datasuficiency tests 1- 6 and explanations

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GMAT DATA SUFFICIENCY TESTS 1-6 WITH EXPLANATIONS THIS FILE IS MEANT FOR USE BY PARTICIPANTS REGISTERED IN OUR PREP COURSES

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DATA SUFFICIENCY TESTS 1 THROUGH 6 WITH EXPLANATIONS

DIRECTIONS: This file includes 6 tests, each containing about 20 questions, that need to be completed by you. Try to complete each test within the allotted time indicated at the beginning of the test. Start timing yourself at the beginning of the test and indicate the time taken to complete each test at the end of each test.



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DATA SUFFICIENCY

EXERCISE 1 CHOOSE: 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 statement TOGETHER are sufficient, but NEITHER statement ALONE IS sufficient. D EACH statement ALONE is sufficient E Statements (1) and (2) TOGETHER are NOT sufficient. 1. In the figure below, is CD > BC ? A B C D (1) AD=20 (2) AB = CD Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 2. How many more men than women are in the room? (1) There is a total of 20 women and men in the room. (2) The number of men in the room equals the square of the number of women in the room. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 3. If n is an integer, is (100-n) an integer? n (1) n > 4 (2) n2 = 25 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( )

4. Last Friday a certain shop sold 3/4 of its inventory of sweaters. Each sweater sold for $20. What was the total revenue last Friday from the sale of these sweaters? (1) When the shop opened last Friday, there were 160 sweaters in the inventory. (2) All but 40 sweaters in the shop’s inventory were sold last Friday Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 5. A jar containing 30 marbles, of which 20 are red and 10 are blue. If 9 of the marbles are removed, how many of the marbles left in the jar are red? (1) Of the marbles removed, the ratio of the number of red ones to the number of blue ones is 2:1. (2) Of the first 6 marbles removed, 4 are red. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 6. Is the triangle ABC, whose angles are x,y and z equilateral? (1) x = y (2) z=60 degrees Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 7. If w + z = 28, what is the value of wz (1) w and z are positive integers (2) w and z are consecutive odd integers Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 8. Is ax = 3 - bx ? (1) x (a+b) = 3 (2) a = b = 1.5 and x = 1 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( )



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9. What is the value of integer x ? (1) x is a prime number (2) 30 < x < 38 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 10. While on a straight road, car X and Car Y are traveling at different constant rates. If car X is now 1 mile ahead of car Y. How many minutes from now will car X be 2 miles ahead of car Y? (1) Car X is traveling at 50 miles per hour and car Y is traveling at 40 miles per hour. (2) 3 minutes ago car X was 1/2 mile ahead of car Y. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 11. In what year was Ellen born? (1) Ellen’s brother Pete, who is 1 1/2 years older than Ellen, was born in 1956. (2) In 1975 Ellen turned 18 years old. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 12. What is the number of female employees in Company X? (1) If company X were to hire 14 more people and all of these people were females, the ratio of the number of male employees to the number of female employees would then be 16 to 9. (2) Company X has 105 more male employees than female employees. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 13. Is the integer x divisible by 36? (1) x is divisible by 12 (2) x is divisible by 9. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( )

14. What is the average (arithmetic mean) of j and k? (1) The average of j + 2 and k + 4 is 11 (2) The average of j, k, and 14 is 10. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 15. What is the value of a - b? (1) a = b + 4 (2) (a - b)2 = 16 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 16. Is rst = 1? (1) rs = 1 (2) st = 1 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 17. In a certain office, 50 percent of the employees are college graduates and 60 percent of the employees are over 40 years old. If 30 percent of those over forty have master’s degrees, how many of the employees over forty have master’s degree? (1) Exactly 100 of the employees are college graduates. (2) Of the employees forty years old or less, 25 percent have master’s degrees. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 18. Is xy < 6 ? (1) x < 3 and y < 2 (2) 1/2 < x < 2/3 and y2< 64 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( )



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19. Is x equal to 2? (1) x, y and x + y are prime numbers (2) y is odd Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) 20. In 1979 Mr. Jackson bought a total of n shares of stock X and Mrs. Jackson bought a total of 300 shares of stock X. If the couple held all of their respective shares throughout 1980, and Mr. Jackson’s 1980 dividends on his n shares totaled $150, what was the total amount of Mrs. Jackson’s 1980 dividends on her 300 shares ? (1) In 1980 the annual dividend on each share of stock X was $0.75 (2) In 1979, Mr. Jackson bought a total of 200 shares of stock X. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) End of Section Test. If you took longer than 25 minutes to complete, record that information here: Additional Minutes.



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DATA SUFFICIENCY EXERCISE 2

20 Questions -- 25 Minutes CHOOSE: 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 statement TOGETHER are sufficient, but NEITHER statement ALONE IS sufficient. D EACH statement ALONE is sufficient E Statements (1) and (2) TOGETHER are NOT sufficient. 1. If the list price of a new car was $12,300, what was the cost of the car to the dealer? (1) The cost to the dealer was equal to 80 percent of the list price (2) The car was sold for $11,070, which was 12.5 percent more than the cost to the dealer. A ( ) B ( ) C ( ) D ( ) E ( ) 2. If p, q, x, y, and z are different positive integers, which of the five integers is the median? (1) p + x < q (2) y < z A ( ) B ( ) C ( ) D ( ) E ( ) 3. A certain employee is paid $6 per hour for an 8-hour workday. If the employee is paid 1 1/2 times this rate for time worked in excess of 8 hours during a single day, how many hours did the employee work today? (1) The employee was paid $18 more for hours worked today than for hours worked yesterday. (2) Yesterday the employee worked 8 hours A ( ) B ( ) C ( ) D ( ) E ( )

4. If n is a member of the set (33, 36, 38, 39, 41, 42), What is the value of n? (1) n is even (2) n is a multiple of 3 A ( ) B ( ) C ( ) D ( ) E ( ) 5. What is the value of x? (1) 2x + 1 = 0 (2) (x + 1 )² = x² A ( ) B ( ) C ( ) D ( ) E ( ) 6. In the figure below, what is the length of AD? A_________B_________C_________D (1) AC = 6 (2) BD = 6 A ( ) B ( ) C ( ) D ( ) E ( ) 7. A retailer purchased a television set for x percent less than its list price, and then sold it for y percent less than its list price. What was the list price of the television set? (1) x = 15 (2) x - y = 5 A ( ) B ( ) C ( ) D ( ) E ( )

8. Is x² greater than x ?

(1) x² is greater than 1 (2) x is greater than -1 A ( ) B ( ) C ( ) D ( ) E ( )



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9. What is the value of r/2 + s/2 ? (1) (r + s)/2 =5 (2) r + s = 10 A ( ) B ( ) C ( ) D ( ) E ( ) 10. If x, y, and z are numbers, is z = 18 ? (1) The average (arithmetic mean) of x, y, and z is 6 (2) x = - y A ( ) B ( ) C ( ) D ( ) E ( ) 11. B C A The circular base of an above ground swimming pool lies in a level yard and just touches two straight sides of a fence at points A and B, as shown in the figure above. Point C is on the ground where the two sides of the fence meet. How far from the center of the pool’s base is point A? (1) The base has area 250 square feet (2) The center of the base is 20 feet from point C. A ( ) B ( ) C ( ) D ( ) E ( )

12. In 1979 Mr. Jackson bought a total of n shares of stock X and Mrs. Jackson bought a total of 300 shares of stock X. If the couple held all of their respective stocks throughout 1980, and Mr. Jackson’s 1980 dividends on his n shares totaled $150, what was the total amount of Mrs. Jackson’s 1980 dividends on her 300 shares ? (1) In 1980 the annual dividend on each share of stock X was $ 0.75 (2) In 1979 Mr. Jackson bought a total of 200 shares of stock X A ( ) B ( ) C ( ) D ( ) E ( ) 13. If Sara’s age is exactly twice Bill’s age, what is Sara’s age? (1) Four years ago, Sara’s age was exactly 3 times Bill’s age. (2) Eight years from now, Sara’s age will be exactly 1.5 times Bill’s age. A ( ) B ( ) C ( ) D ( ) E ( ) 14. What is the value of (x/yz)? (1) x = y/2 and z = 2x/5 (2) x/z = 5/2 and 1/y = 1/10 A ( ) B ( ) C ( ) D ( ) E ( ) 15. An infinite sequence of positive integers is called an “alpha sequence” if the number of even integers in the sequence is finite. If S is an infinite sequence of positive integers, is S an alpha sequence? (1) The first ten integers in S are even. (2) An infinite number of integers in S are odd. A ( ) B ( ) C ( ) D ( ) E ( )



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16. If xy > 0, does (x-1)(y-1) = 1 ? (1) x + y = xy (2) x = y A ( ) B ( ) C ( ) D ( ) E ( ) 17. After winning 50 percent of the first 20 games it played, Team A won all of the remaining games. What was the total number of games that Team A won? (1) Team A played 25 games altogether. (2) Team A won 60 percent of all the games it played. A ( ) B ( ) C ( ) D ( ) E ( ) 18. @ + $ = * In the addition problem above, each of the symbols @, $, and * represents a positive digit. If @ < $, what is the value of $? (1) * = 4 (2) @ = 1 A ( ) B ( ) C ( ) D ( ) E ( )

19. Cancellation Fees Schedule Days prior to Percent of Departure Package Price 46 or more 10% 45 - 31 35% 30 - 16 50% 15 - 5 65% 4 or fewer 100% The table above shows the cancellation fee schedule that a travel agency uses to determine the fee charged to a tourist who cancels a trip prior to departure. If a tourist canceled a trip with a package price of $1,700 and a departure date of September 4, on what day was the trip canceled? (1) The cancellation fee was $595 (2) If the trip had been canceled one day later, the cancellation fee would have been $255 more. A ( ) B ( ) C ( ) D ( ) E ( ) 20. Is 5k less than 1,000? (1) 5k+1 is greater than 3,000 (2) 5k-1 is 500 less than 5k. A ( ) B ( ) C ( ) D ( ) E ( ) END OF SECTION END OF SECTION Did you complete the section in 25 Minutes? If not, indicate how much longer you took to complete this section: Additional Minutes.



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DATA SUFFICIENCY EXERCISE 3

21 Questions -- 27 Minutes

CHOOSE: A Statement (1) alone is sufficient (to give a unique value for the information sought) but statement (2) alone is not sufficient B Statement (2) alone is sufficient but statement (1) alone is not sufficient. C BOTH statement TOGETHER are sufficient, but NEITHER statement ALONE IS sufficient. D EACH statement ALONE is sufficient E Statements (1) and (2), alone or TOGETHER, are NOT sufficient. 1. The results of a certain experiment included 6 data values that were all multiples of the same number c, namely, c, 8c, 2c, 5c, 4c, and 4c. Was the average (arithmetic mean) of the 6 data values greater than 8?

(1) c < 4 (2) c > 2

A ( ) B ( ) C ( ) D ( ) E ( )

2. If n is an integer, is n + 2 a prime number? (1) n is a prime number (2) 30 < n < 40

A ( ) B ( ) C ( ) D ( ) E ( ) 3. If t is not equal to zero, is r a positive number? (1) rt = 12 (2) r + t = 7

A ( ) B ( ) C ( ) D ( ) E ( )

4. If x is an integer, is y an integer? (1) The average (arithmetic mean) of x, y, and y - 2 is x . (2) y - x = 1

A ( ) B ( ) C ( ) D ( ) E ( ) 5. How many minutes does it take for a circular shaped illumination to quadruple its area? (1) The initial diameter of the circular shaped illumination is 10 in. (2) The circumference of the circular shaped illumination increases at the rate of 6 in/minute

A ( ) B ( ) C ( ) D ( ) E ( ) 6. The inside of a rectangular carton is 48 centimeters long, 32 centimeters wide, and 15 centimeters high. The carton is filled to capacity with k identical cylindrical cans of fruit that stand upright in rows and columns as shown below. If the cans are 15 centimeters high, what is the value of k? (1) Each of the cans has a radius of 4 centimeters. (2) 6 of the cans fit along the length of the carton.

A ( ) B ( ) C ( ) D ( ) E ( ) 7. If k is a positive integer, is the value of (b – a) at least twice the value of 3k - 2k? (1) a is equal to 2k+1 and b is equal to 3k+1 (2) k = 3

A ( ) B ( ) C ( ) D ( ) E ( )



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8. If P and Q are each circular regions, what is the radius of the larger of these regions? (1) The area of P plus the area of Q is equal to 90 pi. (2) The larger circular region has a radius that is 3 times that of the smaller circular region.

A ( ) B ( ) C ( ) D ( ) E ( ) 9. Is a+ b divisible by 8? (1) When a is divided by 8, it leaves a remainder of 3. (2) When b is divided by 8, it leaves a remainder of 5.

A ( ) B ( ) C ( ) D ( ) E ( ) 10. What is the ratio of x : y : z? (1) z = 1 and xy = 32 (2) x = 2y and y = 4z

A ( ) B ( ) C ( ) D ( ) E ( ) 11. If 3m = 5n , what is the value of m + n ? (1) 2m + n = 26 (2) If the value of m were increased by 2, then it will be twice the value of n.

A ( ) B ( ) C ( ) D ( ) E ( ) 12. On a Friday Morning, a certain machine ran continuously at a uniform rate to till a production order. At what time did it completely fill the order that morning ? (1) The machine began filling the order at 9:30 a.m. (2) The machine had filled 1/2 of the order by 10:30 a.m. and 5/6 of the order by 11:10 a.m.

A ( ) B ( ) C ( ) D ( ) E ( )

13. A ladder is propped up against a vertical wall such that the bottom of the ladder makes an angle of 60 degrees to the ground. The ladder slips down and the bottom of the ladder moves away from the wall. In this position, the bottom of the ladder makes an angle of 45 deg rees to the ground. How much farther did the ladder move from the wall when it slipped? (1) The ladder is 10 meters long (2) The bottom of the ladder was 5 meters from the wall in its original position.

A ( ) B ( ) C ( ) D ( ) E ( ) 14. Can the positive integer n be written as the sum of two different positive prime numbers? (1) n is greater than 3 (2) n is odd.

A ( ) B ( ) C ( ) D ( ) E ( ) 15. What is the last number in a set comprising six consecutive odd integers? (1) The average (arithmetic mean) of the six consecutive odd integers is 36. (2) Twice the average (arithmetic mean) of the six consecutive odd integers is equal to the sum of the first and the last number in the set.

A ( ) B ( ) C ( ) D ( ) E ( ) 16. Town T has 20,000 residents, 60 percent of whom are female. What percent of the residents were born in Town T? (1) The number of female residents who were born in Town T is twice the number of male residents who were not born in Town T. (2) The number of female residents who were not born in Town T is twice the number of female residents who were born in Town T.

A ( ) B ( ) C ( ) D ( ) E ( )



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17. A right triangular region PQR lies in a rectangular coordinate plane such that each of its sides, PQ and PR, lies parallel to the rectangular coordinate axes. Is the right triangle isosceles? (1) One of the points Q on the hypotenuse RQ has the coordinates (3, 4). (2) One of the points R on the hypotenuse RQ has the coordinates (-2, -1)

A ( ) B ( ) C ( ) D ( ) E ( ) 18. If both x and y are non-zero numbers, what is the value of y/x? (1) x = 8

(2) y³ = x²

A ( ) B ( ) C ( ) D ( ) E ( ) 19. If x = 0.rstu where r, s, t, and u each represent a non-zero digit of x, what is the value of x ? (1) r = 3s = 2t = 6u (2) The product of r and u is equal to the product of s and t

A ( ) B ( ) C ( ) D ( ) E ( ) 20. If x and y are integers between 10 and 99, inclusive, is (x - y) /9 an integer ? (1) x and y have the same two digits but in the reverse order. (2) The tens’ digit of x is 2 more than the units’ digit and the tens’ digit of y is 2 less than the units’ digit.

A ( ) B ( ) C ( ) D ( ) E ( )

21. Is the positive integer n equal to the square of an integer ? (1) For every prime number p, if p is a divisor of n, then so is p-squared.. (2) Square root of n is an integer.

A ( ) B ( ) C ( ) D ( ) E ( ) Did you complete this section in 25 Minutes ? If not, indicate how much longer did you take to complete it ? Minutes.



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Data Sufficiency Exercise 4

Time : 25 Minutes 20 Questions.

Directions: Each of the Data Sufficiency problems below consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you are to fill in oval, A if statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked; B if statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked; C if BOTH statements TOGETHER are sufficient to answer the question asked, but NEITHER statement alone is sufficient. D if EACH statement ALONE is sufficient to answer the question asked; E if statement (10 and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed. All numbers used are real numbers. A figure in the data sufficiency problem will conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2). All figures lie in a plane unless otherwise indicated. In questions 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.

