1

A and B are in a line to purchase tickets. How many people are in the line? (1) There are 15 people behind A and 15 people in front of B. (2) There are 5 people between A and B. (algebra, medium) (T) Suppose there are n people in the line and A is the ath place and B is the bth place in the line. (1) says that a= n 15 and b = 16. (2) says that a = b + 6 (if A is in front of B) or a = b 6 (if A is behind B). Thus n = a +15 has two possible values: n could be either b + 21 = 37 or b + 9= 25. NOT SUFF

2

a , b , and c are integers and a < b < c. S is the set of all integers from a to b, inclusive. Q is the set of all integers from b to c, inclusive. The median of Set S is (3/4) b. The median of set Q is (7/8) c. If R is the set of all integers from a to c , inclusive, what fraction of c is the median of set R? 3/8 1/2 11/16 5/7 3/4 (statistics,hard) Note that for a set of consecutive integers, the median is the the average of the first and the last integer Median of S =(a+b)/2 therefore a=b/2 Median of Q=(b+c)/2 therefore b= (3/4)c Thus a= (3/8)c Median of R = (a+c)/2 = (11/16)c

3

A basket contains 5 apples, of which 1 is spoiled and the rest are good. If Henry is to select 2 apples from the basket simultaneously, what is the probability that the two apples selected will include the spoiled apple?

1/5 3/10 2/5 1/2 3/5 (probability, medium) Since the ratio of the apples chosen to the total number of apples is 2 to 5, the probability that the two apples selected will include the spoiled apple is 2/5. Other ways of arriving at the same result: Pr (first apple is spoiled) + Pr (second apple is spoiled) = 1/5 + (4/5)(1/4) =2/5 (for the second apple to be spoiled, the first must be one of the 4 good apples) 1 Pr (both are good)= 1 (4/5) (3/4) = 2/5

1/5 3/10 2/5 1/2 3/5

4

A boat traveled upstream a distance of 90 miles at an average speed of (v-3) miles per hour and then traveled the same distance downstream at an average speed of (v+3) miles per hour. If the trip upstream took half hour longer than the trip downstream, how many hours did it take the boat to travel downstream? 2.5 2.4 2.3 2.2 2.1 (movement, hard) Remember that distance = rate x time. Thus the trip upstream took 90/(v-3) hours and the trip downstream took 90/(v+3) hours.

Thus )3)(3(2)3(902)3(902

1

3

90

3

90

vvvv

vv

3310899)90)(3(4 22 vvv

(As 1089900302 and only numbers with units digit 3 and 7 have squares that have units digit 9, it is clear that v=33.)

Since the time taken downstream is 3

90

v , the correct answer is

90/36=45/18=5/2=2.5 hours. Another way to solve for v: 180/(v-3) = 180/(v+3) + 1. Take advantage of the fact that in the GMAT, velocity is usually an integer: lLook for two factors of 180 that differ by 6 and whose pairs differ by 1: 30 (30 6) and 36 ( 36 5). Thus v=33

2.5 2.4 2.3 2.2 2.1

5

A bookstore that sells used books sells each of its paperback books for a certain price and each of its hardcover books for a certain price. If Joe, Mary and Paul bought books in this store, how much did Mary pay for one paperback book and one hardcover book? (1) Joe bought 2 paperback books and 3 hardcover books and paid $12.50. (2) Paul bought 4 paperback books and 6 hardcover books and paid $25.00. (algebra, medium) Suppose that p is the price of each paperback book and h is the price of each hardcover book. Mary bought one of each, so we need the value of p+h. (1) says that 2p + 3h = 12.5. Clearly not sufficient: if p=h, p+h=5, but if h=2p, p=5/4 and h=5/2, making p+h=15/4 (2) says that 4p + 6h = 25, an equation that is equivalent to that given by (1). Thus (2) is not sufficient, not even in conjunction with (1). (T) NOT SUFF

6

A box contains 10 light bulbs, fewer than half of which are defective. Two bulbs are to be drawn simultaneously from the box. If n of the bulbs in box are defective, what is the value of n? (1) The probability that the two bulbs to be drawn will be defective is 1/15. (2) The probability that one of the bulbs to be drawn will be defective and

the other will not be defective is 7/15. (probability, hard) (1) The greater the value of n ( a non-negative integer), the higher will be

the probability that both drawn bulbs are defective. Thus, as (1) gives the exact probability, we can determine the value of n:

n/10 x (n-1)/9 = n(n 1) /90 = 1/15. Therefore n(n 1) = 6 and n= 3 SUFF (2) The probability that only the first is defective is n/10 x (10 n)/9. The

proability that only the second is defective is (10 n)/10 x n/9, the same. Thus (2) tells us that 2n (10 n )/90 =7/15

n(10 n) = 21. Since 21 = 3 x 7, n could be 3 or 7. However, as it is given that n < 5, n must be 3 SUFF

7

A can manufacturer has 5 identical machines, each of which produces cans at the same constant rate. How many cans will all 5 machines running simultaneously produce in z hours ? (1) Running simultaneously, 3 of the machines produce 72,000 cans in 2z hours. (2) Running simultaneously, 2 of the machines produce 24,000 cans in z hours. (combined work, hard) Note that the number of can produced is directly prortional to the number of machines working and to the number of hours the machines work. (1) If 3/5 of the 5 machines produce 72,000 cans in twice the z hours, all 5 machines running simultaneously produce 72,000(5/3)/2 cans in z hours. SUFF (2) If 2/5 of the 5 machines produce 24,000 cans in z hours, all 5 machines running simultaneously produce 24,000(5/2) cans in z hours. SUFF

8

A car traveling at a certain constant speed takes 2 seconds longer to travel 1 kilometer than it would take to travel 1 kilometer at 75 kilometers per hour. At what speed, in kilometers per hour, is the car traveling? 71.5 72 72.5 73 73.5 (movement,hard) It would take 1/75 of an hour to travel 1 kilometer at 75 kilometers per hour, and 1/75 of a hour is 60/75= 4/5 of a minute = 48 seconds. Thus the car will take 50 seconds (5/6 of a minute) to travel 1 kilometer. In 1 minute, this car would travel 6/5 of a kilometer, and in 60 minutes , 72 kilometers.

Also note that for a constant distance, tava = tbvb, so va =

=

71.5 72 72.5 73 73.5

9

A cash register in a certain clothing store is the same distance from two dressing rooms in the store. If the distance between the two dressing rooms is 16 feet, which of the following could be the distance between the cash register and either dressing room? I. 6 feet II. 12 feet III. 24 feet I only II only III only I and II II and III (geometry, medium) The placement of the cash registers can be represented by the three vertices of an isosceles triangle. We know that AC= 16, and since the length of any side of a triangle must be less than the sum of the other two sides, the lengths of the other two sides must be greater than 8 feet.

