

Accepted Answer

Since 48 = 22*8 => x+y = 16 1) Given that x2 = 81, then both x= 9 and x= -9 is possible. Therefore, y=7 or y = 25 is possible. NOT SUFFICIENT 2) Given that x-y = 2m ut follows that -x+y=-2. Adding the last equation to x+y = 16 gives 2y=14 or y=7 SUFFICIENT

Question 42

DS: A total of 60000 was invested for one year. Part of the amount earned simple annual interest at the rate of x percent per year. and the rest earned simple annual interest at the rate of y percent per year. If the total interest earned by the 60000 for that year was 4080, what is the value of x? 1) x = 0.75y 2) The ratio of the amount that earned interest at the rate of x percent per year to the amount that earned interest at y percent per year was 3:2 (1,5 y)

Accepted Answer

Since X+Y = 60000, if x * X + y*Y = 4080, determine the value of x! 1) Given that x = 0,75, then y4/3 x. Since X+Y = 60000 => Y = 60000-x Substitution: x*X +y*Y = 4080 becomes x*X + 1,5x (60000-X) = 4080 HOWEVER; the value of X is unknown so it is not possible to determine x NOT SUFFICIENT 2) Given that X = 1,5 Y, then Y= 2/3 X and Y = 60000 = 2/3 ( 60000-Y), from which the values of Y and X can be determined. HOWEVER; the value of y is unknown so it is not possible to determine x NOT SUFFICIENT TAKING BOTH STATEMENTS TOGEHTER IS SUFFICIENT!

Question 43

(0.02)^4/(0.002)^2=

Accepted Answer

When raising a power to another power, multiply the exponents: --> (24·10-8) / (22·10-6) = Separate into fractions of 2 and fractions of 10 --> (24/22) · (10-8/10-6) = When dividing powers with the same base, subtract the exponents: --> 22·10-2 = 0.04

Question 44

DS: If a, b, c and d are integers, is (2^a·3^b)/(2^c·3^d) even? (1) a>c (2) b=c+d

Accepted Answer

(2a·3b)/(2c·3d) = (2a-c·3b-d) Yes\No question => asking whether an expression is odd. SUFFICIENT: Answering a definite "Yes" or a definite "No", answering "it is odd" or "it is even" (1): if a>c, there are more 2s in the numerator than there are in the denominator. However, you know nothing about the 3s. There may be at least as many in the numerator as in the denominator, and then the expression is even. However, they may be more 3s in the denominator, and then it's a fraction, and therefore not even. Stat.(1)->Maybe->BCE. (2): means actually b≥d, i.e., there are at least as many 3s in the numerator as there are in the denominator. In this case the expression could be even (if we plug in a>c). However, does this apply for DOZEN of numbers, too? By plugging a negative c (e.g. c=-1), d could be greater than b, in which case the expression is a fraction and not even. Stat.(1+2): Stat.(1) gives us half the answer with a>c. If b, c, and d are all zero or positive, then there are no 3s in the denominator and the expression is even. However, despite appearances, (2) does not necessarily mean b≥d. Plug in b=0, c=-1, d=1 and you get b<d, and the expression is a fraction, which is not even. NOT SUFFICIENT

Question 45

Prove that y/y-x + x/x-y = 1

Accepted Answer

Prove that y/y-x + x/x-y = 1 y/y-x + x/x-y = 1 y/y-x - x/y-x = 1 y-x/y-x = 1

Question 46

DS: In the equation x2+k·x+1=0, x is a variable and k is a constant. If the quadratic equation has two distinct real roots, which of the following could be true? I) k=0 II) k=−1 III) k=−3

Accepted Answer

x2+k·x+1=0 I) k=0 II) k=−1 III) k=−3 x2+kx+1=0 CHECK THE DISCRIMINANT b2-4ac Plug k=-3 into the equation and see that it satisfies the other requirements presented, i.e. that the equation will have two possible solutions. Do so by checking that the discriminant is positive when k=-3: b2-4ac 9-4 = 5, which is greater than zero.Thus, the equation will have two solutions when k=-3, so k=-3 is a possible value. Thus, the equation will have two solutions when k=-3, so k=-3 is a possible value.

Question 47

If the quadratic equation has two distinct real roots, which of the following could be true?

Accepted Answer

Check the discriminant: b2-4ac = (-3)2-4·1·1 > 0 -> two possible values Thus, the equation will have two solutions when k=-.., so k=.. is a possible value.

Question 48

c

Accepted Answer

C = BOTH STATEMENTS TOGETHER ARE SUFFICIENT Stat. (1): the value of the least number of the set does not tell you how many terms are there, nor their average. Stat.(1)->IS->BCE. Stat. (2): The average of the terms in the set is a useful piece of information, but you're still missing the number of terms in the set. If the set includes 3 terms or 30 terms, the sum of the set will be completely different. Stat.(2)->IS->CE. Stat. (1)+(2): Recall that the average of a set of consecutive integers is also the average of the first and last terms. Therefore, if the least term is 8 and the greatest term is x, then (8 + x)/2 = 16, according to Stat. (2). From here it is possible to find the largest term x, and the number of terms can also be found - simply count the number of consecutive integers between 8 and x. You have what you need to find the sum of the terms in the set, so Stat.(1+2)->S->C.

Question 49

Greatest Common Divisor

Accepted Answer

The GCD of two or more numbers is the product of the common factors of all numbers.

Question 50

What is the sum of the terms in a certain sequence of consecutive integers? (1) The least term in the sequence is 8. (2) The average of (arithmetic mean) terms is 16.

Accepted Answer

The issue is finding the sum of a series of consecutive integers. you need the average of the set, and the number of terms. If the statement(s) provide this information, then it is sufficient. -> missing the number of terms in the set.

GMAT (Subject) / Quantitative - Arithmetic (Lesson)