1. Is x < y? (1) x2 < y2

(2) x < y + 1 Your Answer: 2. Machine A runs at a constant rate and produces a lot consisting of 100 bolts in 30 minutes. How much less time would it take to produce the lot of bolts if both machines A and B were run simultaneously? (1) Both machines A and B produce the same number of bolts per hour. (2) It takes machine A twice as long as it takes machines A and B, running simultaneously, to produce the same lot of bolts. Your Answer: 3. Is p a prime number? (1) p+1 is a prime number. (2) p is even integer. Your Answer: 4. Is the hundredths digit of the decimal number d greater than 5? (1) The tenths digit of the decimal number 10d is 7 (2) The thousandths digit of d/10 is 7. Your Answer: 5. If n is an integer, is (100 - n) an integer? 2n (1) n < 60 (2) n is divisible by 10. Your Answer:



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6. If -10 < k < 10 , is k > 0 ? (1) k2 > 4 (2) 1 - k > 0 Your Answer: 7. Buckets X and Y contained only water and bucket Y was half full. If all the water in bucket X was poured into bucket Y, what fraction of the capacity of Y was then filled with water? (1) Before the water from X was poured into Y, X was 1/4 full. (2) X has twice the capacity of Y. Your Answer: 8. Did Anne pay less than d dollars, including tax, for her mink coat? (1) The price Anne paid for her mink coat was (0.9.d), excluding tax. (2) The tax payable on mink coat sale is 10 percent of the selling price. Your Answer: 9. Is c > d ? (1) 1 - c/d > -1 (2) 0.5 < c/d < 2.0 Your Answer: 10. In a certain health club, are more than 2/3 of the members females? (1) The club has exactly 75 female members. (2) The ratio of female to male members is 3:1 Your Answer:

11. If x is an integer, is y an integer? (1) The average of x and y is NOT an integer. (2) ( x + y ) = 2( x - y ) Your Answer: 12. The price per share of stock X increased by 10 percent over the same time period that the price per share of stock Y decreased by 10 percent. The reduced price per share of stock Y was what percent of the original price per share of stock X ? (1) The increased price per share of stock X was equal to the original price per share of stock Y. (2) The increase in price per share of stock X was 10/11 the decrease in the price per share of stock Y. Your Answer: 13. Any decimal that has only a finite number of non-zero digits is a terminating decimal. Examples: 32, 0.78, and 6.087 are three terminating decimals. If r and s are positive integers, is the ratio r/s a terminating decimal? (1) 90 < r < 100 (2) s = 3 Your Answer: 14. What is the area of the rectangular region of sides L and W? (1) L + W = 14 (2) d = 10 , (d is the diagonal) Your Answer:



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15. Is 2n > 100? (1) 3n > 100 (2) 2n+1 = 128 Your Answer: 16. Your Answer: 17. Your Answer:

18. In a certain office, 60 percent of the employees are college graduates and 50 percent of the employees are over forty years old. If 30 percent of those over forty years of age have a master’s degree, how many employees are college graduates? (1) Exactly 60 employees are over 40 and have a master’s degree. (2) There are four times as many college graduates as there are those over forty with master’s degrees. Your Answer: 19. If n is an integer, is n an odd integer? (1) (3n + 1) is an odd integer (2) (2n - 1) is an odd integer Your Answer: 20. Your Answer: End of Section End of Section Time you took to complete this section: Minutes



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Data Sufficiency Exercise 5



1. What is the value of | n |? (1) n is a negative integer. (2) n2 = 9 Your Answer: 2. If x, y and z are the three internal angles of a triangle, what is the value, in degrees, of angle x ? (all values of x, y and z are in degrees) (1) x + y = 127 (2) x + y = 2z + 21 Your Answer: 3. Of the n people participating in a test preparation program, 60 percent had not taken the test previously. Of the remaining, 25 percent had taken the test more than once previously. How many had taken the test just once previously? (1) 12 people had taken the test more than twice previously. (2) 18 people had taken the test more than once previously. Your Answer: 4. How many more men than women are in the public swimming pool? (1) There are twice as many men as there are women. (2) If six more women came into the pool, there will be an equal number of men and women. Your Answer:



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5. Is the sum of a set of six consecutive odd positive integers a perfect cube? (1) The smallest number in the set is a prime number. (2) The largest number in the set is 41. Your Answer: 6. How many miles long is the route from Thunder Bay to Sarnia? (1) It will take 2 hours less time to travel the entire route at an average rate of 60 miles per hour than at an average rate of 50 miles per hour. (2) The first half of the distance can be traveled in 5 hours at 60 miles per hour rate of speed. Your Answer: 7. If x is divisible by 2, is x + y an even number? (1) y is a factor of x (2) x = 6 Your Answer: 8. All applicants to a military recruitment program must pass both a written test and a physical test. If 60 percent of the applicants passed the written test and 75 percent of the applicants passed the physical test, what percent of the applicants did not pass both or either tests? (1) 55 percent of the applicants passed both tests. (2) 20 percent of the applicants did not pass either test. Your Answer:

9. If an item was marked up 30 percent on its cost and then sold at a discount on the sticker price, what was the selling price, in dollars? (1) The item cost $120 (2) The sticker price was $156 Your Answer: 10. If n is an integer, is n/15 an integer ? (1) 3n/15 is an integer (2) 7n/15 is an integer Your Answer: 11. Mary and Martha received wage increases following their annual performance review? Who received the greater increase, in dollars? (1) Mary received a 10 percent increase on her wages. (2) Martha received a 8 percent increase on her wages. Your Answer: 12. Is John ahead of Paul in the line-up? (1) There are 20 people in the line-up. (2) There are exactly 6 people between Paul and John in the line-up. Your Answer:



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13. Lorraine can drive from her home to a super-market by one of two possible routes. If she must also return by one of the two routes, what is the distance of the shorter route? (1) When Lorraine drives to the supermarket by the shorter route and returns by the longer route, she travels a total of 11 miles. (2) When she drives both ways, from her home to the super-market, by the shorter route, she travels a total distance of 8 miles. Your Answer: 14. If p < 100, is the prime number p equal to 31? (1) p = n3 + 4 , where n is an integer. (2) p + 5 is a perfect square. Your Answer: 15. What is the ratio of x to y ? (1) If the value of y were increased by 7, then the ratio of x to y will be equal to 1. (2) The ratio of x to 4y is 3 to 5. Your Answer: 16. If a television commercial consists of a total of 17,280 frames on film, how long, in minutes, does the commercial run? (1) The commercial runs without interruption at the rate of 24 frames per second. (2) It takes 6 times as long to run the film as it takes to rewind the film, and it takes a total of 14 minutes to do both. Your Answer:

17. The symbol @ represents one of the following operations: addition, subtraction, multiplication, or division. What is the value of 4 @ 3 ? (1) 0 @ 4 = 4 (2) 4 @ 0 = 4 Your Answer: 18. If y = 2x+1, what is the value of y - x? (1) 22x+4 = 64 (2) y = 22x

Your Answer: 19. In a certain group of people, the average (arithmetic mean) I.Q of the males is 128 and of the females 136. What is the average I.Q. of the people in the group? (1) The group contains twice as many females as males (2) The group contains 10 more females than males. Your Answer: 20. If x, y, and z are three integers, are they consecutive odd integers? (1) y - x = 2 (2) x + y is even Your Answer: 21. In triangle ABC, if AB = x, BC = x + 2, and AC = y, then which of the three internal angles of the triangle has the greatest degree measure? (1) y = x + 3 (2) x = 2 Your Answer:

HHHOOOWWW LLLOOONNNGGG DDDIIIDDD YYYOOOUUU TTTAAAKKKEEE TTTOOO

CCCOOOMMMPPPLLLEEETTTEEE::: MMMIIINNNSSS



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DDDAAATTTAAA SSSUUUFFFFFFIIICCCIIIEEENNNCCCYYY

EEEXXXEEERRRCCCIIISSSEEE 666 Time : 25 Minutes 20 Questions.

Directions: Each of the Data Sufficiency problems below consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of mathematics and everyday facts (such as the number of days in July, or the meaning of counterclockwise), you are to fill in oval, A if statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked; B if statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked; C if BOTH statements TOGETHER are sufficient to answer the question asked, but NEITHER statement alone is sufficient. D if EACH statement ALONE is sufficient to answer the question asked; E if statement (10 and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed. All numbers used are real numbers. A figure in the data sufficiency problem will conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2). All figures lie in a plane unless otherwise indicated. In questions 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.

1. Is x greater than 2? (1) x2 > 4 (2) x is a multiple of 2 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 2. In an election to the Secretary of the Club, if each of the 1,000 members voted for either Mary or Michelle (but not both), what percent of the female members voted for Michelle? (1) Eighty percent of male members voted for Michelle. (2) Twice as many male members voted for Michelle as female members. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 3. During 1987, 8.7 percent of the men in the labor force were unemployed in June as against 8.4 percent in May of that year. If the number of men in the labor force was the same in both months, how many more men were unemployed in June than in May? (1) The number of unemployed men in the labor force during May was 1.68 million. (2) The total number of men in the labor force was 20.0 million during the two months - May and June - of 1987. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 4. If the average (arithmetic mean) of 4 numbers is 45, how many numbers are greater than 45? (1) Two of the numbers are 60 and 45. (2) One of the numbers is 25. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )



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5. Is the value of x greater than 5? (1) x3 > 125 (2) 4 < x < 6 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 6. Is n + 1 a prime number? (1) n is a product of two prime numbers. (2) n2 + 2n + 1 = 49 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 7. A rectangular frame encloses a picture. What is the length in inches of the picture? (1) The frame measures 24 inches by 18 inches. (2) Area of the frame = area of the picture it encloses. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 8. How far apart do Jane and June live? (1) The local public library is 5 miles due north of Jane’s house and 12 miles due east of June’s house. (2) The local school is 12 miles due west of Jane’s house and 5 miles due south of June’s house. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

9. If n > 0, is n equal to the sum of two different prime numbers? (1) n is equal to the square of the smallest odd prime number. (2) n + 2 is a prime number. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 10. 2/3 of the N people polled in a survey involving 2 questions said Yes to Question 1. What fraction of the people polled did NOT say yes to both questions? (1) 3/5 of those who answered YES to Question 1, answered YES to Question 2. (2) 1/3 of those polled answered NO to Question 1. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 11. If s, u, and v are positive integers, is s > v ? (1) s > u

(2) 2s = 2u + 2v

Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 12. What is the area of the square region? (1) The diagonal is 10 inches. (2) The perimeter is 20 \/2 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )



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13. What is the measure of an external angle of the triangle ABC? (1) One of the internal angles measures 72 degrees. (2) The triangle ABC is an isosceles triangle. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 14. What is the ratio of the volume of cube X to that of cube Y? (1) The length of an edge of cube X is 6 inches. (2) The ratio of the surface area of cube X to that of cube Y is ¼. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 15. What is the value of k? (1) In the xy-coordinate system, (a,b) and (a+3, b+k) are two points that lie on the line defined by the equation x = 3y - 7. (2) k.k = 1 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 16. What is the average (arithmetic mean) dollar amount of all the paychecks that John Doe received last year? (1) Last year John Doe received 26 paychecks. (2) The average of John Doe’s first thirteen checks during the year was $750. The average of John Doe’s last 13 checks was $800. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

17. How many hours does it take for Machine A to fill a production lot, working alone? (1) Machines A and B, operating simultaneously, can fill the production lot in 2 hours. (2) Machine B, working alone, can fill the lot in 5 hours. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 18. How many students are in the school? (1) 40 more than 1/3 of all the students in the school are taking a science course and, of these, 1/4 are taking Physics. (2) Exactly 1/8 of all the students in the school are taking Physics. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 19. Is c2 + d2 > 1? (1) d > 0 (2) c/d > 1 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( ) 20. @, &, and * are three different positive digits and If @ + & = * , What is the value of &? (1) * = 4 (2) @ = 1 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

End of Section How long did you take to complete this section?



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ANSWER KEY AND EXPLANATIONS TO THE DATA SUFFICIENCY 1 – 6

TESTS



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DATA SUFFICIENCY EXERCISE 1- Answer Key

CHOOSE: 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 statement TOGETHER are sufficient, but NEITHER statement ALONE IS sufficient. D EACH statement ALONE is sufficient E Statements (1) and (2) TOGETHER are NOT sufficient. 1. In the figure below, is CD > BC ? A B C D (1) AD=20 (2) AB = CD Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 alone is not sufficient because we do not get a sense of the absolute values of CD and BC from the given information. Our choices are B,C, or E at this stage. Statement 2 tells us that AB and CD have equal values. That does not mean much either. Our choices are C or E. When we combine the two statements, we can generate multiple scenarios: AB and CD could be both equal to 1 and BC will then be equal to 18. Or, AB and CD could be equal to 8, in which case BC will be equal to 4, and so on. Since we get BC > CD in one scenario and quite the opposite conclusion in the other scenario, we conclude that even the combined information does not provide a unique value for the lengths of CD and BC. We have to pick E.

2. How many more men than women are in the room? (1) There is a total of 20 women and men in the room. (2) The number of men in the room equals the square of the number of women in the room. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 gives us a total value for the number of men and women in the room. We can get a range of possible values of men and women:m1w19,m2w18,m3w17,......m16w4......m19w1. Therefore, we conclude that statement 1 alone is not sufficient to answer the question. The choices before us are B,C, or E. Statement 2 tells us that the number of men in the room is equal to the square of the women in the room. Once again, we get a range of possible values for men andwomen:m1w1,m4w2,m9w3,m16w4,m25w5, and so on. A range of values and not a unique information. We conclude that statement 2 alone is also not sufficient. Our choices narrow to C or E. When we combine the two statements, we can see that there is just one set of values for men and women: w4 and w16 for a total of 20. The combined information is sufficient to answer the question. We pick C as the answer. 3. If n is an integer, is (100-n) / n an integer? (1) n > 4 (2) n.n = 25 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Whenever you see an algebraic expression or equation, try to rearrange things before you go to examine the statements. In this problem, we can write ( 100 - n ) / n as 100/n - 1. If we get a sense from statements 1 and 2 that n is a factor of 100, we will be able to answer the question posed. Statement 1 tells us that n is greater than 4. Which means that if n is 5, then it is a factor of 100. If n is 6, then it is not. If n is 7, then it is not. If n is 10, then it is a factor of 100. And so on. Since we do not know what the true value of n is, we get a range of possible YES/NO scenarios. What do we conclude ? That statement 1 alone is not sufficient to answer the question. Our choices are B, C, or E. Statement 2 tells us that n is equal to +5 or -5. In either case, n is a factor of 100 and 100/n - 1 will be a whole number. Statement 2 alone is sufficient and we pick B.



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4. Last Friday a certain shop sold 3/4 of its inventory of sweaters. Each sweater sold for $20. What was the total revenue last Friday from the sale of these sweaters? (1) When the shop opened last Friday, there were 160 sweaters in the inventory. (2) All but 40 sweaters in the shop’s inventory were sold last Friday Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) To ace the data sufficiency section, you should be able to make the most sense of the given information before you go to examine the statements and determine ahead of time what information you are seeking. In this problem, if N is the number of sweaters in the inventory, we know that 3N/4 sweaters were sold and N/4 were not. At $20 a sweater, the revenue from the sale of 3N/4 sweaters is $60N/4 = $15N. Statement 1 tells us that N = 160. Good enough. We know that 3N/4 sweaters were sold and we can calculate the revenue. Statement 1 alone is sufficient. Our choices are A or D. Statement 2 tells us that the number of sweaters not sold is equal to 40. Which means that N/4 = 40 . We know that 3 times this number was sold. We can determine the quantity sold and compute the revenue from the sales. Statement 2 alone is also sufficient to answer the question. We pick D as the answer.

5. A jar containing 30 marbles, of which 20 are red and 10 are blue. If 9 of the marbles are removed, how many of the marbles left in the jar are red? (1) Of the marbles removed, the ratio of the number of red ones to the number of blue ones is 2:1. (2) Of the first 6 marbles removed, 4 are red. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) We are looking for information pertaining to how many of the 9 marbles removed from the jar were red in color. Statement 1 tells us that the number of red ones removed was 2x and the number of blue ones was 1x . This information tells us that the number of red ones removed was 6 and that of blue was 3. Good enough. We can determine how many red ones were left behind in the jar from this information alone. Therefore, our choices at this juncture are A or D. Statement 2 does not tell us much. We do not know if the last 3 marbles removed were all red or all blue or any combination of the two. Since we are faced with a range of possible values for the removed red marbles, we have to conclude that statement 2 alone is not sufficient. We pick A as the answer.