I only II only III only I and II II and III

10

According to the directions on a can of frozen orange juice concentrate, 1 can of concentrate is to be mixed with 3 cans of water to make orange juice. How many 12-ounce cans of the concentrate are required to prepare 200 6-ounce servings of orange juice? 25 34 50 67 100 (ratios, medium) One can of concentrate generates 4 12-ounce cans of juice. Thus x cans of concentrate generates 4(12)x ounces of juice. As we need 200 6-ounce servings, we can solve )6(200)12)(4( x . Dividing, we see that 502 x and thus

25 cans of concentrate are needed.

25 34 50 67 100

11

A certain basket contains 10 apples, 7 of which are red and three of which are green. If 3 different apples are to be selected at random from the basket, what is the probability that 2 of the apples selected will be red and 1 will be green?

7/40 7/20 49/100 21/40 7/10

(probability, hard) Recall the formula for the number of unordered subsets of size k of a set of size

n. !

)1)...(1(

!)!(

!

k

knnn

kkn

nCkn

. The required probability is 1327 CC

divided by 310C . Thus the answer is 21(3)/120= 21/40

7/40 7/20 49/100 21/40 7/10

12

A certain bank charges a maintenance fee on a standard checking account each month that the balance falls below $1000 at any time during the month. Did the bank charge a maintenance fee on Sue's standard checking account last month? (1) At the beginning of last month, Sue's account balance was $1500.00 (2) During last month, a total of $2000.00 was withdrawn from Sue's checking account. (algebra, medium) (T) No information is given about possible deposits. NOT SUFF

13

A certain business produced x rakes each month from November through February and shipped x/2 rakes at the beginning of each month from March through October. The business paid no storage costs for the rakes from November through February, but it paid storage costs of $0.10 per rake each month from March through October for the rakes that had not been shipped. In terms of x, what was the total storage cost, in dollars, that the business paid for the rakes for the 12 months from November through October? 0.40x 1.20x 1.40x 1.60x 3.20x (algebra, medium) From November through February, 4x rakes are produced. So, as x/2 rakes are shipped at the beginning of each of 8 months starting with March. The business is charged $0.10 for every rake-month. The sum of the rake months is x/2 multiplied by (1+2+3+...+7)=1.4x

0.40x 1.20x 1.40x 1.60x 3.20x

14

A certain candy manufacturer reduced the weight of candy bar M by 20 percent but left the price unchanged. What was the resulting percent increase in the price per ounce of candy bar M? 5% 10% 15% 20% 25% (percents, medium) Suppose that the original weight in ounces and price of candy bar M were w and p. This the original price per ounce was p/w and the new price per ounce was p/0.8w = 5/4 (p/w), 25% higher than the original.

5% 10% 15% 20% 25%

15

A certain car averages 25 miles per gallon of gasoline when driven in the city and 40 miles per gallon when driven on the highway. According to these rates, which of the following is closest to the number of miles per gallon that the car averages when it is driven 10 miles in the city and then 50 miles on the highway? 28 30 33 36 38 (ratios, medium) The car is driven a total of 60 miles and uses 10/25 + 50/40 = 0.4 + 1.25 =1.65 5/3 gallons. Dividing the number of miles by the number of gallons we get about 36 miles per gallon, or to be exact, 3600/165 miles per gallon :

Alternatively, use fractions: For the 10 miles driven in the city, 10/25 = 2/5 of a gallon of gas was used. For the 50 miles driven on the highway, 50/40 =5/4 gallons of gas was used. Thus a total of 33/20 gallons were used, at rate of 60 (33/20) = 2060/33 =400/11 miles per gallon. Note that 400/11 = 3600/99 36

28 30 33 36 38

16

A certain characteristic in a large poplulation has a distribution that is symmetric about the mean m. If 68 percent of the distribution lies within one standard deviation d of the mean, what percent of the distribution is less than m + d? 16% 32% 48% 84% 92% (statistics, medium) As the distribution that is symmetric about the mean m, the percent that is not within one standard deviation about m (32%), can be divided into two equal parts, that above m + d and that below m d . Thus 16% of the values are above m + d, which means that 84% are below m + d.

16% 32% 48% 84% 92%

17

A certain circular area has its center at point P and radius 4, and points X and Y lie in the same plane as the circular area. Does point Y lie outside the circular area? (1) The distance between point P and point X is 4.5. (2) The distance between point X and point Y is 9. (geometry, hard) (1) Tells us nothing about Y. X is 0.5 units outside the circle. NOT SUFF (2) Nothing is said about point P. NOT SUFF (T) Since the diameter of the circle is 8, Y must be at least 0.5 units outside the circle. SUFF

18

A certain city with a population of 132,000 is to be divided into 11 districts, and no district is to have a population that is more than 10% greater than the population of any other district. What is the minimum possible population that the least populated district could have? 10,700 10,800 10,900 11,000 11,100 (percents, hard) Remember that the sum of the population of the 11 districts must be 132,000, so to mimimize the population of the least populated district, we need to maximize that of the 10 other districts. Letting the population of the least populated district be x thousand, the population each of the other 10 could be as large as 1.1x. Thus to mimimize x, solve x+10(1.1x)=132. Thus 12x=132 and x=11

10,700 10,800 10,900 11,000 11,100

19

A certain cloth with a diameter of 20 inches is placed in a circular tray with a diameter of 24 inches. What fraction of the trays surface is not covered by the cloth? 1/6 1/5 11/36 25/36 5/6 (geometry, medium) The trays surface area is = 144 , while that of the cloth is 100 (144 - 100 ) /144 = 11/36. Also, note that the area of a circle is directly proportional to the square of its radius. Thus the ratio of the areas is the square of the ratio of their radii, which is 10:12 = 5:6. The ratio of the areas, then, is 25/36, so 11/36 of the trays surface is not covered by the cloth. 1/6 1/5 11/36 25/36 5/6

20

A certain club has 20 members. What is the ratio of the number of 5-member committees that can be formed from the members of the club to the number of 4-member committees that can be formed from the members of the club? 16 to 1 15 to 1 16 to 5 15 to 6 5 to 4 (combinatronics, medium)

We are asked for the ratio of 520C : 420C , which is equal to

!5

1617181920 divided by

!4

17181920 = 16 to 5. .

16 to 1 15 to 1 16 to 5 15 to 6 5 to 4

21

A certain company assigns employees to offices in such a way that some of the offices can be empty and more than one employee can be assigned to an office. In how many ways can the company assign 3 employees to 2 different offices? 5 6 7 8 9 (combinatronics, medium) Suppose that the offices are A and B. Each of the 3 employees can be assigned to either office, so there 2 x 2 x 2 = 8 ways the company can assign the 3 employees. It would be more time-consuming to consider 2 cases: Case I- all 3 employees go to the same office- 2 ways Case II- 1 employee goes to one office, the other 2 go to another. Choose which employee is alone (3 ways) and then which office she goes to (2 ways)- 2 x 3 = 6 ways Total 8 ways

5 6 7 8 9

22

A certain company charges $6 per package to ship packages weighing less than 2 pounds each. For a package weighing 2 pounds or more, the company charges an initial fee of $6 plus $2 per pound. If the company charged $38 to ship a certain package, which of the following was the weight of the package, in pounds.

16 17 19 20 22 (algebra, medium) If a package weighs x pounds, the amount charged is 6 + 2x. For the amount charged to be $38, 2x + 6 = 38, so x=16

16 17 19 20 22

23

A certain company divides its total advertising budget into television, radio, newspaper, and magazine budgets in the ratio 8 : 7 : 3 : 2, respectively. How many dollars are in the radio budget? (1) The television budget is $18,750 more than the newspaper budget. (2) The magazine budget is $7,500 (ratios, medium) For some positive number x, the television (t), radio (r), newspaper (n) and magazine (m) budgets are 8x, 7x, 5x and 2x. Any information that allows us to find the value of x will permit us to determine how many dollars are in the radio budget.

(1) t n =18,750 8x 3x = 5x = 18,750 SUFF (2) m = 2x = 7,500 SUFF

24

A certain company expects quarterly earnings of $0.80 per share of stock, half of which will be distributed as dividends to shareholders while the rest will be used for research and development. If earnings are greater than expected, shareholders will receive an additional $0.04 per share for each additional $0.10 of per share earnings. If quarterly earnings are $1.10 per share, what will be the dividend paid to a person who owns 200 shares of the companys stock? $92 $96 $104 $120 $240 (ratios, medium) If earnings were $0.80 per share, $0.40 per share would be paid as dividends As earnings per share are actually $1.10 = $0.80 + 3($0.10), dividends per share will be $0.40 + 3($0.04) =$0.52 . For 200 shares, the dividend paid will be $104. $92 $96 $104 $120 $240

25

A certain economics report defines a middle-income family as a family whose income is at least half, but no more than twice, the median family income. According to this report, is a family whose income is $73,000 considered a middle-income family? (1) The median family income is $37,152 (2) The mimimum income of a middle-class family is $18,576 (inequalities, medium) (1) SUFF (2) From (2), we can deduce (1). Half the median income is $18,576, according to the definition. SUFF

26

A certain farmer pays $30 per acre per month to rent farmland. How much does the farmer pay per month to rent a rectangular plot of farmland that is 360 feet by 605 feet? (43,560 square feet = 1 acre) $5,330 $3,630 $1,350 $360 $150 (ratios, medium) The number of square feet is 360(605), so the number of acres rented is 360(605)/(43560) = 5 acres, or 5(30) = 150 dollars per month.

$5,330 $3,630 $1,350 $360 $150

27

A certain group of car dealerships agreed to donate x dollars to a Red Cross chapter for each car sold during a 30-day period. What was the total amount that was expected to be donated? (1) A total of 500 cars were expected to be sold. (2) 60 more cars were sold than expected, so that the total amount actually donated was $28,000. (algebra, medium) Suppose that the number that were expected to be sold is n. We are asked about the value of xn . (1) n = 500. Without knowing the value of x, we cannot answer the question. NOT SUFF (2) Suppose that the number that were expected to be sold is n. Then (n + 60)x = 28,000. xn = 28,000 60x. Without knowing the value of x, we cannot answer the question. NOT SUFF (T) 560x = 28000, so x= 28000/560. As we know the value of x and the value of n, we can answer the question. SUFF

28

A certain jar contains only b black marbles, w white marbles and r red marbles. If a marble is picked at random from the jar, is the probability that the marble chosen will be red greater than the probability that the marble chosen will be white? (1) r/(b + w) > w/(b + r) (2) b - w > r (probability, hard) The probability of getting a red marble will be greater than that of getting a white marble if and only if there are more reds than whites, that is, if r>w. ............................................................................................................ (1) The ratio of the number of reds to the number of non-reds is greater than the ratio of the number of whites to the number of non-whites. This indicates that there are indeed more reds than whites. One way to see that this must be so is to simplify the inequality given in (1) by using T (total) = r+w+b:

wrwTrTrTwwTrwT

w

rT

r

)()( SUFF

.................................................................................................................... (2) b-w > r , in other terms b > r+w. This means that there are more black marbles than red and white combined. Thus more than half of the marbles are black. However, we have no means of comparing r and w. NOT SUFF

29

A certain kennel will house 24 dogs for 7 days. Each dog requires 10 ounces of dog food per day. If the kennel purchases dog food in cases of 30 cans each and if each can holds 8 ounces of dog food, how many cases will the kennel need to feed all of the dogs for 7 days? 5 6 7 8 9 (ratios, medium) The number of ounces of food needed is 24 x 7 x 10 Each case contains 30 x 8 ounces of food. Dividing, we get (24 x 7 x 10)/ 30 x 8 = 7

5 6 7 8 9

30

A certain law firm consists of 4 senior partners and 6 junior partners. How many different groups of 3 partners can be formed in which at least one member of the group is a senior partner? (Two groups are considered different if at least one group member is different.)

48 100 120 288 600

(combinatronics, hard) Rather than consider the four cases in which at least one senior partner is chosen, it is faster to count the number of groups that do not include a senior

member, 36C = 6 x 5 x 4 / 3! = 20, and subtract this from the total number of

groups, 310C = 10 x 9 x 8/ 3 x 2 x 1 = 120.

48 100 120 288 600

31

A certain library assesses a fine for overdue books as follows. On the first day the book is overdue, the total fine is $0.10. For each additional day that the book is overdue, the total fine is either increased by $0.30 or doubled, whichever results in the lesser amount. What is the total fine for a book on the fourth day it is overdue?

$0.60 $0.70 $0.80 $0.90 $1.00 (algebra, medium) The fine in cents on the first day is 10. On the second, it is the lesser

of 10+30 and 10(2) , i.e. 20. On the third, it is the lesser of 20+30 and 20(2), i.e. 40. On the fourth, it is the lesser of 40+30 and 40(2), i.e. 70.