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6. Is the triangle ABC, whose angles are x,y and z equilateral? (1) x = y (2) z=60 degrees Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) An equilateral triangle has all sides of equal length and all three angles equal to 60 degrees. Do we get a sense of this from the statements given ? Statement 1 tells us that two of the angles are equal. We know that ABC is at best an isosceles triangle. We have no other information to lead us to conclude that ABC is equilateral or not. Statement 1 alone is, therefore, not sufficient. Our choices are B, C or E. Statement 2 tells us that one of the angles is 60 degrees. The other two angles could have a range of possible values such that they add up to 120 degrees. Not good enough. Our choices narrow to C or E. When we combine the two statements, we can conclude that if z=60 and x=y, then x and y must also be equal to 60 degrees. Bingo. We pick C as the answer. 7. If w + z = 28, what is the value of wz? (1) w and z are positive integers (2) w and z are consecutive odd integers Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 tells us that w and z can take on a range of possible values such that they add up to 28. For instance, if w is 1, then z is 27; if w is 2, then z is 26; if w is 10, then z is 18 and so on. Not good enough for a unique value for the variables. Our choices are B, C, or E at this stage of the game. Statement 2 tells us that w and z are consecutive odd integers. Can we think of two consecutive odd numbers such that they add up to 28? We can: 13 and 15. Since we are interested in the value for wz, it does not matter what value w takes and what value z takes. We know that if w is 13, then z is 15; if w is 15, then z is 13. In either case, the product wz will have a unique value. Statement 2 alone is sufficient and statement 1 alone is NOT sufficient. So we pick B as the answer.

8. Is ax = 3 - bx ? (1) x (a+b) = 3 (2) a = b = 1.5 and x = 1 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Once again, we have an algebraic equation staring at us. Let us play around with the equation and manipulate it into another form: ax = 3 - bx ? or ax + bx = 3 ? Or x (a + b) = 3 ? Statement 1 tells us exactly what we are looking for. That x(a+b) = 3. Good enough to answer our question precisely. Statement 1 alone is sufficient. Our choices are A or D. Statement 2 gives us values for a, b, and x. If we plug these values into the expression x (a+b), we get 1 (1.5 + 1.5 ) = 3. Once again, the information in statement 2 alone is also sufficient to answer the question. Therefore, we pick D as the answer. 9. What is the value of integer x? (1) x is a prime number (2) 30 < x < 38 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 tells us that x can have any prime number value and does not give us a unique value for x. Therefore, statement 1 alone is not sufficient to answer the question. Our choices are B,C, or E. Statement 2 tells us that x can have any value in the range 31 through 37. Not precise enough. Statement 2 alone is also not sufficient to answer the question precisely. Our choices narrow to C or E. When we combine the statements 1 and 2, we get two possible values for x : 31 and 37, both prime integers . Once again, we have more than one possible value from the given set of statements. Not good enough. We pick E.



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10. While on a straight road, car X and Car Y are traveling at different constant rates. If car X is now 1 mile ahead of car Y. How many minutes from now will car X be 2 miles ahead of car Y ? (1) Car X is traveling at 50 miles per hour and car Y is traveling at 40 miles per hour. (2) 3 minutes ago car X was 1/2 mile ahead of car Y. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) To determine the number of minutes it takes for car X to get 1 more mile ahead of Car Y, we need the relative speed (difference of the two speeds) of car X with respect to car Y. Statement 1 gives us the exact same information we are seeking. Since the relative speed of Car X is 10 miles with respect to Car Y, we know that it will take another 6 minutes for Car X to gain a mile ahead of Car Y. Statement 1 alone is sufficient. Our choices are A, or D. Statement 2 gives us information about the relative speed in terms of time. Since the cars are traveling at constant rates, if car X was 1/2 mile ahead of car Y, 3 minutes ago, then we know that it takes car X 6 minutes to gain an extra mile ahead of car Y. Statement 2 alone is also sufficient to answer the question. We pick D. 11. In what year was Ellen born? (1) Ellen’s brother Pete, who is 1 1/2 years older than Ellen, was born in 1956. (2) In 1975 Ellen turned 18 years old. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 tells us that if Pete was born during January - June of 1956, then Ellen was born in 1957. If Pete was born during July-Dec. 1956, then Ellen was born in 1958. Since we do not have any indication of when Pete was born in terms of month, we conclude that statement 1 alone is not sufficient to answer the question. Our choices are B, C, or E. Statement 2 tells us that Ellen must have been born in 1957 so that she will turn 18 in 1975. Good enough to answer the question. Statement 2 alone is sufficient and statement 1 alone is not sufficient. We pick B.

12. What is the number of female employees in Company X? (1) If company X were to hire 14 more people and all of these people were females, the ratio of the number of male employees to the number of female employees would then be 16 to 9. (2) Company X has 105 more male employees than female employees. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 tells us that ratio of Males (M) to Females+14 (F+14) is 16/9. This does not mean much because this ratio will be valid for values of M=32 and F = 4 or M=160 and F = 76. Since we get a range of possible values, we have to conclude that statement 1 alone is not sufficient. Our choices are B, C, or E. Statement 2 tells us that M = F + 105. Again, this gives rise to a range of unlimited values for F and M. Not good enough. Our choices Narrow to C or E. When we combine the two statements, we notice that we have two equations with two variables. We can solve the two equations to get the values for M and F. Good enough. We pick C. The equations are: M / (F+14) = 16/9 Or 9M = 16F + 224 ............ (1) M = F + 105 ............(2) 2 variables and 2 independent equations. Good enough.



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13. Is the integer x divisible by 36 ? (1) x is divisible by 12 (2) x is divisible by 9. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 tells us that x can have values that are multiples of 12, values such as 12,24,36,48,etc. If x is 12, 24, 48, 60 etc. Then x is not divisible by 36. If x is 36,72, 108, and so on, then x is divisible by 36. Since we do not know the true value of x, we have to conclude that statement 1 alone is not sufficient. Our choices are B, C, or E at this stage. Statement 2 tells us that x is a multiple of 9. Not good enough. Because if x is 9, then it is not divisible by 36. If it is 36, then it is . Therefore statement 2 alone is also not sufficient. Our choices are C or E. When we combine the two, we notice that the least common multiple of 12 and 9 is 36 which means that x must have values that are multiples of 36. Good enough to answer the question with the combined information. We pick C. 14. What is the average (arithmetic mean) of j and k? (1) The average of j + 2 and k + 4 is 11 (2) The average of j, k, and 14 is 10. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 tells us that j+2+k+4 = 22. We can find out what the value for j+k / 2 will be. (8 . You are not required to solve for the value in the test. Do not waste time solving for the average value. It is enough for you to recognize that the equation is amenable to a unique solution) . Therefore, statement 1 alone is sufficient to answer the question. Our choice is A or D. Statement 2 tells us that j+k+14 = 30. Once again, we can compute the value for (j+k) / 2 from this equation. Statement 2 alone is also sufficient to answer the question with a unique value. We pick D.

15. What is the value of a - b? (1) a = b + 4 (2) (a - b) (a - b) = 16 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 gives us the value for a - b as 4. A unique value and good enough information. Our choice is either A or D. Statement 2 tells us that a-b is either +4 or -4 . More than one possible value. Not good. We pick A. 16. Is rst = 1 ? (1) rs = 1 (2) st = 1 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 does not give us any clue as to the value for t. T could have any value, including fractional value. We cannot answer the question posed using information in statement 1 alone. Our choices are B, C , or E. Statement 2 does not give us a value for r. Once again, r can have any number of values, including fractional. Statement 2 alone is also not sufficient to answer the question. Our choice is either C or E. When we combine the two information , we can conclude only the following: That r, s, and t are each equal to 1, in which case rst will be equal to 1 or that r and t are equal in value while s is the reciprocal of the value of r or t. For instance if r and t are equal to 1/3 and s is equal to 3, then the conditions specified in statements 1 and 2 will be met but the value for rst will be 1/3. Since we get a range of possible values from the two information given, we have to conclude that we cannot answer the question precisely even if we combine the two statements. We have to pick E.



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17. In a certain office, 50 percent of the employees are college graduates and 60 percent of the employees are over 40 years old. If 30 percent of those over forty have master’s degrees, how many of the employees over forty have master’s degree? (1) Exactly 100 of the employees are college graduates. (2) Of the employees forty years old or less, 25 percent have master’s degrees. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) With word problems this long, you must reduce the information to equations you can work with. Let N be the number of employees in the company. We have College Grads = 0.5N ...... (1) 40 + = 0.6N.........(2) 40+ & Master’s = 0.3. (0.6N) = 0.18N.... (3) We are required to get a unique value for 0.18N. If the statements give us numbers pertaining to any of these categories, we can work with that. Let us examine statement 1. Statement 1 tells us that the number of college grads is equal to 100. Which means that 0.5N=100. We can determine the value for N from this equation and plug the value in 0.18N to get the information sought. Statement 1 alone is sufficient to answer the question. Our choice is either A or D. Statement 2 tells us that 25% of 0.4N = 0.1N employees are under 40 years of age and have master’s degree. This is information pertaining to a new category and the information is in terms of N. Not good enough to get a number value for 0.18N. Statement 2 alone is not sufficient We pick A.

18. Is xy < 6 ? (1) x < 3 and y < 2 (2) 1/2 < x < 2/3 and y.y < 64 Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 tells that the values of x can be 2 or -10. Similarly, the values of y can be 1.5 or -5. If x is 2 and y is 1.5, then xy is 3, which is less than 6. On the other hand, if x is -10 and y is -5, then xy is +50, which is greater than 6. Since we do not know the true values for x and y, we have to conclude that statement 1 alone is not sufficient to answer the question. Our choices must be B,C, or E. Statement 2 gives a range within which x values must lie and also tells us that y is either +8 or -8. Since the greatest value that x can have is less than 2/3, with a value of y = 8, we determine that the maximum possible value for xy is 8.2/3 or 5 1/3. Which is less than 6. Therefore, we conclude that the information in statement 2 alone is sufficient to answer the question. We pick B. 19. Is x equal to 2? (1) x, y and x + y are prime numbers (2) y is odd Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 tells us that either x or y must be equal to 2. Why is this so? We know that the only even prime number is 2 and all the other prime numbers are odd numbers. If x+y is a prime number, then it must be odd. How do we get an odd number? By adding an even number to an odd number. Therefore, when we add x and y, two prime numbers, we conclude that one of them must be an even value so that the resulting number x+y will be odd. The only conclusion we can make from statement 1 is that either x is 2 or y is 2. Not good enough. Our choices are B, C, or E. Statement 2 tells us that y is an odd number. We have no information pertaining to x in statement 2. Statement 2, therefore, is not sufficient. Our choice is either C or E. When we combine the two statements, we reason that either x or y is equal to 2, and if y is an odd number, then x must be equal to 2. Bingo. We pick C.



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20. In 1979 Mr. Jackson bought a total of n shares of stock X and Mrs. Jackson bought a total of 300 shares of stock X. If the couple held all of their respective shares throughout 1980 , and Mr. Jackson’s 1980 dividends on his n shares totaled $150, what was the total amount of Mrs. Jackson’s 1980 dividends on her 300 shares ? (1) In 1980 the annual dividend on each share of stock X was $0.75 (2) In 1979, Mr. Jackson bought a total of 200 shares of stock X. Your Choice: A ( ) B ( ) C ( ) D ( ) E ( ) Statement 1 tells us that each share received $0.75 in dividends. We can determine how many dollars Mrs. Jackson received on her 300 shares. Statement 1 alone is sufficient to answer the question. Our choice is either A or D. Statement 2 tells that n = 200 which means that if Mr. Jackson received $150 in dividend income, then each share was worth $0.75 in dividends. We can determine how many dollars Mrs. J received on her holding of 300 shares. Statement 2 alone is also sufficient to answer the question. We must choose option D.



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DATA SUFFICIENCY EXERCISE 2- Answer Keys

20 Questions -- 25 Minutes CHOOSE: 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 statement TOGETHER are sufficient, but NEITHER statement ALONE IS sufficient. D EACH statement ALONE is sufficient E Statements (1) and (2) TOGETHER are NOT sufficient. 1. If the list price of a new car was $12,300, what was the cost of the car to the dealer? (1) The cost to the dealer was equal to 80 percent of the list price (2) The car was sold for $11,070, which was 12.5 percent more than the cost to the dealer. A ( ) B ( ) C ( ) D ( ) E ( )

2. If p, q, x, y, and z are different positive integers, which of the five integers is the median? (1) p + x < q (2) y < z A ( ) B ( ) C ( ) D ( ) E ( ) 3. A certain employee is paid $6 per hour for an 8-hour workday. If the employee is paid 1 1/2 times this rate for time worked in excess of 8 hours during a single day, how many hours did the employee work today? (1) The employee was paid $18 more for hours worked today than for hours worked yesterday. (2) Yesterday the employee worked 8 hours A ( ) B ( ) C ( ) D ( ) E ( )

Either statement alone is sufficient to answer the question. Statement 1 tells us that cost is 80% of list price ( a known value). Good enough. Our choices are A or D. Statement 2 tells us that the selling price was 12.5% higher than cost. Good enough to answer the question precisely. We pick D.

To get the median value, we require information pertaining to the order of values for the integers p, q, x, y and z. Statement 1 tells us that q is bigger than both p and x. But we do not know whether p is greater than x or not. We also do not know about the other two values, y and z. Not good enough. Our choices are B,C or E. Statement 2 tells us that y is less than z. Not good enough because we do not know about the other values. Our choices are C or E. When we combine the two statements, we are not any wiser because we do not know in what order we should place x, p, y, and z. We must give up and pick E as the answer.

Statement 1 alone is not sufficient to answer the question because because $18.00 could mean 3 hours of regular time or 2 hours of extra time. And we do not know how many hours the employee worked “yesterday”. Our choices are B, C or E. Statement 2 tells us that the employee worked 8 hours yesterday. Not good enough on its own. We cannot answer the question: “How many hours did the employee work today?” from this information. Choices are C or E. When we combine the two statements, we know that the employee worked 8 hours yesterday and that the employee worked 2 “overtime” hours today for a total of 10 hours. We pick C.



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4. If n is a member of the set (33, 36, 38, 39, 41, 42), What is the value of n ? (1) n is even (2) n is a multiple of 3 A ( ) B ( ) C ( ) D ( ) E ( ) 5. What is the value of x ? (1) 2x + 1 = 0 (2) (x + 1 )² = x² A ( ) B ( ) C ( ) D ( ) E ( )

6. In the figure below, what is the length of AD? A_________B_________C_________D (1) AC = 6 (2) BD = 6 A ( ) B ( ) C ( ) D ( ) E ( ) 7. A retailer purchased a television set for x percent less than its list price, and then sold it for y percent less than its list price. What was the list price of the television set? (1) x = 15 (2) x - y = 5 A ( ) B ( ) C ( ) D ( ) E ( )

Statement 1 tells us that n is even. We have 3 even values in the set: 36, 38, and 42. Not a unique solution. We conclude that statement 1 alone is NOT sufficient. Our choices are B, C or E. Statement 2 tells us that n is a multiple of 3: We have 4 values: 33,36,39, and 42 for n. Not a unique solution either. Our choices narrow to C or E. When we combine the two statements, we conclude that n must be either 36 or 42. We do not get a unique value for n even with the combined information. We give up and pick E.

Statement 1 is good enough to get the value for x. (X = - ½ ) . Our choices are A or D. Statement 2 must be manipulated to the form: x² + 2x + 1 = x² or 2x + 1 = 0 or x = - ½ . Good enough. Either statement alone is sufficient to answer the question definitively. We pick D.

Statement 1 alone is NOT sufficient because we do not get a value for AD from the given information. Choices are B, C or E at this point. Statement 2 is not useful either because we still do not get a clue as to the value for AD. Choices narrow to C or E. When we combine the two statements, we are no better off because we do not know the relationship of line segments AB, BC and CD to use the two statements to good use. We pick E.

Statement 1 tells us the discount percentage when the retailer bought the television. That is not good enough to get the list price. Choice B, C or E. Statement 2 gives us the difference in the discount rates. Once again, not sufficient to answer the question. Even if we combine the two statements, we are no better off. We need either the purchase price or the selling price along with discount %. We pick E.



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8. Is x² greater than x ?

(1) x² is greater than 1 (2) x is greater than -1 A ( ) B ( ) C ( ) D ( ) E ( ) 9. What is the value of r/2 + s/2 ? (1) (r + s)/2 =5 (2) r + s = 10 A ( ) B ( ) C ( ) D ( ) E ( ) 10. If x, y, and z are numbers, is z = 18 ? (1) The average (arithmetic mean) of x, y, and z is 6 (2) x = -y A ( ) B ( ) C ( ) D ( ) E ( )

11. B C A The circular base of an above ground swimming pool lies in a level yard and just touches two straight sides of a fence at points A and B, as shown in the figure above. Point C is on the ground where the two sides of the fence meet. How far from the center of the pool’s base is point A ? (1) The base has area 250 square feet (2) The center of the base is 20 feet from point C. A ( ) B ( ) C ( ) D ( ) E ( )

In this problem, we are required to find the value for the radius of the circular base from the information given in statements 1 and 2. Statement 1 tells us that the area of the circular base is 250 sq. Ft. Or π . R² = 250. Good enough. We can get the value for R. Our choices are A or D. Statement 2 is not much use because we get the following picture: B C 20’ A We have the hypotenuse value in the above right triangle BCO, but we need one other information such as an anlge to determine the radius. We conclude that statement 2 alone is not sufficient and pick A.