$0.60 $0.70 $0.80 $0.90 $1.00

32

A certain list consists of 5 different integers. Is the average (arithmetic mean) of the two greatest integers greater than 70? (1) The median of the integers in the list is 70. (2) The average of the integers in the list is 70. (statistics, hard) (1) The median of the integers is 70, so 70 is the third greatest integer. The two integers greater than the median are both greater than 70, so the average of these two integers will be greater than 70. SUFF (2) If the average of the integers in the list is 70, we can think of the list as two separate groups of integers, one that consists of the two greatest integers and the other consisting of the three smallest integers. The average of the first group is greater than the average of the second. As the average of all of the integers is 70, it must be that the average of the first group must be greater than 70. SUFF Alternatively, if the average of the two greatest is not greater than 70, the sum of these two integers is at most 140. As the sum of all five integers is 70(5)=350, the sum of the three smallest is at least 210, so the average of the three smallest is at least 70. However, the average of the three smallest must be less than that of the two greatest, as all the integers are different. Thus the average of the two greatest must be greater than 70. Some prefer to think as follows: if you remove the smallest element from a set of numbers, the average of the remaining elements will be higher than that of the numbers of the original set, 70. Remove the smallest of the remaining elements, and the average of the rest will rise.

33

A certain list consists of several different integers. Is the product of all the integers in the list positive? (1) The product of the greatest and smallest of the integers in the list is

positive. (2) There is an even number of integers in the list. (algebra, medium) In other words, the question is asking whether the number of negative integers in the list is an even number. (1) We know from (1) that either all the integers are positive, in which case, the number of negative integers is indeed an even number (0 is an even number), or all the integers are negative, in which case the number of negative integers is equal to the number of integers in the set, a number that may be odd or even. NOT SUFF. (2) Alone, this tells us nothing about the number of negative integers in the set. NOT SUFF (T) We know that the number of negative integers in the set must be an even number, either 0 (in the case that all of the integers are positive) or the number of elements in the set, an even number, according to (2). SUFF

34

A certain list of 100 data has a mean of 6 and a standard deviation of D, where D is positive. Which of the following pairs of data, when added to the list, must result in a list of 102 data with standard deviation less than D? -6 and 0 0 and 0 0 and 6 0 and 12 6 and 6 (statistics, medium) The standard deviation is a measure of how far the elements are from the mean. If to a set with at least two distinct numbers is added an element equal to the mean of that set, the standard deviation of the new set be less than that of the old set. We are told that the standard deviation the list of 100 data is positive, so if 6, the mean of the list of 100 data, is added twice, the resulting list of 102 data will have a standard deviation that is less than D.

-6 and 0 0 and 0 0 and 6 0 and 12 6 and 6

35

A certain manufacturer of cake, muffin, and bread mixes has 100 buyers, of whom 50 purchase cake mix, 40 purchase muffin mix, and 20 purchase both cake mix and muffin mix. If a buyer is to be selected at random from the 100 buyers, what is the probability that the buyer selected will be one who purchases neither cake mix nor muffin mix? A. 1/10 B. 3/10 C. 1/2 D. 7/10 E. 9/10

36

A certain meter records voltage between 0 volts and 10 volts, inclusive. If the average value of 3 recordings on the meter was 8 volts. What is the smallest possible recording, in volts ? 2 3 4 5 6 (algebra, medium) We know that the sum of the three recordings must be 3 x 8= 24. To minimize the value of one recording, find the maximum value of the sum of the other two recordings: 10+10=20. Thus, one recording could be as low as 24-20= 4

2 3 4 5 6

37

A certain movie depicted product A in 21 scenes, product B in 7 scenes, product C in 4 scenes, and product D in 3 scenes. The four product manufacturers paid amounts proportional to the number of scenes in which their product was depicted in the movie. If each manufacturer paid x dollars per scene, how much did the manufacturer of product D pay for this advertising? (1) The manufacturers of product A and B together paid a total of $560,000 for this advertising. (2) The manufacturer of product B paid $60,000 more for this advertising than the manufacturer of product C paid. (ratios, medium) We need to find the value of 3x. Any means of finding the value of x would be sufficient. (1) 21x + 7x= 28x =560,000. 3x= 3(560,000)/28 SUFF (2) 7x 4x =3x = 60,000 SUFF

38

A certain one-day seminar consisted of a morning session and an afternoon session. If each of the 128 people attending the seminar attended at least one of the two sessions, how many of the people attended the morning session only? (1) 3/4 of the people attended both sessions. (2) 7/8 of the people attended the afternoon session. (sets, medium) We can say that 128= |morning only| + |afternoon only| + |both morning and afternoon| From (1), we can deduce that at most 1/4 of the 128 people attended the morning session only. As no information is given about the number of people that attended the afternoon session only, we cannot say how many attended the morning session only. NOT SUFF (2) says that 7/8 of the people attended the afternoon session. Thus 1/8 did not attend the afternoon session. As each of the 128 people attended at least one of the two sessions, the number that attended the morning session only is 1/8 of 128. SUFF

39

A certain quantity is measured on two different scales, the R-scale and the S-scale, that are related linearly. Measurements on the R-scale of 6 and 24 correspond to measurements on the S-scale of 30 and 60, respectively. What measurement on the R-scale corresponds to a measurement of 100 on the S-scale? 20 36 48. 60 84 (coordinate geometry, medium) If the S scale were plotted on the y-axis, the relation between S (y) and R (x) would be a line with a slope of (60-30)/(24-6)= 5/3. Thus if (x, 100) is one this line, (100-60)/(x 24) =5/3, so 5(x -24) = 3(40) and x -24 = 24. Thus x=48.

20 36 48. 60 84

40

A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

8 12 15 21 27

(combinatronics, medium) We need to consider not only platters that have two kinds of fruit and two kinds

of cheese , 2622 CC = 1 x 15= 15, but also platters that have one kind of fruit

and one kind of cheese (2 x 6=12).

8 12 15 21 27

41

A certain roller coaster has 3 cars, and a passenger is equally likely to ride in any 1 of the 3 cars each time that passenger rides the roller coaster. If a certain passenger is to ride the roller coaster 3 times, what is the probability that the passenger will ride in each of the 3 cars? 0 1/9 2/9 1/3 1 (probability, medium) For this to happen, the car assigned on the second ride has to be different from the one assigned on the first and the one assigned on the third has to be the one car not assigned on either of the first two rides. Thus the required probability is 2/3 1/3 = 2/9. Alternatively, there are 33=27 equally probable car assignments (aaa,aba,aab,caa,,ccc), of which 3! involve rides on the 3 different cars: 3!/27 = 2/9.

0 1/9 2/9 1/3 1

42

A certain right triangle has sides of length x, y and z where x < y < z. If the area of this triangular region is 1, which of the following indicates all of the possible values of y?

2y 22

3 y

2

3

3

2 y

3

2

4

3 y

4

3y

(geometry, hard) The smaller the value of x, the larger the value of y. As there is no limit as to how how close to 0 x is, there is no limit as to how large y can be. Only the first choice reflects this fact. More formally, as this is a right triangle, x and y are the lengths of the legs, and z is the length of the hypotenuse. The area of this triangle is xy/2 < y2 / 2 since x < y . Thus y2 / 2 > 1 and y > 2

2y 22

3 y

2

3

3

2 y

3

2

4

3 y

4

3y

43

A certain state has a sales tax of 2 percent on the purchase price of all products. In addition, a city within this state imposes its own 0.5 percent sales tax on the purchase price of all products. If the sales tax on a particular product purchased in this city was $2.80, what was the purchase price of this product? 40 56 112 137 140 (percents, medium) A purchase p in this city is levied a total sales tax of 2.5%. Thus 2.5% of the p is 2.80, so 25% of p = 28 . Multiplying both sides by 4, p= 112

40 56 112 137 140

44

A certain stock exchange designates each stock with a one , two, or three letter code where each letter is selected from 26 letters of the alphabet. If the letters may be repeated and if the same letters used in a different order constitute a different code, how many different stocks is it possible to uniquely designate with these codes? 2,951 8,125 15,600 16,302 18,278 (combinatronics, hard) We need to count one, two and three-letter codes: 26 + 26x26+ 26x26x26. Note that the units digit of the sum will be 8, an observation that can save calculating time, but only if you are short of time!

2,951 8,125 15,600 16,302 18,278

45

A certain store sells chairs individually or in sets of 6. The store charges less for purchasing a set of 6 chairs than for purchasing 6 chairs individually. How much does the store charge for purchasing a set of 6 chairs? (1) The charge for purchasing a set of 6 chairs is 10 percent less than the

charge for purchasing the 6 chairs individually. (2) The charge for purchasing a set of 6 chairs is $20 more than the charge

for purchasing the 5 chairs individually. (percents, medium) (1) If the charge for each chair bought individually is x, the charge for the

set is 0.9(6x). However, no information about x is given. NOT SUFF (2) The set price is 5x + 20, where x is the price of one chair bought

individually. No information is given about x, though. NOT SUFF (T) We know 0.9(6x)= 5x +20, so x and thus the set price can be

found. SUFF

46

A certain telephone company offers two plans, A and B. Under plan A, the company charges a total of $0.70 for the first 7 minutes of each call and $0.06 per minute thereafter. Under plan B, the company charges $0.08 per minute of each call. What is the duration of the call, in minutes, for which the company charges the same amount under plan A and under plan B? 2 9 14 21 30 (algebra, medium) Consider an x minute call, where x is greater than 7: Under plan A, 0.7 +0.06(x-7) is charged, and under plan B, 0.08x is charged. Equating the two charges and solving for x: 0.02x=0.28 , 2x=28, x=14

2 9 14 21 30

47

A certain theater has a total of 884 seats, of which 500 are orchestra seats and the rest are balcony seats. When tickets for all the seats in the theater are sold, total revenue from ticket sales is $34,600. What was the theaters total revenue from ticket sales for last nights performance? (1) The price of an orchestra ticket is twice the price of a balcony ticket. (2) For last nights performance, tickets for all the balcony seats but only 80 percent of the tickets for the orchestra were sold. (algebra, medium) (1) Suppose the price of a balony seat is p dollars. Then 500(2p) + 384(p) =34,600. p can be found, so the price of each type of ticket can be found. However, we know nothing about last nights sales. NOT SUFF (2) We cannot determine the exact price of each type of ticket. NOT SUFF (T) The total revenue, 334p + 0.8(500)2p can be found, as (1) gives us the value of p. SUFF

48

A certain triangle has two angles measuring 45 and 75 If the side opposite the 45 angle has length 6, what is the length of the side opposite the 75 angle?

332 )13(3 6 36 262

(geometry, hard) Remember the two right isosceles triangles we saw in class!

332 )13(3 6 36 262 CORRECT

49

A child selected a three-digit number, XYZ, where X, Y and Z denote the digits of the number and X + Y + Z= 10. If no two of the three digits were equal, what was the three-digit number? (1) X < Y < Z (2) The three-digit number selected was even. (factors and multiples, easy) (1) XYZ could be 127, 136, 145 or 235 NOT SUFF (2) XYZ could be 136 or 226 or one of many other possibilities NOT SUFF (T) XYZ must be 136

50

A circular jogging track forms the edge of a circular lake that has a diameter of 2 miles. Johanna walked once around the track at the average rate of 3 miles per hour. If t represents the number of hours it took Johanna to walk completely around the lake, which of the following is a correct statement? 0.5 < t < 0.75 1.75 < t < 2.0 2.0 < t < 2.5 2.5 < t < 3.5 3 < t < 3.5 (geometry, movement, medium) Remember that distance = rate x time, so the time Johanna took (in hours) is the distance she walked (in miles), the circumference of a circle with diameter 2, divided by the rate at which she walked ( in miles per hour)-

So t= 32 . As is slightly more than 3, t is slightly more than 2.

0.5 < t < 0.75 1.75 < t < 2.0 2.0 < t < 2.5 2.5 < t < 3.5 3 < t < 3.5

51

A clock store sold a certain clock to a collector for 20 percent more than the store had originally paid for the clock. When the collector tried to resell the clock to the store, the store bought it back at 50 percent of what the collector had paid. The shop then sold the clock again at a profit of 80 percent on its buy-back price. If the difference between the clock's original cost to the shop and the clock's buy-back price was $100, for how much did the shop sell the clock the second time? $270 $250 $240 $220 $200

(percents, hard)

original cost = x collector price = 1.2x buy back price = 0.6x resell price = 1.08x x- 0.6x = 100 => x = 100/0.4 = 250 resell price = 1.08 250 = 270

$270 $250 $240 $220 $200

52

A clothing store acquired an item at a cost of x dollars and sold the item for y dollars. The stores gross profit from the item was what percent of its cost for the item? (1) y x = 20 (2) xy = 45 (algebra, percents, medium) We are asked for the value of ((y x)/x)100%. It is sufficient to know the value of y/x (1) NOT SUFF (2) NOT SUFF (T) With the two equations, we can say that (y 20)y =45 or y2 20y 45 = 0, a quadratic equation that has two solutions of opposite signs, Since the selling price y must be positive, we can determine the one possible value of y and thus of x. SUFF

53

A collection of 36 cards consists of 4 sets of 9 cards each. The 9 cards in each set are numbered 1 through 9. If one card has been removed from the collection, what is the number on that card? (1) The units digit of the sum of the numbers on the remaining 35 cards is 6. (2) The sum of the numbers on the remaining 35 cards is 176. Note that we can find the sum of the entire collection. One need not do so, but it is 4(1 + 2 + ... + 8 + 9) = 4(9)(5) = 180 (an arthemetic sequence with first and last terms a and b has a sum of (a+b)n/2, where n is the number of terms. (1) If we know the units digit of the sum of the remaining 35 cards, we can determine the units digit of the card removed. Since the number on each card is a one-digit number. SUFF (2) SUFF

54

A college admissions officer predicts that 20 percent of the students who are accepted will not attend the college. According to this prediction, how many students should be accepted to achieve a planned enrollment of x students? 1.05x 1.1x 1.2x 1.25x 1.8x (percents, medium) Suppose that n are accepted. 0.2n will not enroll. Thus 0.8n= x and n = x/0.8 = x/(4/5) = 5x/4 =1.25x.