For x² to be greater than x, |x| must be greater than 1. If 0 <x <1, then x² will be less than x. Statement 1 tells us that |x| is greater than 1. Good enough. A or D. Statement 2 tells us that x could be a negative fraction such as -1/4, in which case x² is greater than x. If x is 1/4, then x will be greater than x². If x is greater than 1, x²will be greater than x. Too many possibilities. Not a unique solution from statement 2. We give up on statement 2 and pick A.

We are required to get the average value for r and s. Statement 1 gives us that. Good enough. A or D. Statement 2 also lets us calculate the average value for r and s. Either statement alone is sufficient. We pick D.

Statement 1 tells us that (x +y +z) ÷ 3 = 6 or x + y + z = 18 . Not much use, because we cannot determine the value for z from this equation. We have a range of possible values for z, and not a unique one. Our choices are B,C, or E. Statement 2 tells us that the absolute values of x and y are equal and that x and y have the opposite



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12. In 1979 Mr. Jackson bought a total of n shares of stock X and Mrs. Jackson bought a total of 300 shares of stock X. If the couple held all of their respective stocks throughout 1980, and Mr. Jackson’s 1980 dividends on his n shares totaled $150, what was the total amount of Mrs. Jackson’s 1980 dividends on her 300 shares ? (1) In 1980 the annual dividend on each share of stock X was $ 0.75 (2) In 1979 Mr. Jackson bought a total of 200 shares of stock X A ( ) B ( ) C ( ) D ( ) E ( ) 13. If Sara’s age is exactly twice Bill’s age, what is Sara’s age? (1) Four years ago, Sara’s age was exactly 3 times Bill’s age. (2) Eight years from now, Sara’s age will be exactly 1.5 times Bill’s age. A ( ) B ( ) C ( ) D ( ) E ( )

14. What is the value of (x/yz) ? (1) x = y/2 and z = 2x/5 (2) x/z = 5/2 and 1/y = 1/10 A ( ) B ( ) C ( ) D ( ) E ( ) 15. An infinite sequence of positive integers is called an “alpha sequence” if the number of even integers in the sequence is finite. If S is an infinite sequence of positive integers, is S an alpha sequence? (1) The first ten integers in S are even. (2) An infinite number of integers in S are odd. A ( ) B ( ) C ( ) D ( ) E ( )

Statement 1 alone is not sufficient because we get x/yz = 5/9x from the information specified. B, C or E Statement 2 tells us that x/yz = 5/20 or 1/4. Good enough. We pick B as the answer.

To qualify as an alpha sequence, S must have a finite number of even integers. Statement 1 does not tell us how many integers are even in the sequence. We simply know that the first 10 are even. That is not sufficient. B, C or E. Statement 2 tells us that S contains an infinite number of odd integers. Not useful information. We require information about even integers. C or E. Even when we combine the two statements, we are clueless about how many even integers there are in S. We cannot, therefore, answer the question. We pick E.

We need to know the value of the dividend to answer the question. Statement 1 tells us precisely that: That each dividend was worth $0.75. Good enough. A or D. Statement 2 tells us that Mr. J had 200 shares. The stem tells us that he received $150 in dividends. We can conclude that each share paid $0.75 in dividends, and answer the question posed. Either statement is sufficient alone and we pick D.

The information in the stem is that Sara= 2. Bill. Statement 1 tells us that S-4 = 3(B-4). We have two equations and we can solve for Sara’s age. A or D Statement 2 tells us that S + 8 = 1.5(B+8) Good enough in conjunction with the stem information. Either statement alone is sufficient to answer the question, and we pick D.



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16. If xy > 0, does (x-1)(y-1) = 1 ? (1) x + y = xy (2) x = y A ( ) B ( ) C ( ) D ( ) E ( )

17. After winning 50 percent of the first 20 games it played, Team A won all of the remaining games. What was the total number of games that Team A won ? (1) Team A played 25 games altogether. (2) Team A won 60 percent of all the games it played. A ( ) B ( ) C ( ) D ( ) E ( ) 18. @ + $ = * In the addition problem above, each of the symbols @, $, and * represents a positive digit. If @ < $, what is the value of $? (1) * = 4 (2) @ = 1 A ( ) B ( ) C ( ) D ( ) E ( )

Statement 1 tells us that @ + $ = 4. Knowing that @ < $, we can conclude that @ must be 1 and $ must be 3. Good enough. A or D. Statement 2 tells us that @ = 1. Not good enough. $ can have a wide range of values. We conclude that statement 1 alone is, but statement 2 alone is NOT, sufficient and pick A.

When we get an algebraic expression such as this one, we will try to expand it.

(X-1)(Y-1) = XY-(X+Y) + 1. Statement 1 tells us that X+Y = XY. Which means that tbe above expression is equal to 1. A or D Statement 2 tells us that x = y, which does not give a unique value for the expression. (When you replace x with y, you get (y-1)2 and you not know what value it has.) We pick A.

We know that the team A won 10 of the 20 games it played. We need to know how many more it won, knowing that it won all of the remaining games. Statement 1 tells us that the team played 25 games. Good enough. A or D. Statement 2 tells us that the team won 60% of all the games it played. If n were the number of games, then the team won 0.6n which is equal to 10 + n - 20. We can determine how many games the team A won from both statements independently. We pick D.



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19. Cancellation Fees Schedule Days prior to Percent of Departure Package Price 46 or more 10% 45 - 31 35% 30 - 16 50% 15 - 5 65% 4 or fewer 100% The table above shows the cancellation fee schedule that a travel agency uses to determine the fee charged to a tourist who cancels a trip prior to departure. If a tourist canceled a trip with a package price of $1,700 and a departure date of September 4, on what day was the trip canceled? (1) The cancellation fee was $595 (2) If the trip had been canceled one day later, the cancellation fee would have been $255 more. A ( ) B ( ) C ( ) D ( ) E ( )

20. Is 5K less than 1,000? (1) 5K+1 is greater than 3,000 (2) 5K-1 is 500 less than 5 to the power of k. A ( ) B ( ) C ( ) D ( ) E ( )

From statement 1, we know that the cancellation amount was 35% of the tour charge. Which means that cancellation took place during 30-45 days. That is a broad range of days and not precise enough to answer the question. B, C, or E. Statement 2 tells us that if the cancellation had taken place one day later, then the additional fee would be 15% of the tour fee. Which means that the cancellation occurred on the 31st or the 16th day. Not precise enough. C or E. When we combine the two statements, we can conclude that the cancellation must have occurred on day 31. We pick C.

We need to know whether k is less than 5 because 55 = 3125 and 54 = 625. Statement 1 tells us that 5k+1 > 3000 Or 5. 5k > 3000 or 5k > 600 The fact that 5k > 600 does not tell us definitively that 5k < 1000 . Greater than 600 could mean 900 or 5000 Statement 1 is not good enough for a unique determination of the value for 5k. We must, therefore, move on to examine statement 2. Our choices are B, C or E at this point in time. Statement 2 tells us that 5k-1 = 5k – 500 or 5-1.5k = 5k – 500 or 1/5 . 5k = 5k – 500 Let us multiply both sides by 5 to get rid of 1/5 on the left. We get 5k = 51.5k – 2500 Let us group the 5k terms together to get: 4. 5k = 2500 or 5k = 625 = 54 We conclude that 5k < 1000 because we know that 5k = 625. Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question definitively. We must pick B as the answer.



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DATA SUFFICIENCY EXERCISE 3-Answer Keys

20 Questions -- 25 Minutes CHOOSE: A Statement (1) alone is sufficient (to give a unique value for the information sought) but statement (2) alone is not sufficient B Statement (2) alone is sufficient but statement (1) alone is not sufficient. C BOTH statement TOGETHER are sufficient, but NEITHER statement ALONE IS sufficient. D EACH statement ALONE is sufficient E Statements (1) and (2) TOGETHER are NOT sufficient. 1. The results of a certain experiment included 6 data values that were all multiples of the same number c, namely, c, 8c, 2c, 5c, 4c, and 4c. Was the average (arithmetic mean) of the 6 data values greater than 8?

(1) c < 4 (2) c > 2

A ( ) B ( x ) C ( ) D ( ) E ( )

2. If n is an integer, is n + 2 a prime number ? (1) n is a prime number (2) 30 < n < 40

A ( ) B ( ) C ( X ) D ( ) E ( )

Let us compute the average of the six values by adding the values and dividing the sum by the number of values. The average in terms of c = 1/6 (c+8c+2c+5c+4c+4c) = 24c/6 = 4c The question is: Is 4c definitely more than 8 or definitely NOT more than 8? Or, paraphrasing the question, is c definitely more than 2 or definitely not greater than 2? Statement 1 alone is not sufficient because if c is less than 4, it could be 3, 2 ½ , 2, 1 ½ and so on. In some scenarios, c is greater than 2, and in others, it is not. We are not logically certain whether c is definitely greater than 2 or not. Statement 2 alone is sufficient to answer the question. If c is greater than 2, according to statement 2, then that ought to definitively answer the question is 4c > 8. We must choose option B. Try to simplify the stem question so that you can have a better handle on the statements presented to you.

We need some additional information about n in order to determine whether n+2 is a prime number or not. Statement 1 tells us that n is a prime integer. How many prime integers can we think of? Lots and lots. Is this a unique solution? Hardly. Statement 1 is not sufficient by itself. B,C, or E. Statement 2 tells us that n lies in the range between 30.0001 and 39.999, inclusive. Once again, a range of possible values for n. Not good for a unique solution. C or E. When we combine the two statements, we notice that n is a prime integer and lies in the range : 30 < n < 40. How many prime numbers can we find in this range? Two: 31 and 37. We notice that n can be either of these values. But the question is not about n, but about n+2. For either of these values for n: 31 or 37, (n+2) is not a prime number. (Because 33 and 39 are not prime numbers.) Therefore, we conclude that we can answer the question definitively by combining the two statements. Notice how we get 2 possible values for n, but a unique solution for (n+2) in terms of a definite answer confirming that n+2 is NOT a prime integer. We must pick choice C.



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3. If t is not equal to zero, is r a positive number? (1) rt = 12 (2) r + t = 7

A ( ) B ( ) C ( X ) D ( ) E ( ) 4. If x is an integer, is y an integer ? (1) The average (arithmetic mean) of x, y, and y - 2 is x . (2) y - x = 1

A ( ) B ( ) C ( ) D ( X ) E ( )

4. How many minutes does it take for a circular shaped illumination to quadruple its area? (1) The initial diameter of the circular shaped illumination is 10 in. (2) The circumference of the circular shaped illumination increases at the rate of pi. In/minute

A ( ) B ( ) C ( ) D ( ) E ( )

Statement 1 tells us that (x+y+y-2) = x 3 Or x + 2y - 2 = 3x or 2(y-1) = 2x or y-1 = x What do we conclude from this? That x and y are consecutive integers. If x is an integer, then y MUST be an integer. Our choices are A or D. Statement 2 tells us that y - x = 1 or y = x+1 Once again, what this means is that x and y are consecutive integers. If x is an integer, so is y. Statements 1 and 2 independently are sufficient to answer the question. We must pick D. Notice how we set up an equation using the information in statement 1 before concluding that the information is useful for a unique answer. Be sure to express in mathematical terms any information in verbal form, before you conclude one way or the other.

When will the area of a circular shaped object quadruple? When the radius doubles, because the area is a function of (radius)2. To answer the question, we need to know the initial value of the radius, and the rate at which the radius is changing. Statement 1 tells us what the radius is, but does not tell us at what rate the radius is increasing. We cannot answer the question using statement 1 alone. Our choices are B, C or E. Statement 2 tells us at what rate the radius is increasing. If the circumference is increasing at the rate of pi inches/ minute, then we know that the radius is increasing at the rate of 1/2 inch/min. We get one half the information we are seeking. Our choices are C or E. We can answer the question by combining the two statements, because we get the radius information from 1, and the rate of change from 2. We must pick C.

We learn that t is not equal to 0, but it could be positive or negative, integer or “not-an-integer,” We need to get a handle on r. Statement 1 tells us that rt = 12. Is this good enough to let us know what r is about? Hardly. R and t could be both negative or both positive, and we cannot answer the question definitively. Our choices narrow to B, C, or E. Statement 2 tells us that r+t = 7. Is this good enough information to conclude that r is positive? No. Why? Because r could be -2 and t could be +9 or r could be +11/3 and t could be +52/3. We get solutions that are “all over the map”. Our choices are C or E. When we combine the two statements, we notice that we have two variables, and two independent equations. We can determine what r is all about. C is the answer. (In fact, the values for r and t are 4 and 3 or 3 & 4, both positive).



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5. The inside of a rectangular carton is 48 centimeters long, 32 centimeters wide, and 15 centimeters high. The carton is filled to capacity with k identical cylindrical cans of fruit that stand upright in rows and columns as shown below. If the cans are 15 centimeters high, what is the value of k ? (1) Each of the cans has a radius of 4 centimeters. (2) 6 of the cans fit along the length of the carton.

A ( ) B ( ) C ( ) D ( X ) E ( )

We know the dimensions of the box, and if we know the radius of the cans, we can answer the question. Statement 1 tells us that each can has a radius of 4 cm. We can determine how many cans of 8 inches diameter can be placed along the length (48 cm), and width (32 cm). Good enough. Our choices are A or D. Statement 2 tells us that 6 cans can be placed along the length 48 cm. This means that the diameter of each can is 8 cm. We can determine how many we can place along the width (32 cm), and answer the question definitively. Either statement alone is sufficient to answer the question. D is it.



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6. If ™ is a positive integer, is the value of b - a at least twice the value of 3k - 2k ? (1) a is equal to 2 to the power of ™+1 and b is equal to 3 to the power of ™+1 (2) ™ = 3

A ( X ) B ( ) C ( ) D ( ) E ( ) 7. If P and Q are each circular regions, what is the radius of the larger of these regions? (1) The area of P plus the area of Q is equal to 90 pi. (2) The larger circular region has a radius that is 3 times that of the smaller circular region.

A ( ) B ( ) C ( X ) D ( ) E ( ) 8. Is a+ b divisible by 8 ?

(1) When a is divided by 8, it leaves a remainder of 3. (2) When b is divided by 8, it leaves a remainder of 5.

A ( ) B ( ) C ( X ) D ( ) E ( )

We need to determine whether b-a μ (3k - 2k) Statement 1 tells us that a = 2k+1 and b = 3k+1 So. What is b - a ? b - a = 3k+1 - 2k+1 We can write b - a = 3. 3k - 2.2k

Or b - a = 2 { 1.5 (3k - 2k)} What did we do here? Simply factored the expression, so that we can determine if (b-a) is twice something or not. What do we see here? That (b-a) is indeed more than twice 3k - 2k. Our choices are A or D. Statement 2 gives us a value for k, but does not tell us what a and b are. We cannot determine what (b-a) will be in the absence of any information about a and b. We conclude that statement2 alone is not sufficient, but statement 1 alone is sufficient. We must pick A.

Statement 1 tells us that pi.rp2 + pi.rq

2 = 90.pi Or rp

2 + rq2 = 90

Is this good enough information to answer the question? Nope. We need to know the values for these radii. We conclude that statement 1 alone is not sufficient. Our choices narrow to B, C or E. Statement 2 tells us that one of the radius is 3 times the other value. If rp is the larger of the two, then rp = 3.rq (it does not matter which is the larger of the two radii) Is this good enough? No. We need to know the number value for the radii. Our choices are C or E.

h bi h h i d d i d

Statement 1 tells us that a = 8.p + 3, where p represents all the other factors of (a-3). Why do we write the information in this manner? Let us say that a = 19. We can write 19=8.2 + 3 We notice that we do not have any information about b in this statement. Our choices are B, C or E. Statement 2 tells us that b = 8.q + 5. Is this statement sufficient to answer the question about (a+b)? We don’t think so. C or E. When we combine the two statements we can write that (a+b) = 8(p+q) + 8. Is (a+b) divisible by 8? You bet. Why is that? Because (a+b) = p+q+1 8 p+q+1 is a whole number and we must conclude that 8 is a factor of (a+b). If the problem was specified in such a manner that we got an expression for a+b = 8(p+q) + 7, we can still answer the question in the negative: That (a+b) is not divisible by 8. As long as the information is good for a definite answer, the information alone or combined is good enough. We pick C in this problem.



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9. What is the ratio of x : y : z ? (1) z = 1 and xy = 32 (2) x = 2y and y = 4z

A ( ) B ( X ) C ( ) D ( ) E ( ) 10. If 3m = 5n , what is the value of m + n ? (1) 2m + n = 26 (2) If the value of m were increased by 2, then it will be twice the value of n.

A ( ) B ( ) C ( ) D ( X ) E ( )

11. On a Friday Morning, a certain machine ran continuously at a uniform rate to till a production order. At what time did it completely fill the order that morning ? (1) The machine began filling the order at 9:30 a.m. (2) The machine had filled 1/2 of the order by 10:30 a.m. and 5/6 of the order by 11:10 a.m.