1.05x 1.1x 1.2x 1.25x 1.8x

55

A combined total of 55 lightbulbs are stored in two boxes; of these, a total of 7 are broken. If there are exactly two broken bulbs in the first box, what is the number of bulbs in the second box that are not broken? (1) In the first box, the number of bulbs that are not broken is 15 times the number of broken bulbs. (2) The total number of bulbs in the first box is 9 more than the total number of bulbs in the second box. (sets, medium) BOX 1 BOX 2 TOTAL BROKEN 2 5 7 NOT BROKEN 48 - x x 48 TOTAL 50 - x x + 5 55 We are asked for the value of x. (1) 48 x = 15(2) SUFF (2) 50 x = (x + 5) + 9 SUFF

56

A company has 2 types of machines, type R and type S. Operating at a constant rate, a machine of type R does a certain job in 36 hours and a machine of type S does the job in 18 hours. If the company used the same number of each type of machine to do the job in 2 hours, how many machines of type R were used? 4 5 6 8 10 (combined work, medium) Suppose x of each type of machine were used to do the job in 2 hours. Thus the fraction of the job that all machines do in 1 hour is 1/2. We can write:

62

1

1236

3

2

1

1836 x

xxxx

4 5 6 8 10

57

A company wants to spend equal amounts of money for the purchase of two types of computer printers costing $600 and $375 per unit, respectively. What is the fewest number of computer printers that the company can purchase?

13 12 10 8 5

(ratios, factors and multiples, hard) If x $600 printers and y $375 printers are to be pruchased, it must be that x:y=375:600=75:120=15:24=5:8. Thus we know that the company will buy 5k $600 printers and 8k $375 printers. The number of printers bought, then is 13k, a multiple of 13. 13 is the smallest multiple of 13.

13 12 10 8 5

58

A company wants to buy printers and computers for a new branch office and the number of computers can be at most 3 times the number of printers Computers cost $1500 each and printer cost $300 each what is the greatest number of computers that a company can buy if it has a total of $9100 to spend on computers and printers ?

2 3 4 5 6 (ratios, hard) Ideally, each printer would be accompanied by 3 computers. This bundle

costs $4,800. Two such bundles (a total of 2 printers and 6 computers) would cost $9,600, $500 more than the limit. 2 printers and 5 computers would cost $1000 less than the limit.

2 3 4 5 6

59

Shipment Number of defective chips in the shipment

Total number of chips in the shipment

S1 2 5,000

S2 5 12,000

S3 6 18,000

S4 4 16,000

A computer chip manufacturer expects the ratio of the number of defective chips to the total number of chips in all future shipments to equal the corresponding ratio for shipments S1, S2, S3 and S4 combined, as shown in the following table. What is the expected number of defective chips in a shipment of 60,000 chips?

14 20 22 24 25

(ratios, medium)

Overall there were 17 defective chips out of a total of 51,000 chips. The ratio of the number of defective chips to the total number of chips in the shipment is 17/51,000 = 1/3,000. Thus in 60,000 =3,000 x 20 chips, 20 defective chips can be expected.

14 20 22 24 25

60

A construction company was paid a total of $500,000 for a construction project. The companys only costs for the project were for labor and materials. Was the companys profit for the project greater than $150,000? (1) The companys total cost was three times its cost for materials. (2) The companys profit was greater than its cost for labor. (inequalities, hard) The profit P = 500 L M The question is whether P> 150 i.e. if L+M < 350 (1) L + M = 3M, i.e. L=2M NOT SUFF (2) P=500-L-M> L 2L + M < 500 NOT SUFF (1) and (2) 5M < 500 M< 100 L = 2M < 200 L + M < 300 < 350 SUFF

61

A contractor combined x tons of a gravel mixture that contained 10% gravel G, by weight, with y tons of a mixture that contained 2% gravel G, by weight, to produce z tons of a mixture that was 5% gravel, by weight. What is the value of x? (1) y = 10 (2) z = 16 (percents, ratios, hard) Note that the information before the question does not give us the means to find x, y or z, but does allow us to find the ratios of these variables: 0.1x + 0.02y=0.05(x+y) so y/x is 5/3. As z=x+y, x/z= x/(y+x)= 3/8

Thus each of (1) and (2) provides sufficient information to find x.

62

A contest will consist of n questions, each of which is to be answered either "True" or "False". Anyone who answers all n questions correctly will be a winner. What is the least value of n for which the probability is less than 1/1000 that a person who randomly guesses the answer to each question will be a winner?

5 10 50 100 1000

(probability, hard) The probability that a contestant will answer a given question correctly is 1/2. Therefore the probability that this person will answer n questions in succession is (1/2)n. If (1/2)n < 10 -3, 2n> 103. Given that 29= 512 and 210= 1024, n must be greater than or equal to 10.

5 10 50 100 1000

63

According to a survey, 93 percent of teenagers have used a computer to play games, 89 percent have used a computer to write reports, and 5 percent have not used a computer for either of these purposes. What percent of the teenagers in the survey have used a computer both to play games and to write reports?

82% 87% 89% 92% 95% (sets, medium) Suppose that n teenagers were surveyed. 0.93n have used a computer to play games, 0.89n have used a computer to write reports, and 0.05n percent have not used a computer for either of these purposes. If y teenagers have used computers for both purposes, n=0.93n + 0.89n + 0.05n y, so y=0.87n

82% 87% 89% 92% 95%

64

A craftsman made 126 ornaments and put them all into boxes. If each box contained either 6 ornaments or 24 ornaments, how many of the boxes contained 24 ornaments? (1) Fewer than 4 of the boxes contained 6 ornaments. (2) More than 3 of the boxes contained 24 ornaments. (inequalities, factors and multiples, hard) Suppose there were x boxes containing 6 ornaments and y containing 24 ornaments. 6x + 24y = 126 i.e. x +4y =21. We are asked for the value of y = 21- x) /4. Note that as x and y are integers, x must be 1 greater than a multiple of 4 (21 is 1 greater than a multiple of 4) (1) As x < 4, x must be 1, and we can calculate the value of y. SUFF (2) Note that x = 21- 4y. As y > 3, y can be either 4 or 5. NOT SUFF

65

Adam and Beth each drove from Smallville to Crown City by different routes. Adam drove at an average speed of 40 miles per hour and completed the trip in 30 minutes. Beths route was 5 miles longer, and it took her 20 minutes more than Adam to complete the trip: How many miles per hour was Beths average speed on this trip? 24 30 48 54 75 (movement, medium) Adams route is 40(1/2) = 20 miles, so Beths is 25 miles. As Beth drove 30+20=50 minutes, 5/6 of one hour, her average speed was 25/(5/6) = 30 miles per hour.