A ( ) B ( ) C ( ) D ( ) E ( )

We have some information specified in terms of m and n, along with the question. We must use this information in conjunction with information in statements 1 and 2 to determine whether we can answer the question or not. Statement 1 tells us that 2m+ n = 26. When we combine this with 3m = 5n, we have unique solution for m and n staring us in the face. A or D Statement 2 tells us that m+2 = 2n. Again, when we combine this information with 3m = 5n, we can determine the values for m and n, and answer the question definitively. D is the answer.

The question is: When did the machine FINISH filling the order? Statement 1 tells us the start time, but does not tell us at what rate the machine is working. We cannot determine at what time the machine completed work. Statement 1 is useless, and our choices are B, C or E. Statement 2 tells us that the machine, working at a constant rate, completed (5/6 - 1/2) = 1/3 of the order in 40 minutes. We know that the machine had done 5/6 of the order at 11:10 am, and needs to finish 1/6 of the order. If it does 1/3 of the order in 40 minutes, the machine will take 20minutes more to do 1/6 of the order. We can conclude that the machine will complete the order at 11:30 am. Statement 2 alone is sufficient, but 1 alone is not. We must pick B.

Statement 1 is not sufficient because we do not know what x and y are. If xy = 32, then x and y could be 1 and 32, or 2 and 16, or 4 and 8, or 8 and 4, or 16 and 2, or 32 and 1. Not a unique solution here. Our choices are B, C or E. Statement 2 tell us exactly what we need to know: A relationship between x , y and z. If x = 2y and y = 4z, then we can write x = 2y = 8z. We know that x:y:z must be 8:4:1. Statement 2 alone is sufficient to answer the question, and we must pick B as the answer.



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12. A ladder is propped up against a vertical wall such that the bottom of the ladder makes an angle of 60 degrees to the ground. The ladder slips down and the bottom of the ladder moves away from the wall. In this position, the bottom of the ladder makes an angle of 45 degrees to the ground. How much farther did the ladder move from the wall when it slipped? (1) The ladder is 10 meters long (2) The bottom of the ladder was 5 meters from the wall in its original position.

A ( ) B ( ) C ( ) D ( X ) E ( )

13. Can the positive integer n be written as the sum of two different positive prime numbers ? (1) n is greater than 3 (2) n is odd.

A ( ) B ( ) C ( ) D ( ) E ( X )

Statement 1 tells us that n > 3. If n is 7, then we can write 7 = 2 + 5, the sum of two different prime integers. If n is 11, then we cannot write 11 as the sum of two different positive prime numbers. Since we do not know what specific value n has, we cannot answer the question definitively from the statement 1 information. B, C or E. Statement 2 tells us that n is odd. We have seen that for values 7, and 11, we get two different scenarios: If n is 7, then we can write the value as the sum of two primes; If n is 11, then we cannot. Statement 2 also gives us “all-over-the- map” solution. Not good enough. C or E. When we combine the two statements, we notice that we have covered all bases with 7 and 11, both values greater than 3, and get conflicting answers. Even the combined information is not good for a unique answer. We must pick E.

This is the picture emerging from the information in the problem: We need to know the value for “x” in the diagram. 45o 60o x 5 m 1 We also notice that the triangles are 60-30-90 before the ladder slipped, and 45-45-90 after the ladder slipped. If we know the value for any one side, we can determine all the other side values because the sides of a 45-45-90 triangle are in proportion 1:1:\/2 and those of 30-60-90 are in proportion 1:\/3:2. Statement 1 tells us that the bottom of the ladder is 5m from the wall. We know that the length of the ladder is 10 feet, and in the new position (45-45-90), the bottom of the ladder must be 10/\/2 = 5\/2 meters from the vertical wall. The slippage, x = 5\/2 - 5 = 5(\/2 -1) = 2 feet(approx). Statement 1 alone is sufficient. Our choices are A or D. Statement2 tells us that the length of the ladder is 10 feet. We can use this information along with the properties of 30-60-90 and 45-45-90 triangles to determine the value for x in the diagram. Statement 2 is also independently sufficient to answer the question. We must pick D as the answer.



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14. What is the last number in a set comprising six consecutive odd integers ? (1) The average (arithmetic mean) of the six consecutive odd integers is 36. (2) Twice the average (arithmetic mean) of the six consecutive odd integers is equal to the sum of the first and the last number in the set.

A ( ) B ( ) C ( ) D ( ) E ( )

15. Town T has 20,000 residents, 60 percent of whom are female. What percent of the residents were born in Town T ? (1) The number of female residents who were born in Town T is twice the number of male residents who were not born in Town T. (2) The number of female residents who were not born in Town T is twice the number of female residents who were born in Town T.

A ( ) B ( ) C ( X ) D ( ) E ( ) This is a problem best dealt with using a matrix that will look like this:

MEN WOMEN TOTAL BORN 6000

c 2N1 X2 10000c

4000 2

NOT BORN

N1 2000

c 2X2

8000

2 10000

c

TOTAL 8000 12,000 20,000

We need to know how many men and women were born in Town T. Statement 1 tells us that Men/Not Born is 1/2 the number for Women/Born. The information is shown in the matrix with the suffix 1. But this information is not sufficient to answer the question. Statement 2 tells us that number of women born in town T is 1/2 the number not born there. This information helps us determine that the number of women born in Town T is 1/3 of 12,000 = 4000; number not born in Town T is 2/3 the total number for women = 2/3 or 12,000 = 8000. But this information alone does not tell us how many men were born in Town T. When we combine the two statements, we can determine that the number of men born in Town T is 6000, and the total number of men and women born in town T is 10000. Choice C.

How do we represent 6 consecutive odd integers?2n +1, 2n+3, 2n+5, 2n+7, 2n+9, 2n+11What is the sum of these integers?12n + 36.What is the average of these integers?12n+36

6Or the average = 2n + 6.Statement 1 tells us that the averageis 36.The equation for this information is:

2n + 6 = 36 or n = 15.We can determine that the last number in theset must be 41. Statement 1 alone is sufficient.Our choices are A or D.Statement 2 is not so precise.

Twice the average = First integer + last integerOr 2 ( 2n+ 6) = 2n+ 1 + 2n+ 11Or 4n + 12 = 4n + 12.Is this information leading us anywhere?Up the garden path, for sure.What do we make of statement 2?Not good enough to answer the question.We have determined that statement 1 alone issufficient, but statement 2 alone is not.What choice do we pick under thecircumstances?Choice A as in “amen.”



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16. A right triangular region PQR lies in a rectangular coordinate plane such that each of its sides, PQ and PR, lies parallel to the rectangular coordinate axes. Is the right triangle isosceles ? (1) One of the points Q on the hypotenuse RQ has the coordinates (3, 4). (2) One of the points R on the hypotenuse RQ has the coordinates (-2, -1)

A ( ) B ( ) C ( X ) D ( ) E ( )

17. If both x and y are non-zero numbers, what is the value of y/x ? (1) x = 8

(2) y³ = x²

A ( ) B ( ) C ( X ) D ( ) E ( )

Q(3,4)1 (-2,-1)2 R P(3,-1)c Statement 1 gives us the coordinates for Q. That is not sufficient information because we need to know the values for R to be able to answer the question. Our choices are B, C or E. Statement 2 gives us the coordinates of R. We do not have any information about Q. Our choices narrow to C or E. When we combine the two statements, we can determine that P must be (3, -1) and that PQ must be 5, and PR must be 5 as well. We can conclude that the triangle is isosceles right triangle using the combined information. We must pick choice C.

We need to know the values for x and y to answer the question. Statement 1 alone is not sufficient because we do not know what y is. Our choices are B, C or E. Statement 2 alone is also not sufficient because we cannot establish a specific set of values for x and y from the given information. Our choices narrow to C or E. When we combine the two statements, we can determine that y must be 4, and x is 8, and the ratio is 4/8 = 1/2. Answer C. (How do we know that y is 4? Because y3= x2

and if x = 8, then y3 = 82 = 64 or y = 4)



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18. If x = 0.rstu where r, s, t, and u each represent a non-zero digit of x, what is the value of x ? (1) r = 3s = 2t = 6u (2) The product of r and u is equal to the product of s and t

A ( ) B ( ) C ( ) D ( ) E ( )

19. If x and y are integers between 10 and 99, inclusive, is (x - y) /9 an integer ? (1) x and y have the same two digits but in the reverse order. (2) The tens’ digit of x is 2 more than the units’ digit and the tens’ digit of y is 2 less than the units’ digit.

A ( ) B ( ) C ( ) D ( ) E ( )

Statement 1 tells us that x and y have the same digits but in the reverse order. You must know that if you take the difference of any two digit integers whose digits are the same but in the reverse order, the difference is always divisible by 9. (See module II for explanation of how this is so) For example, the numbers could be 13 and 31, 24 and 42, or 56 and 65. As long as the two numbers have the same two digits but in the reverse order, the difference is always divisible by 9. Our choices are A or D. Statement 2 tells us x and y could be any of the following values: x: 20, 31,42,53,64,75,86,97 y: 13, 24, 35, 46, 57, 68, 79 If x is 20 and y is 13, then the difference is not divisible by 9. But if x is 42 and y were 24, then the difference is. We are getting “all-over-the-map” solution with this information. We must conclude that statement 2 is not good for a unique solution. However, statement 1 alone is good for a definite answer. We must pick choice A.

X is defined as a decimal number, and the digits r,s,t, and u represent the tenth, hundredth, thousandth, and ten-thousandth digits. Can we determine what the values for these digits must be using the information in statements 1 and 2 independently or combined? Let us see. Statement 1 tells us that there is a relationship between the digits. Because we know that a digit is a positive integer having a value between 0 and 9, and because r,s,t, and u are defined as non-zero digits, they must have values that lie between 1 and 9. The value for u is constrained by the relationship r = 6u. We conclude that u must be 1, because if u is 2, then r must be 12, and r cannot be a digit with a value of 12. If u=1, then r = 6, s = 2, and t = 3. (We have used the relationship specified to determine the values for r,s t, and u.). Now that we know what r,s,t, and u are, we know that x = 0.6231 Choices are limited to A or D. Statement 2 is not good for a unique solution. ru=st is valid for a range of values for r,s,t, and u. We must pick choice A.



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20. Is the positive integer n equal to the square an integer ? (1) For every prime number p, if p is a divisor or n, then so is p-squared.. (2) Square root of n is an integer.

A ( ) B ( ) C ( ) D ( ) E ( )

The question is: can we write n = k2 where k is any integer? Statement 1 tells us that p2 is a factor of n. We know that p2 is one factor, and there are other factors which we will collectively represent by the letter r. We get n = r. P2 Since we do not know whether we can write r as a square of an integer, we cannot determine from statement 1 alone whether n can be expressed as a perfect square. Our choices are B, C or E. Statement 2 tells us that \/n = integer When we square both sides, we get n = (integer)2 Voila! We got the answer we are looking for. Statement 2 alone is sufficient to answer the question definitively, but statement 1 alone is not. The corresponding choice is B.



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DATA SUFFICIENCY 4



1. Is x < y ? (1) x2 < y2

(2) x < y + 1

Statement 1 tells us that x < y or x > -y. We cannot conclude from this statement that x < y. In real life terms, x could be -2 and y could be 3, in which case x < y. Or x could be +3 and y could be -8, in which case x2 < y2 , but x > y. We are getting “all-over-the-map” solution with this statement. Our choices are B, C or E. Statement 2 tells us that x < y+1. This information is not sufficient to conclude that x < y because x could be y + 0.2 or x could be y - 5. Both values for x are consistent with the specification x < y+1. Our choices are C or E. When we combine the two statements, we conclude that x must lie in the range: -y < x < y For some values of x in this range, x will be greater than y; for others, x will be less than y. We cannot conclude on any basis that x < y even when we combine the two statements. We must pick E. On a number line, we will represent this information as follows: -y 0 +y X If x is to the left of 0 ( -Y < X < 0), then x > y. If x is to the right of 0 (0 < X < Y), then x < y. Can you see the possibilities here? E is the best answer.


Not to be reproduced without our express written consent Name of the Registrant: PARTICIPANTS Registration Number: 0098TURBO-2452USA

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2. Machine A runs at a constant rate and produces a lot consisting of 100 bolts in 30 minutes. How much less time would it take to produce the lot of bolts if both machines A and B were run simultaneously? (1) Both machines A and B produce the same number of bolts per hour. (2) It takes machine A twice as long as it takes machines A and B, running simultaneously, to produce the same lot of bolts. 3. Is p a prime number? (1) p+1 is a prime number. (2) p is even integer.

4. Is triangle ABC equilateral? (1) Two of the angles are 60 degrees each. (2) One side measures 6 inches.

We know the rate at which machine A is running. We need to know the rate at which machine B is running so that we can determine how long it will take for both machines to complete the job working together at their respective constant rates. Statement 1 tells us that Machine B runs at the same rate as machine A. We can conclude that both machines running together can finish the job in 15 minutes or in 15 fewer minutes. Statement 1 alone is sufficient. Our choices are A or D. Statement 2 tells us that Machine A takes twice as long as Machines A and B together. This information can mean only one thing: That machine B operates at the same rate as Machine A. We can live and work with this information also. We must pick D.

Statement 1 tells us that p+1 is a prime number. If p+1 is 3, then p is 2, a prime number. But if p+1 is 5, then p is 4, not a prime number. Can you see that we are beginning to get conflicting results with just two values tested for p+1? What must we conclude? That statement 1 alone is not sufficient. Our choices are B, C or E. Statement 2 tells us that p is even integer. If p is 2, then p+1 is a prime integer, and so is p. If p is any other even integer, then it cannot be a prime integer, because the only even value that is prime integer is 2. Since there is no basis for us to conclude that p is equal to 2, we must conclude that statement 2 is also not sufficient. Choices narrow to C or E. When we combine the two statements, we notice that we are dealing with the same uncertainties encountered under the two statements independently. We must go with E.

We need to know if the angles are 60 degrees each or if the sides are the same length. We need to use our knowledge of the sum of the internal angles of a triangle. We know that the internal angles add up to 180o. Statement 1 tells us that two angles are 60o each. We can conclude that the third angle must be 60o as well. The triangle must be equilateral. Our choices are A or D. Statement 2 tells us that one side is 6 inches. We have no information about the other two sides. We cannot conclude on any basis that the triangle is an equilateral one using statement 2 alone. We conclude that statement 1 alone is, but not statement 2 alone is, sufficient. We pick A.



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5. If n is an integer, is (100 - n) / 2n an integer? (1) n < 60 (2) n is divisible by 10. 6. If -10 < k < 10 , is k > 0 ? (1) k2 > 4 (2) 1 - k > 0

7. Buckets X and Y contained only water and bucket Y was half full. If all the water in bucket X was poured into bucket Y, what fraction of the capacity of Y was then filled with water? (1) Before the water from X was poured into Y, X was 1/4 full. (2) X has twice the capacity of Y.

K is specified in a certain range of values. The question is: Does k lie to the right of 0 on the number line ? (Is K positive?). Statement 1 tells us that k > 2 or k < -2. For example, k could be +5 or -7, and both these values will . the condition specified, viz. K2 > 4. We cannot conclude definitively that k is positive using statement 1 alone. Our choices are B, C or E. Statement 2 tells us that 1 - k > 0 or 1 > k. If k < 1, k could be 0.9 or -5. Both these values are less than 1. But one value is positive, and the other is not. We are getting “all-over-the-map” kind of solution with this information in statement 2. Our choices narrow to C or E. When we combine the two statements, we notice that we have two possible range of values from statement 1; k > 2 or k < -2 Statement 2 tells us that k < 1. The only range of values that is consistent with both specifications is k < -2. We can conclude using the combined information that k is not positive. We can answer the question definitively now. We must pick choice C.

We need to know the relative capacity of buckets X and Y, and also to what capacity each was filled before the transfer. (We have information for Y, but we need the information for X in this regard.) If we do not have these two pieces of information, we cannot answer the question. Statement 1 tells us that X was filled to some fraction of its capacity before the transfer. This information alone is not sufficient. We need to know the relative sizes of the two buckets. Our choices are B, C or E. Statement 2 tells us that the proportion information, but we do not know to what capacity X was filled before the transfer. Choices narrow to C or E. When we combine the two statements, we have all the information we are looking for to answer the question. We must go with choice C.

Statement 1 tells us that n < 60. If n = 10, then the expression being tested is not an integer . (We get a value of 4.5). But if n = 20, then the expression has a value of 2, an integer. But we have no means of knowing whether n is 10 or 20, because both these values are consistent with the specification n < 60. We must conclude that statement 1 alone is not sufficient. Our choices are B, C or E. Statement 2 tells us that n is a multiple of 10. If n is 10, then the expression is not an integer. If n is 20, then the expression is an integer. We are getting “all-over-the-map” solution with this information. We must conclude that statement 2 alone is not sufficient to answer the question. When we combine the two statements, we must still contend with 10 or 20 or 30 or 40 or 50. Unless n is 20, the expression cannot be an integer. There is no basis for concluding that n is 20, We must pick E.



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8. Did Anne pay less than d dollars, including tax, for her mink coat? (1) The price Anne paid for her mink coat was (0.9.d), excluding tax. (2) The tax payable on mink coat sale is 10 percent of the selling price.

9. Is c > d ? (1) 1 - c/d > -1 (2) 0.5 < c/d < 2.0

What do we need to know to be able to answer the question? We need to know the selling price before or after tax, and the tax rate. Statement 1 tells us the selling price before tax. We need to know the rate of tax as well to be able to answer the question. Our choices are B, C or E. Statement 2 tells us the tax rate, but not the selling price before or after tax. Choices narrow to C or E. When we combine the two statements, we have all the information we need to answer the question. (We have the selling price before tax, and the tax rate). We must pick C.

Statement 1 tells us that 1 - c/d > -1 or - c/d > -2 Or -c > -2d or c < 2d We must manipulate the inequality to the form obtained in order to make sense of the given information. What do we see here? That c < 2d. Is this good enough to conclude that c > d ? Hardly. Because c could be 1.5 d ( in which case c > d) or c could be 0.5d in which case c < d. We are getting “all-over-the-map” solution with this statement. We must move on to examine statement 2. Our choics are B, C or E. Statement 2 tells us that 0.5d < c < 2d If C is 0.7d, then c < d. If x is 1.8x, then c > d. Not a unique solution. Statement 2 alone is not sufficient. Choices narrow to C or E. When we combine the two statements, we deal with the inequality 0.5d < c < 2d (because the first statement tells us that c < 2d, and the second one tells us that 0.5d < c < 2d). We have examined this inequality, and concluded that we cannot get a unique solution. Neither statement alone or combined is good for a unique solution or a definitive answer. We must pick E.



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10. In a certain health club, are more than 2/3 of the members females? (1) The club has exactly 75 female members. (2) The ratio of female to male members is 3:1 11. If x is an integer, is y an integer? (1) The average of x and y is NOT an integer. (2) ( x + y ) = 2. ( x - y )

12. The price per share of stock X increased by 10 percent over the same time period that the price per share of stock Y decreased by 10 percent. The reduced price per share of stock Y was what percent of the original price per share of stock X ? (1) The increased price per share of stock X was equal to the original price per share of stock Y. (2) The increase in price per share of stock X was 10/11 the decrease in the price per share of stock Y.

We are looking for some “proportion” information here. Statement 1 tells us the number of female members. We do not know the value for total membership, or the number of males. We cannot answer the question using this information alone. Our choices are B, C or E. Statement 2 tells us exactly what we need to know. If the ratio of female to male is 3:1, then we know that 3/4 of all members in the club are females, and 1/4 are females. Is 3/4 greater than 2/3? You bet. Can we answer the question posed definitively? Certainly. We pick B.

We learn that the price of stock X went from X to 1.1X, and at the same time, the price of stock Y went from Y to 0.9Y. The question is: What is the ratio of 0.9Y to X? We can determine the percent information by multiplying this ratio by 100. Statement 1 tells us that 1.1X = Y. Or Y/X = 1.1 or 0.9 (Y/X) = 0.99 We can determine the percent value by multiplying 0.99 by 100. Statement 1 alone is sufficient to answer the question. Our choices are A or D. Statement 2 tells us that 0.1X = 10/11 (0.1Y) Or Y/X = 1.1 We can live with this information, and answer the question about (0.9Y/X). Statement 2 alone is also sufficient to answer the question. We must pick D.

We have to take a “wait and see” attitude, and see what information comes along in statements 1 and 2 about x and y. Statement 1 tells us that 1/2(X+Y) is not an integer. This information is true if X and Y are odd and even integers respectively, or the other way around. (The sum of odd and even integers is odd integer, and the odd integer is not divisible by 2). Or y could be a fraction, and 1/2(X+Y) will not be an integer in this instance also. We cannot definitively answer whether Y is an integer or not using statement 1 alone. Choices are B, C or E. Statement 2 tells us that x + y = 2x - 2y or x = 3y X is specified as an integer, but y could be 1/3 or 2/3 or 3/3 or 4/3 or any such values so that x will be an integer. As you can see, y does not have to be an integer so that x will be an integer. On the other hand, y could very well be an integer so that x = 3y will be an integer. We cannot be sure on the basis of this information. Choices narrow to C or E. When we combine the two statements, we notice that we have x = 3y and 1/2(x+y) = Not an integer. Let us plug in 3y for x. We get 1/2 (3y + y) = 2y = Not an integer. If 2y is not an integer, what conclusion can we draw? That y is not an integer. We can answer the question by combining the two statements. We must pick C.



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13. Any decimal that has only a finite number of non-zero digits is a terminating decimal. Examples: 32, 0.78, and 6.087 are three terminating decimals. If r and s are positive integers, is the ratio r/s a terminating decimal? (1) 90 < r < 100 (2) s = 3 14. What is the area of the rectangular region of sides l and w ? (1) l + w = 14 (2) d = 10 , (d is the diagonal)

15. Is 2n > 100 ? (1) 3n > 100 (2) 2n+1 = 128

We need to know the length and the width of a rectangle so that we can determine its area. Statement 1 tells us that the perimeter is 28. This information is not enough to determine what values L and W have. For example, L could be 1 and W could b 13, in which case the area is 13. But if L is 2 and W is 12, then the area is 24. We are not getting a unique solution here, are we? Our choices are B, C or E. Statement 2 tells us that the diagonal is 10 units. We cannot determine the area for a rectangle from its diagonal information. (We can determine the area for a square using the diagonal length alone). The only determination we can make is that L2 + W2 = 102 Choices narrow to C or E. When we combine the two statements, we have two equations, and two variables: L + W = 14 And L2 + W2 = 102 We can solve for L and W, and determine the area. We can answer the question using the combined information. We must pick C.

If 2n > 100, then n must be at least 7. Statement 1 tells us that 3n > 100. The only determination we can make from this information is that n is at least 5. Is this useful information? No. Because, if n is 5, then 25 is 32. But if n is 10, then 2n > 100. “At least equal to 5” is not a useful information. We have to reckon with B, C or E. Statement 2 tells us that 2n+1 = 128 = 27 We can determine the true value for n definitively: n = 6. Once we know that n = 6, we can conclude that 2n = 26 = 64. We can answer the question definitively that 2n is not greater than 100. We must pick B.

We need to know the values for r and s so that we can answer the question definitively. Statement 1 tells us that r lies in a certain range. But this information alone is not sufficient to determine the ratio r/s. Our choices are B, C or E. Statement 2 gives us the value for s, but is silent about r. We must conclude that statement 2 is not sufficient alone. C or E. When we combine the two statements, we notice that if r is 93 or 96 or 99, then r/s will be a terminating decimal. If r is any ohter value in the range specified for r, then r/s will not be a terminating decimal. Since we do not know what value r has, we cannot conclude definitively that r/s is a terminating decimal or not. The combined information is not good for a unique solution or a definitive answer. We must pick choice E.



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16. YOUR ANSWER: B

17. YOUR ANSWER: D

X is defined as a PRIME INTEGER THAT IS A FACTOR OF Y. The question is: does that factor X repeat twice as factors of Y? We need to be able to say definitively one way or the other: Yes, X2 is a factor of Y or No, X2 is definitely not a factor of Y. Statement 1 is not sufficient to come up with a definitive answer because it tells us nothing about Y. All we can say is that X is either 2, 3, or 5 but the question was whether X repeats with a FREQUENCY of 2 as a factor of Y? Because we do not know anything about Y, we cannot tell whether X occurs JUST ONCE as a factor or Twice or more frequently. We must eliminate options A and D now. Statement 2 tells us that Y = 36. Let us PRIME FACTOR OUT 36 as 36 = 22 times 32. Given that X is a PRIME FACTOR OF Y, we must conclude that X is either 2 or 3, the prime factors of Y. If the question was: what is X, then we must give up and say that we cannot uniquely compute X but the question is whether X occurs with an exponent of 2. Regardless of whether X is 2 or 3, it does occur with an exponent or frequency of 2. Statement 2 alone is sufficient to logically conclude that X2 is indeed a factor of Y. We must, therefore, choose option B.

For a regular Hexagon that is equilateral and equiangular, the longest diagonal is 2 times the side and the shortest diagonal (that forms the base of a triangle connecting the two sides that make a V) is \/3 times side. We can see that length is made up of 3 such short diagonals and the length is made up of 2 ½ such long diagonals. Therefore, length = 5 times X where X is the value for the side of the hexagon, and Width = 3 times \/3 times X. Statement 1 tells us that 5X = 5 or X = 1. Therefore, width = 3 times SQRT(3) Area of the tile is 5 times 3 SQRT(3) = 15 times SQRT3 Statement 1 alone is sufficient to answer the question uniquely. We must stay with options A and D. Statement 2 tells us that the width = 3\/3 (X) = 3\/3 or X = 1. This, again, leads us to conclude that length is 5 so that the area is 15 times SQRT(3). Each statement allows us to uniquely and independently compute the area of the tile. We must, therefore, choose option D.



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18. In a certain office, 60 percent of the employees are college graduates and 50 percent of the employees are over forty years old. If 30 percent of those over forty years of age have a master’s degree, how many employees are college graduates? (1) Exactly 60 employees are over 40 and have a master’s degree. (2) There are four times as many college graduates as there are those over forty with master’s degrees.

19. Is n an odd integer ? (1) (3n + 1) is an odd integer (2) 2n - 1 is an odd integer

If N is the number of employees, then we have: College graduates = 0.6N 40+ age = 0.5N Master’s & 40+ age = 30% of (40+age value) = 0.3 ( 0.5N) = 0.15N We need to determine the value for N so that we can determine what is the value for 0.6N. Statement 1 tells us that 0.15N = 60. We can determine that N must be 400, and 0.6N must be 240. We can answer the question using statement 1 alone. Our choices are A or D. Statement 2 does not provide us with new information. If we set up the information in statement 2 in mathematical terms, we get: 0.6N = 4 ( 0.15N) (College grads = 4 times (40+ and Master’s)) We already know this information at the outset, and we cannot get a value for N using this information. We must conclude that statement 2 alone is NOT sufficient to answer the question, but statement 1 alone is. We must pick choice A. Can you see how we need to translate the “verbal” information in the stem to mathematical terms before we can make sense of the question?

Statement 1 tells us that 3n+1 is an odd integer. How do we get an odd integer by adding two integers? One of the integers must be odd, and the other even. Since 1 is an odd integer, we must conclude that 3n must be even integer. Because 3 is NOT even, n must be even integer. We can answer the question definitively using statement 1 alone. Our choices are A or E. Statement 2 tells us that 2n - 1 is odd integer. All that we can conclude is that 2n must be even integer because 1 is an odd integer. (Even integer minus odd integer gives us odd integer). But 2n will be even for all values of n - both odd and even values - because the factor 2 will make 2n even under all circumstances. We conclude that n is a wild card here, and we cannot definitively answer whether n is odd or even. We must conclude that statement 2 alone is not sufficient, but staement 1 alone is. We must pick A.



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20. YOUR ANSWER: A

The question is whether the integer whose units digit is the square and the tens digit is the triangle is definitely divisible by 4 or definitely not divisible by 4. We know (or, should know) that in order for an integer to be divisible by 4, the last two digits must be such that they make up a two-digit integer that is divisible by 4 or that is a multiple of 4. For example, the last two digits must form integers such as 00, 04, 08, 12, 16, 20, 24, …. 96. Statement 1 tells us that the same two digits – triangle as tens and square as units – are such that they constitute a two digit integer that is divisible by 4. We must logically conclude that the Triangle-Square occurring as the last two digits of the integer in the stem integer must also make that integer divisible by 4. We must now stay with options A and D, and eliminate B, C, and E. Statement 2 alone is NOT sufficient because, the triangle and square could be 4 and 8 so that the stem integer is divisible by 4. They could also be 3 and 9 so that the stem integer is NOT divisible by 4. As you can see, we cannot make a decision having logical certainty using statement 2 alone but statement 1 was definitive. We must, therefore, choose option A.



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DATA SUFFICIENCY 5 ANSWER KEY

AND EXPLANATIONS


1. What is the value of | n | ? (1) n is a negative integer. (2) n

2 = 9

We should remember that the absolute value of a number is the distance from the center on a number line, and that distance is always a positive quantity. Statement 1 tells us that n is a negative integer. This information is not good for a unique solution, because we can think of an endless number of negative integer values. Our choices narrow to B, C or E. Statement 2 tells us that n = +3 or n = -3. Either way, the absolute value of n is 3. We can determine the precise value for the absolute value of n, an answer the question definitively. We conclude that statement 2 alone is sufficient, but statement 1 alone is not. We must pick choice B.



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2. If x, y and z are the three internal angles of a triangle, what is the value, in degrees, of angle x ? (all values of x, y and z are in degrees) (1) x + y = 127 (2) x + y = 2z + 21 3. Of the n people participating in a test preparation program, 60 percent had not taken the test previously. Of the remaining, 25 percent had taken the test more than once previously. How many had taken the test just once previously? (1) 12 people had taken the test more than twice previously. (2) 18 people had taken the test more than once previously.

4. How many more men than women are in the public swimming pool? (1) There are twice as many men as there are women. (2) If six more women came into the pool, there will be an equal number of men and women.

We get the following picture from the information given in the stem: Number taking the test 0 time(s) = 0.6n Number taking the test 1+ time(s) = 25% (0.4n) = 0.1n The question is: How many have taken the test just once? We know that 0.3n have taken the test just once previously (Difference of n and (0.6n+0.1n)). We need a number value corresponding to 0.3n. Statement 1 tells us the number corresponding to “more than twice”. We do not have any information given to us so that we can relate this number to an expression. We have no use for this statement. Our choices narrow to B, C or E. Statement 2 tells us that 18 people took the test more than once. We can relate this number to 0.1n and conclude that n = 18/(0.1) = 180. We can determine what 0.3n is from this information: 54 people. We conclude that statement 2 alone is sufficient, and pick B as the answer.

We know that x + y + z = 180o. The question is: What is x ? Statement 1 tells us that x + y = 127. This information is not sufficient for a unique solution to the value for x. X and y could take a range of values consistent with the specification. X could be 30 and y could 97 or x could 27 and y could 100, and so on. Our choices are B, C or E. Statement 2 tells us that x + y = 2z + 21. This equation tells us that 180-z = 2z + 21 or z = 159/3 = 53o. When we plug in this value for z in x + y = 2z + 21, we get x+y=127. This information is the same as the one in statement 1, and is not good for a unique solution for x. C or E. When we combine the two statements, we are stuck with the same equation twice, and we cannot solve for two variables with just one equation. We must pick choice E.

Statement 1 tells us that the number of men in the pool is 2N and the number of women is N. If N is 1, then there is just 1 more man in the pool than woman. If N is 10, then there are 10 more men than women in the pool. Not a unique picture emerging from this statement. We have to deal with B, C or E. Statement 2 tells us that if 6 more women came into the pool, there will be an equal number of men andwomen. This can mean only one thing: There are 6 more men than women in the pool. Choice B.



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5. Is the sum of a set of six consecutive odd positive integers a perfect cube? (1) The smallest number in the set is a prime number (2) The largest number in the set is 41 A perfect cube is an integer, which can be expressed as the third power of another integer. For example, 27 is a perfect cube because we can write 27 = 33. (Similarly, a perfect square is an integer that can be expressed as the square of another integer. 4 isa perfect square because 4 = 22). The question is: Does the sum of six consecutive odd integers add up to a value that can be expressed as a perfect cube? We need to know the values in the set so that we can make this determination. Statement 1 tells us that the smallest number in the set of consecutive odd integers is a prime number. We could be looking at 3,5,7,9, 11, 13, in which case the sum is 39, which cannot be expressed as a perfect cube. On the other hand, if the numbers were 31,33,35,37,39, and 41, then the sum is 216, which can be expressed as 63. As you can see, statement 1 is not good for a unique solution because we are getting “all-over-the-map” solution with the information provided. Our choices narrow to B, C or E. Statement 2 tells us that the last number in the set is 41. We can work

backwards and conclude that the numbers in the set must be 31,33,35,37,39, and 41. We can answer the question one way or the other by stating that “yes, theh sum is a perfect cube” oor “no, the sum is NOT a perfect cube”. Why? Because we know the precise values in the set. In fact, in this set of values, the sum is 216, which is a perfect cube. (63). We conclude that statement 2 alone is sufficient to answer the question, and pick B. Do you understand the type of reasoning you have to apply to come to a determination of whether the information makes sense , or is good for a unique solution or not. As you might have noticed, we have to invoke our conceptual understanding of what a perfect cube is before we go to deal with the informatin in the statements. You have to pre-determine what information will suffice, and then go to examine the information in the statements. .