24 30 48 54 75

66

A driver completed the first 20 miles of a 40-mile trip at an average speed of 50 miles per hour. At what average speed must the driver complete the remaining 20 miles to achieve an average speed of 60 miles per hour? 65 68 70 75 80 (movement, medium) To achieve an average speed of 60 miles an hour for the 40 mile trip, the distance must be covered in 40/60= 2/3 hour. The first 20 miles took 20/50=2/5 hour, so the remaining 20 miles must be covered in 2/3 -2/5 =4/15 hour. Thus the average speed required for the remaining 20 miles is 20(4/15) =75 miles per hour.

65 68 70 75 80

67

A family-size box of cereal contains more cereal and costs more than the regular-size box of cereal. What is the cost per ounce of the family-size box of cereal? (1) The family-size box of cereal contains 10 ounces more than the regular- size box of cereal. (2) The family-size box of cereal costs $5.40. (algebra, ratios, easy) (T) As no information is given about the number of ounces in a regular box of cereal, more information is needed. NOT SUFF

68

A farm used two harvesting machines, h and k, to harvest 100 acres of wheat. Harvesting machine h, working alone at its constant rate harvested 40 acres of wheat in 8 hours. Then harvesting machine k was brought in, and harvesting machines h and k, working together at their respective constant rates, harvested the remaining acres of wheat in 5 hours. Harvesting machine k harvested how many acres of wheat per hour? 7 8 12 13 15 (combined work, medium) Machine h harvested 40 acres in 8 hours, so it harvests 5 acres per hour. In the 13 hours it worked, it harvested 65 of the 100 acres. The remaining 35 acres were harvested in 5 hours by k, at a rate of 7 per hour.

7 8 12 13 15

69

A five-member committee is to be formed from a group of five military officers and nine civilians. If the committee must include at least two officers and two civilians, in how many different ways can the committee be chosen? 119 1200 3240 3600 14400 (combinatronics, hard) Such a committee must consist of either 3 officers and 2 civilians or 2 officers and 3 civilians. There are 5C3 x 9C2 committees that can be formed with 3 officers and 2 civilians 10 x 36 = 360 There are 5C2 x 9C3 committees that can be formed with 2 officers and 3 civilians 10 x 84 = 840 Thus there are 1200 different ways the committee can be chosen. A common mistake is to choose two officers and two civilians and then choose a fifth person from among the 10 people not chosen. This results in double counting.

119 1200 3240 3600 14400

70

A folk group wants to have one concert on each of the seven consecutive nights starting January 1 of next year. One concert is to be held in each of cities A, B, C, D and E. Two concerts are to be held in city F, but not on consecutive nights. In how many ways can the group decide on the venues for these seven concerts? 10 x 5! 14 x 5! 15 x 5! 20 x 5! 21 x 5! (combinatronics, hard) Step I: Decide which two non-consecutive concerts will be be held in F:

This can be done in 27C - 6 = 15 ways. (Exclude the 6 placements in which the

two dates are consecutive. Step II: Assign venues A, B, C, D and E to the remaining 5 concerts: 5! ways. Answer: 15 x 5!

10 x 5! 14 x 5! 15 x 5! 20 x 5! 21 x 5!

71

A 4-person task force is to be formed from the 4 men and 3 women who work in company G's human resources department. If there are to be 2 men and 2 women on this task force, how many task forces can be formed? 14 18 35 56 144 (combinatronics, medium)

There are 64

2 C ways to choose the men for the task force and 3 ways to

decide which woman will not serve on this task force. Therefore there are 3 x 6 =18 ways to form this task force.

14 18 35 56 144

72

A glass was filled with 10 ounces of water, and 0.01 ounce of the water evaporated each day during a 20-day period. What percent of the original amount of water evaporated during this period? 0.002% 0.02% 0.2% 2% 20% (percents, medium) Over the 20 days, 0.20 ounces evaporated, and (0.2/10) x 100% = 2%

0.002% 0.02% 0.2% 2% 20%

73

Alices take-home pay last year was the same each month, and she saved the same fraction of her take-home pay each month. The total amount of money that she had saved at the end of the year was 3 times the amount of that portion of her monthly take-home pay that she did not save. If all the money that she saved last year was from her take-home pay, what fraction of her take-home pay did she save each month? 1/2 1/3 1/4 1/5 1/6 (algebra, hard) Suppose that she she saved fraction f of her take-home pay p, where 0

74

A jar contains 16 marbles, of which 4 are red, 3 are blue, and the rest are yellow. If 2 marbles are to be selected at random from the jar, one at a time without being replaced, what is the probability that the first marble selected will be red and the second marble selected will be blue? 3/64 1/20 1/16 1/12 1/8 (probability, hard) Let R1 and B2 be the events that the the first marble drawn is red and the second marble drawn is blue.

20

1

15

3

16

4)|Pr()Pr()Pr( 12121 RBRBR

3/64 1/20 1/16 1/12 1/8

75

All of the stocks on the over-the-counter market are designated by either a 4-letter or a 5-letter code that is created by using the 26 letters of the alphabet. Which of the following gives the maximum number of different stocks that can be designated with these codes?

2(265) 26(264) 27(264) 26(265) 27(265) (combinatronics, hard) We need to consider both 4-letter and 5-letter codes. There are 264 four-letter codes and 265= 26(264) five-letter codes, so the maximum number of stocks that can be so designated is 27(264).

2(265) 26(264) 27(264) 26(265) 27(265)

76

A furniture dealer purchased a desk for $150 and then set the selling price equal to the purchase price plus a markup that was 40 percent of the selling price. If the dealer sold the desk at the selling price, what was the amount of the dealer's gross profit from the purchase and the sale of the desk?

$40 $60 $80 $90 $100

(percents, medium) Suppose the selling price was s dollars. Then the gross profit was s 150. As s= 150 + 0.4s, 0.6s=150 and s=250. The gross profit, therefore, is 250-150=100.

$40 $60 $80 $90 $100

77

A glass was filled with 10 ounces of water, and 0.01 ounce of the water evaporated each day during a 20-day period. What percent of the original amount of water evaporated during this period? 0.002% 0.02% 0.2% 2% 20% (precents, medium) In all 20/100 =0.2 ounces of the 10 ounces of water evaporated. Since 0.2/10 = 2/100, 2% of the 10 ounces of water evaporated.