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6. How many miles long is the route from Thunder Bay to Sarnia? (1) It will take 2 hours less time to travel the entire route at an average rate of 60 miles per hour than at an average rate of 50 miles per hour. (2) The first half of the distance can be traveled

in 5 hours at 60 miles per hour rate of speed.

7. If x is divisible by 2, is x + y an even number? (1) y is a factor of x (2) x = 6

We are dealing with a “distance” problem, and we must know that distance is a function of speed and time of travel. Statement 1 tells us that if it took “t” hours to travel the distance at an average speed of 50 mph, then it would take (t-2) hours to travel the same distance traveling at 60 mph. Our equation is; 50. T = 60 (T-2) We can solve for T: T = 12 hours. We can plug this value back into one of the equations and conclude that the distance is 50X12 = 600 miles or 60 (12-2) = 600 miles. Statement 1 is good enough to answer the question definitively. Our choices narrow to A or D. Statement 2 tells us that one half the total distance is equal to 5 times 60 = 300 miles. It does not take an I.Q. of one trillion to figure out that the total distance must be twice 300 miles, or 600 miles. Statement 2 alone is also sufficient to answer the question definitively. Statements 1 and 2 are independently

We know that x is even integer. If x+y is to be even, then y must be even as well. Let us see if we can get a sense of what y is about from the statements. Statement 1 tells us that y is a factor of x. All this means is that we can write x = k.y where k collectively represents all the other factors. It is possible that k is even and y is odd, in which case x will still be even. It is also possible that y is even and k is odd, giving an even value for x. But we are not sure which is which, and we conclude that statement 1 alone is not sufficient. Our choices narrow to B, C or E. Statement 2 tells us that x = 6. We do not know what y is about from this statement. We conclude that statement 2 alone is not sufficient. When we combine the two statements, we notice that x is 6, and y is a factor of x. We can write 6 as a product of 1 and 6, or of 2 and 3. Y could be 1 or 3; or it could be 2 or6. In other words, y could be even just as easily as it can be odd. Even after we combine the two statements, we are not sure about y in terms of whether it is even or odd. We are getting a “may be” kind of a solution, and are not in a position to conclude definitively that y is even after combining the statements. We must pick E.



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8. All applicants to a military recruitment program must pass both a written test and a physical test. If 60 percent of the applicants passed the written test and 75 percent of the applicants passed the physical test, what percent of the applicants did not pass both or either tests? (1) 55 percent of the applicants passed both tests. (2) 20 percent of the applicants did not pass either test.

We have two types of tests, and two possible outcomes for each test. We must conclude that it is a “matrix” problem with which we are dealing. Let us set up the matrix using the information in the stem:

WRITTEN PASS

WRITTEN FAIL

TOTAL

PHYSICAL PASS

55%1 X 75%

PHYSICAL FAIL

X X 25%

TOTAL 60% 40% 100% We need to determine the values in cells marked X to be able to answer the question. Statement 1 tells us that 55% passed both tests. We have marked in the corresponding cell with the suffix 1 to show that the information is obtained from statement 1. We can compute the values for the other remaining cells so that the column and row totals will agree. We can answer the question using statement 1 alone. Our choices narrow to A or D. Statement 2 tells us that 20% failed both tests. We have marked this information in the corresponding cell with the suffix 2, to indicate that this information was obtained from statement 2.

WRITTEN PASS

WRITTEN FAIL

TOTAL

PHYSICAL PASS

55%1 X 75%

PHYSICAL FAIL

X 20%2 25%

TOTAL 60% 40% 100% Once again, we can determine all the values in the cells of the this matrix, and answer the question definitively using this statement 2 alone. We conclude that statements 1 and 2 are independently sufficient to answer the question. We must pick D.



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9. If an item was marked up 30 percent on its cost and then sold at a discount on the sticker price, what was the selling price, in dollars? (1) The item cost $120 (2) The sticker price was $156

10. If n is an integer, is n/15 an integer ? (1) 3n/15 is an integer (2) 7n/15 is an integer

If the Cost is C, then the sticker price was 1.3C. The question is: What is the selling price after the discount? We require the discount percentage, and the cost in dollars to be able to answer the question: Sticker ( 1- d% ) = Selling price 100 1.3C ( 1- d% ) = Selling price 100 Statement 1 tells us what the cost is. We can determine that the sticker price was 1.3 times $120. Can we determine the selling price using this information alone? We cannot. Why not? Because we do not know the discount applied on the sticker price. Our choices are B, C or E. Statement 2 tells us that the sticker was $156. We can go back and compute the cost, but we are beating about the bush, and moving in circles not unlike a dog chasing its own tail. We still do not have the rate of discount to be abole to answer the question. Our choices are limitted to C or E. When we combine the two statements, we are no better off because we still do not know what discount rate was applied on the sticker price. We must pick E.

The question is: is n divisible by 15 or is 15 a factor of n? Statement 1 tells us that 3n/15 is an integer . We can conclude that n/5 is an integer or that n is a multiple of 5. n can be any of the following values: 5, 10, 15, 20, 25, 30, 35, and so on. Unless n is 15 or 30 or 45, or any other multiple of 15, n/15 cannot be an integer. Since we do not know what value n will have in this range of possible values, we must conclude that statement 1 alone is not sufficient. Choices narrow to B, C or E. Statement 2 tells us that 7n/15 is an integer. Since 7 is a prime integer, and not a factor of 15, we must conclude that n must be a multiple of 15 so that 7n/15 will be an integer. Therefore, n/15 must be an integer. We must pick choice B.



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11. Mary and Martha received wage increases following their annual performance review? Who received the greater increase, in dollars? (1) Mary received a 10 percent increase on her wages. (2) Martha received a 8 percent increase on her wages.

12. Is John ahead of Paul in the line-up? (1) There are 20 people in the line-up. (2) There are exactly 6 people between Paul and John in the line-up. 13. Lorraine can drive from her home to a super-market by one of two possible routes. If she must also return by one of the two routes, what is the distance of the shorter route? (1) When Lorraine drives to the supermarket by the shorter route and returns by the longer route, she travels a total of 11 miles. (2) When she drives both ways, from her home to the super-market, by the shorter route, she travels a total distance of 8 miles.

We need to know the greatest “dollar value” increase received by one of the ladies. Statement 1 tells us that Mary received 10% increase on her wages. All that we know is that her wages went from $X to $1.1X. Do we get a sense of how many dollars worth of increase Mary received? Hardly. Besides, we have no information pertaining to the other lady in the picture here. We must conclude that statement 1 alone is not sufficient. Choices narrow to B, C or E, and your odds have improved significantly , odds of picking the right answer. Statement 2 tells us Martha’s wages went from $Y to $1.08Y. We are clueless in Chicago about what Y is all about. We cannot determine a dollar value for the increase that Martha received. Statement 2 is silent about Mary. Our choices are C or E. When we combine the two statements, we are still not making progress, because we do not have any information in either statement that gives us a handle on the dollar values involved in the raises. We must conclude that the statements alone or combined are about as useful as a cheap umbrella during a hurricane onslaught. We must pick E.

Statement 1 does not tells us the relative positions of Paul and John in the line-up. All that we know is that there are 20 people in the line-up. Not good. Choices are B, C or E. Statement 2 tells us that there are 6 folks between Paul and John. That does not tell us whether Paul is ahead of John or the other way around. Choices narrow to C or E. When we combine the two statements, we do not have any information that lets us get a handle on the relative positions of Paul and John in terms of who is ahead of whom. We must pick E.

If S is the length of the shorter route, and L that of the longer route, then we know that S+L = 11 miles from statement 1. This does not tell us what the value for S is. As you can see, you have two variables, and one equation. Not good enough. Choices narrow to B, C or E.Statement 2 tells us that S + S = 8 miles or S = 4 miles. This is the answer we are looking for. We can answer the question definitively using statement 2 alone, but statement 1 alone is not sufficient. We must pick B.



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14. If p < 100, is the prime number p equal to 31 ? 3 (1) p = n + 4 , where n is an integer. (2) p + 5 is a perfect square.

P is defined as a prime number having a value less than 100. Is p equal to 31? Can we answer this question using the information in statement 1 or 2 or combined? Statement 1 tells us that p = n3 + 4,where n is an integer. The values for n are subject to the constraint that p be a prime integer less than 100. Can n be 1? If n = 1, then p = 5, a prime integer. Can n be 2? If n=2, then p = 23 + 4 = 12, not a prime integer. Therefore, n cannot be 2. Can n be 3 ? if n is 3, then p = 33 + 4 = 31, a prime integer. Can n be 4? If n is 4, then p = 43 + 4 = 68, not a prime integer. n cannot be 4. Can n be 5? If n is 5, then n3 is greater than 100, and will not satisfy the constraint specified. Therefore, for values of n equal to 1 and 3, p could be 5 or 31. Two possible values. Not a unique solution. Statement 1 is not good for a unique solution. We must deal with choices B, C, or E. Statement 2 tells us that (p+5) is a perfect square. A perfect square is an integer that can be expressed as the square of another integer. Example: 4 is a perfect square because 4 can be written as 22. Knowing that p is a prime integer, the only values for p that will give a perfect square value for (p+5) are 11, 31, and 59. If p is 11, thenp+5 is 16, a perfect square. If p is 31, then p+5 is 36, a perfect square. If p is 59, then p+5 is 64, a perfect square. Using the information in statement 2, we conclude that p can have 3 possible values: 11, 31, and 59. Is this a unique solution? Hardly. What do we conclude? That statement 2 alone is not sufficient for a unique solution that lets us answer the question definitively. Our choices narrow to C or E. When we combine the two statements, we notice that statement 1 gives two possible values for p : 5 and 31 Statement 2 gives us 3 possible values for p: 11, 31, and 59. Is there a common value in both sets? You bet. It is 31. When we combine the two statements, we can conclude that p must be 31 so that this value will satisfy both conditions stipulated in the two statements. We must pick choicee C.



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15. What is the ratio of x to y ? (1) If the value of y were increased by 7, then the ratio of x to y will be equal to 1. (2) The ratio of x to 4y is 3 to 5.

16. If a television commercial consists of a total of 17,280 frames on film, how long, in minutes, does the commercial run? (1) The commercial runs without interruption at the rate of 24 frames per second. (2) It takes 6 times as long to run the film as it takes to rewind the film, and it takes a total of 14 minutes to do both.

Statement 1 tells us that x = y+7. Can we determine the ratio in terms of a specific value for x/y using this information? No. Because, the equation x = y+7 is true for a multitude of vlaues of y. If y is 1, then x is 8, and the ratio of x to y is 8/1. If y is 2, then x is 9, and the ratio of x/y is 9/2. You are beginning to see “all-over-the-map” solutions already. We must conclude that statement 1 alone is not sufficient. We have choices B, C and E left. Statement 2 tells us exactly what we want to know: if x/4y = 3/5, then x/y = 12/5. We have a unique solution, and we can answer the question definitively. We cannot ask for a better deal than that. We conclude that statement 1 alone is not good for a unique solution, but statement 2 alone IS. We must pick choice B.

We need to know the number of frames per second or per minute that run so that we can determine the time. Remember, we know already how many frames the commercial has. Statement 1 tells us the number of frames per second that run. We can determine how long it will take to run the commercial with 17,280 frames at the rate of 24 frames per second. We conclude that statement 1 alone is sufficient, and our choices are A or D. Statement 2 states that it takes 6t minutes to run the film, and t minutes to rewind the film for a total of 7t minutes, which is equal to 14 minutes. We can conclude that it must take 12 minutes to run the film. Statement 2 is also independently sufficient to answer the question. We must pick choice D.



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17. The symbol @ represents one of the following operations: addition, subtraction, multiplication, or division. What is the value of 4 @ 3 ? (1) 0 @ 4 = 4 (2) 4 @ 0 = 4

18. If y = 2x+1 , what is the value of y - x ? (1) 22x+4 = 64 (2) y = 22x

Statement 1 tells us clearly that @ symbol must stand for “addition”. It cannot be “subtraction” because, if it were, 0 @ 4 will be -4. Multiplication and Division of 0 by 4 will likewise yield 0. Therefore @ must stand for addition. Statement 1 is good for a unique solution. Our choices narrow to A or D. Statement 2 is not so clear. @ could be “addition” or “subtraction” because, in both cases, 4 @ 0 will be equal to +4. We conclude that statement 2 alone is not sufficient, but statement 1 alone is sufficient. We must pick A.

Statement 1 tells us that 22x+4 = 26 or x = 1. Knowing the value for x, we can compute what y must be: y = 2x+1 = 22 = 4. Statement 1 alone is sufficient to answer the question. Our choices narrow to A or D. Statement 2 tells us that y = 22x. When we combine this information with the information provided at the outset, viz. Y = 2x+1, we can conclude that 2x = x+1 or x = 1. Once again, we have a unique value for x, and we can determine what y = 2x+1 must be. It is equal to 4. Statement 2 also is independently sufficient to answer the question definitively. We must pick D.



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19. In a certain group of people, the average (arithmetic mean) I.Q of the males is 128 and of the females 136. What is the average I.Q. of the people in the group? (1) The group contains twice as many females as males (2) The group contains 10 more females than males.

20. If x , y , and z are three integers, are they consecutive odd integers? (1) y - x = 2 (2) x + y is even 21. In triangle ABC, if AB = x, BC = x + 2, and AC = y, then which of the three internal angles of the triangle has the greatest degree measure? (1) y = x + 3 (2) x = 2

The angle opposite the longest side will have the largest measure. Statement 1 tells us that AC is the longest side, because x+3 is the largest of the three values for the sides of the triangles. We can conclude that the angle opposite AC must be the largest angle in the triangle. Statement 1 alone is sufficient. A or D. Statement 2 is not so precise. We have some value for x, but we are clueless about y. We cannot answer the question definitively using the information in statement 2 alone. We conclude that statement 2 alone is not sufficient, but statement 1 alone is sufficient. We pick A.

This is a weighted average problem. Statement 1 gives us the proportion of men and women in the group. We can work with this information. How? If N is the number of males, then 2N is the number of females. What is the average I.Q. for the group? Average = 128N + 136(2N) = 400N = 133 N+2N 3N As you can see, statement 1 is good for a unique solution, and a definitive answer. You must set up the problem in the above manner before concluding whether the information is good for a solution or not. Our choices narrow to A or D. Statement 2 tells us that F = M + 10 If N is the number of males, then F = N + 10. Let us see what happens when we try to set up the equation for average: Average = N(128) + 136(N+10) N + N + 10 = 264N + 1360 2N + 10 As you can see, we cannot get a number value for this expression. We must conclude that statement 2 is not good for a unique solution. Rememeber; you must set up the problem in the manner suggested before concluding that the information is useless. We must pick A, because statement 1 alone is sufficient, but statement 2 alone is NOT.

We need information about x, y, and z to answer the question. We notice that statements 1 and 2 do not provide any information about z. We cannot answer the question using the statements 1 and 2 independently, or combined. We pick E.



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DATA SUFFICIENCY 6 EXPLAINED 1. Is x greater than 2? (1) x2 > 4 (2) x is a multiple of 2 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

We are required to compare X and 2 and decide which is bigger. This is how you must understand the question posed. Statement 1 tells us that X > 2 or X < -2. This is how you must interpret X2 > 4. Under the first likely scenario, X is indeed more than 2. Under the second equally likely scenario, X is not more than 2. We have a conflict here and cannot make a decision in the face of conflict. We must eliminate options A and D, and work with the three remaining options B, C, E. Statement 2 tells us that X is a multiple of 2. Likely values for X are: -4, -2, 0,2,4,6,………….. This is how you must read “multiple of 2” statement. As you can see, some of the likely values are more than 2, others are equal to 2, and still others are less than 2. We have another major conflict here, and conflict is not good for a definite yes or no decision. We must eliminate option B and work with the remaining C and E. If we combine the two statements, we notice that –2, 0, and 2 are not permissible values for X but the other permissible values satisfying both statements give rise to a conflict: some are more than 2 and others are less than –2. Once again, we cannot answer the question posed with a unique yes or no decision. We must pick option E. Remember: If X2 > 4, then X > 2 or X < -2. IF X2 < 4, then X < 2 and X > -2. In other words, -2 < X < 2.

2. In an election to the Secretary of the Club, if each of the 1,000 members voted for either Mary or Michelle (but not both), what percent of the female members voted for Michelle? (1) Eighty percent of male members voted for Michelle. (2) Twice as many male members voted for Michelle as female members. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

Our “Need to Know” information is: How many females there are in the group, and how many of those females voted for Michelle. Statement 1 tells us that 80% of males voted for Michelle. This information does not provide any of the two pieces of information we set to seek out in our “N.T.K”. We have to eliminate options A and D because Statement 1 is not good enough to make a decision with. We have B, C, and E left to choose from. Statement 2 tells us that Mmichelle voters = 2 FMichelle voters This ratio information does not tell us any of the two pieces of information required in our N.T.K to answer the question posed. We cannot work with statement 2 alone also. We have to eliminate option B, and move to combine the two statements. When we combine the two statements, we notice that the Females who voted for Michelle represent 40% of the Male members in the club. But then, if the males represent 50% of the group, then the females voting for Michelle are 40% of 50% = 20% of the group. If the males represent 70% of the group, then the females voting for Michelle are 40% of the 70% = 28% of the group. This conflict is bad enough. We are also unable to answer the question posed: What percent of Females voted for Michelle. We have to pick Answer choice E.



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3. During 1987, 8.7 percent of the men in the labor force were unemployed in June as against 8.4 percent in May of that year. If the number of men in the labor force was the same in both months, how many more men were unemployed in June than in May? (1) The number of unemployed men in the labor force during May was 1.68 million. (2) The total number of men in the labor force was 20.0 million during the two months - May and June - of 1987. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

If X represents the number of men in the labor force, then we are required to find out the difference between 0.087X and 0.084X or the value for 0.03X. We will be able to answer the question posed if we can glean some information about the value for X. This is our N.T.K. Statement 1 tells us that 0.084X = 1.68 Million. We can determine that X = 20 million, and we can determine what 0.03 of X is using this information. Statement 1 alone is sufficient to answer the question posed. We must keep options A and D, and eliminate options B, C, and E. If statement 2 is also sufficient, we will pick D. Otherwise, we will go with A. Statement 2 is even better. It tells us that X = 20 million. We can compute what 0.03X is using this information. We notice that statements 1 and 2 are independently sufficient to make a decision. We must choose option D. Note that your ability to predetermine your “Need to Know” information, and to process a lot of verbiage into simple algebraic terms is a critical ability in this part of the GMAT.

4. If the average (arithmetic mean) of 4 numbers is 45, how many numbers are greater than 45? (1) Two of the numbers are 60 and 45. (2) One of the numbers is 25. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

We are told that the sum of the four numbers is 180. We are required to make a decision about how many numbers are greater than 45. Notice that the problem talks about “numbers”, not about “integers”. Statement 1 tells us that the first two numbers add up to 105. The remaining two numbers must add up to 75. Can we tell how many numbers are greater than 45? May be one of them is if the two numbers are 51.5 and 23.5, or NONE if the numbers are 37 and 38. There is a conflict and we cannot make a unique decision. WE must conclude that statement 1 is not sufficient to make a unique decision, and eliminate options A and D. Statement 2 tells us that one of the four numbers is 25. All that we know is that the other three numbers add up to 155. Can we tell how many of these three remaining numbers are more than 45? We cannot. May be all three are greater than 45 if the numbers happen to be 50, 51, 54. Or, just two are more than 45 if the numbers happen to be 60, 50, and 45. Can you see the conflict here? We cannot make a unique decision by using statement 2 also. We must eliminate option B and move on to combine the two statements. When we combine the two statements, we notice that we know what the three of the four numbers are, and we can determine what the fourth number is and answer the question in a unique fashion. The fourth number must be 50, and given that another one is 60, there are two numbers greater than 45. We must choose option C.



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5. Is the value of x greater than 5? (1) x3 > 125 (2) 4 < x < 6 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

We are required to compare X and 5, and decide in a unique fashion which is better. This is how you must understand the question posed. Statement 1 tells us that X > 5. We can tell that X is indeed greater than 5 because that is what this statement leads us to believe. We must keep options A and D, and eliminate options B, C, and E. Statement 2 is not good for a unique decision. X could be 4.5 or 5.9. WE cannot decide whether X is definitely more than 5 or less than 5. We must conclude that statement 2 alone is not sufficient to make a decision. We must pick answer A. 6. Is n + 1 a prime number? (1) n is a product of two prime numbers. (2) n2 + 2n + 1 = 49 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

We need to know what N is so that we can determine what N+1 ought to be. Statement 1 tells us that N could be 6, which is the product of 2 and 3 – both prime integers -, and in this scenario N + 1 is 7, a prime number. But then, N could also be 15 – product of 3 and 5, both prime integers – and N + 1 is 16, which is not a prime number. Can you see the conflict here? We cannot make a unique decision by using statement 1 alone. We must eliminate options A and D, and keep options B, C, and E only. Statement 2 tells us that (N+1)2 = 49 or N + 1 = 7 or N + 1 = -7. Of course, you were required to recognize that N2 + 2N + 1 is the same as (N + 1)2. “Recognition” is an essential skill you will require if you want to do well in the GMAT.

Because we get two likely values for N + 1, +7 and – 7, we cannot make a unique decision about whether N + 1 is prime or not. If N+1 is 7, then it is prime. If N+1 is –7, it is not prime. We must eliminate option B, and move on to combine the two statements. When we combine the two statements, we notice that N + 1 must be 7 because the product two prime numbers cannot be negative, and N + 1 cannot be a negative value (because N cannot be negative according to statement 1). When we combine the two statements, we can make a decision that N + 1 is indeed prime. We must pick option C. 7. A rectangular frame encloses a picture. What is the length in inches of the picture? (1) The frame measures 24 inches by 18 inches. (2) Area of the frame = area of the picture it encloses. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

We need to look for information allowing us to compute the length of the picture in a unique way. Statement 1 tells us about the frame size. We are not able to determine the length of the picture by knowing the size of the frame. We cannot make a decision by using statement 1 alone. Let us eliminate options A and D. Statement 2 says that the area of the frame is the same as the area of the picture it encloses. Obviously the total frame area is divided in two equal halves, but we are not able to determine what the picture area is on the basis of this statement alone. Let us eliminate choice B. When we combine the two statements, we notice that the picture area is ½ of 24 X 18 square inches. But we cannot determine the length if we know the picture area unless we know what the width is or unless the problem tells us that the picture is a square shaped one. We tried all the tricks in the book, and still cannot make a unique decision about the length of the picture. We must go with option E.



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8. How far apart do Jane and June live? (1) The local public library is 5 miles due north of Jane’s house and 12 miles due east of June’s house. (2) The local school is 12 miles due west of Jane’s house and 5 miles due south of June’s house. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

Statement 1 gives us the following picture. June 12 Library 5 Jane We get a 5:12:13 triangle here, and Jane and June must live 13 miles apart. We must keep options A and D, and eliminate choices B, C, and E. Statement 2 gives us the following picture: JUNE 5 SCHOOL 12 JANE Once again, we see a 5:12:13 triangle, and can determine that June and Jane live 13 miles apart. Each statement independently allows us to answer the question posed in a unique sort of way. We must choose option D. Remember to recognize the standard triple proportions when you deal with the righ triangles: 3:4:5 and 5:12:13, and occasionally 7:24:25.

9. If n > 0, is n equal to the sum of two different prime numbers? (1) n is equal to the square of the smallest odd prime number. (2) n + 2 is a prime number. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

The question is: Can we determine whether we can write N as the sum of two different prime integers? In order to answer this question, we need to know what N is. Statement 1 tells us that N is equal to 32 = 9. We can write 9 as the sum of 2 and 7. Yes, N can be written as the sum of two different prime integers. Statement 1 allows us to make a unique decision. We will keep options A and D, and kill options B, C, and E. Statement 2 is not as precise as statement 1 was. If N+2 is a prime integer, N could be 3 and N+2 could be 5, a prime integer. But 3 cannot be written as the sum of two different prime digits. But then N could be 5 and N + 2 is 7, another prime integer. Here, we can write 5 as the sum of two different prime integers – 2 and 3. We are dealing with two different conflicting scenarios now allowing us to make a unique decision. We will pick option A.



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10. 2/3 of the N people polled in a survey involving 2 questions said Yes to Question 1. What fraction of the people polled did NOT say yes to both questions? (1) 3/5 of those who answered YES to Question 1, answered YES to Question 2. (2) 1/3 of those polled answered NO to Question 1. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

We have two questions, and each question has two mutually exclusive response possibilities. We can represent this scenario in terms of a table as follows. According to the question stem, the total for Q1 YES is 2N/3. This tells us that the number answering NO to Q1 is 1/3 N.

Q1 YES Q1 NO TOTAL

Q2 YES 2N/5(ST.1)

Q2 NO 4N/15(ST.2) ?????? TOTAL 2N/3(stem) N/3(stem) N

According to statement 1, 3/5th of 2N/3 answered YES to Q2. This gives us 2/5th of N for Q1 YES and Q2YES. This information allows us to compute the value for those answering Yes to Q1 but NO to Q2. (2N/3 – 2N/5= 4N/15). We cannot Compute the value for Q1NO and Q2 NO cell on the basis of statement 1 information alone. We must eliminate options A and D. Statement 2 does not tell us anything new. If the stem tells us that 2/3rd of N people answered Yes to Q1, by default we know that 1/3rd of N answered NO to Q1. We cannot determine the value for the shaded cell on the basis of this statement alone. Even if we combine the two statements, we are not looking at any new piece of information allowing us to compute the value for the shaded cell shown in the table above. We must pick option E.

11. If s, u, and v are positive integers, is s > v? (1) s > u

(2) 2s = 2u + 2v

Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

WE are required to compare S and V, and decide which is bigger. Statement 1 tells us that there is a connection between S and U. We cannot use this information to arrive at any conclusion about S and V. We must eliminate options A and D. Statement 2 tells us that S = V+1 and S = U+1. If S =2, then U = 1 and V = 1. If S = 3, then U = 2, and V = 2. If S = 4, then U = 3, and V = 3 and so on. This statement clearly tells us that S is bigger than V. We can make a unique decision by using statement 2 alone but not by using statement 1 alone. We must pick option B. 12. What is the area of the square region? (1) The diagonal is 10 inches. (2) The perimeter is 20 \/2 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

We can compute the area of a square in terms of its side (side2) or in terms of its diagonal (diagonal2 / 2). Statement 1 tells us what the diagonal is. We know that the area must be 102/2 = 50. We can make a unique decision by using statement 1 alone. Let us keep options A and D, and eliminate options B, C, and E. Statement 2 tells us that the side of the square is 1/4th of the perimeter or 5\/2. We can square the side and get the area using statement 2 alone also. We must choose option D.



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13. What is the measure of an external angle of the triangle ABC? (1) One of the internal angles measures 72 degrees. (2) The triangle ABC is an isosceles triangle. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

We are asked whether we can determine the value for “any one” of the external angles of a triangle. We should know that the external angle is the adjacent angle to an internal angle. Statement 1 tells us that one of the internal angles is 72 degrees. We know that the corresponding external angle is 180 – 72 = 108 degrees. We must keep options A and D, and eliminate options B, C, and E. Statement 2 tells us that two angles are equal in the given triangle. But we do not know what the angles are, and cannot determine the corresponding external angle. We must pick option A. 14. What is the ratio of the volume of cube X to that of cube Y? (1) The length of an edge of cube X is 6 inches. (2) The ratio of the surface area of cube X to that of cube Y is ¼. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

The volume of a cube is a function of its side3. If X is the side of cube X and Y that of cube Y, we are required to compute a unique value for the ratio X3/Y3. Statement 1 does not allow us to compute this ratio because it does not tell us anything about cube Y. We must eliminate options A and D. Statement 2 tells us that 6X2/6Y2 = ¼ or X2/Y2 = ¼ or X/Y = ½. IF we cube (X/Y) we get the ratio of the volume of X to that of Y. We can detemine that the ratio of volume of X to that of Y is (1/2)3 = 1/8. Statement 2 allows us to get a unique value for the ratio, whereas statement 1 does not. We must pick option B.

15. What is the value of k? (1) In the xy-coordinate system, (a,b) and (a+3, b+k) are two points that lie on the line defined by the equation x = 3y - 7. (2) k.k = 1 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

Statement 1 allows us to compute the slope of the line defined by the equation x = 3y – 7 by rearranging the equation in the standard form: Y = 1/3 X + 7/3. We can see that the slope is 1/3. We can also compute the slope by using the two points (a, b) and (a+3, b+k) as Slope = (b – b –k)/(a – a –3) = k/3 We know that the two expressions for the slope are identical and we can set them equal: K/3 = 1/3 or K = 1. This is a unique value for K and statement 1 alone is sufficient. Let us keep options A and D. Statement 2 tells us that K = 1 or K = -1. Not a unique value for K. We cannot make a decision by using statement 2 alone whereas statement 1 alone was sufficient. We must pick option A.



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16. What is the average (arithmetic mean) dollar amount of all the paychecks that John Doe received last year? (1) Last year John Doe received 26 paychecks. (2) The average of John Doe’s first thirteen checks during the year was $750. The average of John Doe’s last 13 checks was $800. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

We need to know the total value of his paychecks and how many he received during the year so that we can compute the average value. Statement 1 tells us that John received 26 paychecks but we do not know the total value of those checks. We cannot use statement 1 alone to make a decision. Let us kill options A and D. Statement 2 tells us that the total of his first 13 checks was 13 times 750, and the total of his last 13 checks was 13 times 800. WE can compute the total of 26 paychecks but we are not going to assume that John did not receive another 5 checks in between the first 13 and the last 13. Remember: No unwarranted assumptions in GMAT. Reluctantly, we must conclude that statement 2 does not allow us to make a unique decision. Let us eliminate option B. By combining the two statements, we know that John received exactly 26 checks, and using statement 2 we can determine the total value of those checks so that we can determine the average value of those checks. Choice C corresponds to this scenario. 17. How many hours does it take for Machine A to fill a production lot, working alone at a constant rate? (1) Machines A and B, operating simultaneously at their respective constant rates, can fill the production lot in 2 hours. (2) Machine B, working alone at its constant rate, can fill the lot in 5 hours. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

Statement 1 tells us that TATB/(TA+TB) = 2. We cannot determine either TA or TB. We must conclude that statement 1 does not allow us to determine the time Machine A will take to do the job working alone at its constant rate. Let us kill options A and D. Statement 2 tells us that TB = 5. We cannot determine TA by using this information alone. Let us kill option B.

When we combine the two statements, knowing TB = 5, and knowing that TATB/(TA+TB) = 2, we can determine TA.

The combined information is good for making a unique decision but each statement independently was not. We must pick option C. 18. How many students are in the school? (1) 40 more than 1/3 of all the students in the school are taking a science course and, of these, 1/4 are taking Physics. (2) Exactly 1/8 of all the students in the school are taking Physics. Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

If N is the number of students, we are required to verify whether it is possible to determine N. Statement 1 tells us that Number taking Science = N/3 + 40 And Number taking Physics = ¼(N/3 + 40) Or, Physics number = N/12 + 10 WE cannot compute N on the basis of the above information alone. Let us kill A and D. You must bear in mind that you need to set up the stated information along the above lines before you give up. Statement 2 tells us that the number taking physics = N/8. Once again, we cannot determine N on the basis of this statement alone. Let us kill option B. When we combine the two statements, we can set up the following equation: N/8 = N/12 + 10. We notice that we area dealing with a single variable equation that allows us to determine a unique value for N. We must pick option C. Note: Be sure to set up equations before deciding whether you can make a decision or not. Data sufficiency requires that you do minimal implementation before giving up. Do not attempt to decide whether you can make a decision or not by simply reading the statements.



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19. Is c2 + d2 > 1? (1) d > 0 (2) c/d > 1 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

Statement 1 is not sufficient to answer the question because we do not know what C is about. Let us eliminate options A and D. Statement 2 tells us that C > D. This is not sufficient information because if c = ½ and D = ¼, then C2 + D2 = 5/16, not more than 1. But if C = 2 and D = 1, then the expression is more than 1. We have conflicting scenarios consistent with the statement 2 and cannot make a unique decision. We must eliminate option B. Even when we combine the two statements, we cannot make a unique decision because anything greater than 0 could be a fraction less than 1, and the first scenario might as well play out as could the second scenario discussed above. We must choose option E.

20. @, &, and * are three different positive digits and If @ + & = * , What is the value of &? (1) * = 4 (2) @ = 1 Your Answer; A ( ) B ( ) C ( ) D ( ) E ( )

The two most important questions that will stand you in good stead when you take the test are: ♦ What do I know? ♦ What do I need to know? In this case, we are dealing with three positive digits. We know that the positive digits could be any value from 1 through 9. We are asked whether we can determine a unique value for the & digit on the basis of information in statements 1 and 2. Statement 1 tells us that the sum of @ and & is 4 because the stem tells us that the value for * is the same as the sum of the values of @ and &. Our reasoning will establish that @ and & could be any combinations of 1 and 3, or 3 and 1. This means that & could be 1 or 3. Notice that 2 and 2 is not a good scenario because the digits are “different”. The test is going to ask you to show that you are paying attention. We notice that the first statement gives us 2 possible values for &, and that is not a unique situation. We must conclude that statement 1 is not sufficient to make a decision. Let us eliminate options A and D. Statement 2 tells us that the other digit @ is 1. We cannot determine a unique value for & because & could be 2 through 8. Notice that the limiting value for * is 9, and if @ is 1, then & could be any value 2 through 8. Once again, there are 7 different likely values for &, and we cannot work with statement 2 alone also. Let us eliminate option B. When we combine the two statements, we know that @ is 1, and, consequently, & must be 3. The combined information lets us determine a unique value for &. We must choose option C.

datasuficiency tests 1- 6 and explanations

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