0.002% 0.02% 0.2% 2% 20%

78

A grocer has 400 pounds of coffee in stock, 20% of which are decaffeinated. If the grocer buys another 100 pounds of coffee of which 60% is deffeinated, what percent, by weight, of the grocers stock of coffee is decaffeinated ? 28 % 30 % 32 % 34 % 40 % (percents, medium) Of the 500 pounds of coffee in stock, 0.2(400) + 0.6(100) = 140 are decaffeinated. 140/500 = 280/1000 = 28%

28 % 30 % 32 % 34 % 40 %

79

A grocer stocks oranges in a pile.The bottom layer was rectangular with 3 rows of 5 oranges each. In the second layer from the bottom, each orange rested on 4 oranges from the bottom layer and in the third layer, each orange rested on 4 oranges from the second layer. Which of the following is the maximum number of oranges that could have been in third layer? 5 4 3 2 1 (geometry, medium)

The second layer from the bottom could have as many as 2 x 4 = 8 oranges, and the third layer from the bottom could have as many as 1 x 3 =3 oranges.

5 4 3 2 1

80

A hiker walking at a constant rate of 4 miles per hour is passed by a cyclist travelling in the same direction along the same path at a constant rate of 20 miles per hour. The cyclist stops to wait for the hiker 5 minutes after passing her, while the hiker continues to walk at her constant rate. How many minutes must the cyclist wait until the hiker catches up?

3

26 15 20 25

3

226

(movement, hard) Remember that d (distance) =v (velocity) multiplied by t (time). In the five minutes (1/12 hour) that elapses between the moment when the cyclist passed the hiker and the moment when the cyclist stopped, the distance between the two people increased at a rate of 20 4 = 16 miles per hour. Thus the distance between the two people when the cyclist stopped was 16/12 = 4/3 mile. It will take the hiker 4/3 4 = 1/3 hour (20 minutes) to cover this distance.

3

26 15 20 25

3

226

81

A lawyer charges her clients $200 for the first hour of her time and $150 for each additional hour. If the lawyer charged her new client $1550 for a certain number of hours of her time, how much was the average (arithmetic mean) charge per hour? $155 $160 $164 $172 $185 (algebra, medium) Suppose that she charged for x hours, so that she charged a total of 200 + 150( x 1) = 1550. Thus x 1 = 9 and x = 10. The average charge per hour, then, was 1550/10 = $155. $155 $160 $164 $172 $185

82

Amys grade was the 90th percentile of the 80 grades for her class. Of the 100 grades from another class, 19 were higher than Amys, and the rest were lower. If no other grade was the same as Amys grade, then Amys grade was what percentile of the grades of the two classes combined?

72nd 80th 81st 85th 92nd

(percents, medium) Amy did as well as or better than 90% of the 80 students in her class and 100-19=81 students in the other class. Thus she did as well as or better than 72+81=153 of the 180 students. As 153/180 =17/20=85%, Amys grade was in the 85th percentile of the two classes combined. Note that as the two classes are fairly equal in size, we expect the percentile to be between roughly halfway between 81 and 90.

72nd 80th 81st 85th 92nd

83

All points (x,y) that lie below the line l , shown above, satisfy which of the following inequalities? y < 2x + 3 y < -2x + 3 y < -x +3 y < x/2 +3 y< -x/2 +3 (coordinate geometry, medium) Note that the line has a slope of (0-3)/(6-0)= -1/2 and a y intercept of 3, so the equation of the line is y= -1/2x + 3. All points below the line are given by the inequality y < -x/2 + 3. y < 2x + 3 y < -2x + 3 y < -x +3

y < x/2 +3 y< -x/2 +3

84

A lighthouse blinks regularly 5 times a minute. A neighboring lighthouse blinks regularly 4 times a minute. If they blink simultaneously, after how many seconds will they blink together again?

20 24 30 60 300

(factors and multiples, medium) The first lighthouse blinks every 12 seconds, whereas the neighbouring lighthouse does so every 15 seconds. We see that the least common multiple of 12 and 15 is 60, so they will not blink together until 60 seconds have passed. You could also reason that, as the greatest common divisor of 4 and 5 is 1, their blinking will not coincide at any time less than one minute.

20 24 30 60 300

85

A list of measurements in increasing order is 4,5,6,8,10, and x. If the median of these measurements is 6/7 times their arithmetic mean, what is the value of x?

16 15 14 13 12

(statistics, medium) As there are 6 measurements, the median is the average of the 3rd and 4th elements when the elements are listed in increasing order. Thus the median is 7. The arithmetic mean of these measurements is (33+x)/6. Thus 7=6/7 of (33+x)/6 and so x=16.

16 15 14 13 12

86

Al, Pablo, and Marsha shared the driving on a 1,500-mile trip. Which of the three drove the greatest distance on the trip?

(1) Al drove 1 hour longer than Pablo but at an average rate of 5 miles per hour lower than Pablo.

(2) Marsha drove 9 hours and averaged 50 miles per hour. (movement, hard) (T) Marsha drove 450 miles, so Al and Pablo drove the remaining 1050 miles. Either Al or Pablo drove the greatest distance, as they drove an average of 525 miles each If they drove quickly, the additional hour that Al drove, even at a slightly lower speed, would make him the one who drove the greatest distance. However, it they drove slowly, the faster speed at which Pablo drove would offset the shorter duration behind the wheel. NOT SUFF Suppose Pablo drove for t hours at a rate of v miles per hour Then Al drove for t + 1 hours at a rate of v 5 miles per hour. Pablo drove farther than Al if vt > (t + 1)(v 5) = vt + v 5t 5, i.e. if v 5t < 5

87

A manufacturer conducted a survey to determine how many people buy products P and Q. What fraction of the people surveyed said that they buy neither product P nor product Q? (1) 1/3 of the people surveyed said that they buy product P but not product Q. (2) 1/2 of the people surveyed said that they buy product Q. (sets, medium) If one can find the fraction f of people surveyed that said that they buy at least one of the two products, 1 f is the fraction that said that they buy neither product. f = p + q + b, where p is the fraction that said that they bought P only, q is the fraction that said that they bought q only, and b is the fraction that said that they bought P and Q. (1) p = 1/3. No information is given about q or b. INSUFF (2) q + b = 1/2. No information is given about p. INSUFF (T) p + q + b = 1/3 + 1/2. SUFF

88

A manufacturer produced x percent more video cameras in 1994 than in 1993 and y percent more video cameras in 1995 than in 1994. If the manufacturer produced 1,000 video cameras in 1993, how many video cameras did the manufacturer produce in 1995? (1) xy = 20 (2) x + y + xy/100 = 9.2 (percents, algebra, hard) We know that the manufacturer produced x percent more video cameras in 1994 than it did in 1993, so the number of cameras produced in 1994 is 1000(1+x/100). As the number of cameras produced in 1995 is y% greater than the number produced in 1994, the number produced in 1995 is 1000(1+x/100)(1+y/100) =1000 (1 + (x+y)/100 + xy/10000)=1000 + 10(x+y)+ xy/10. (1) Knowing that xy=20 does not allow us to find x+y. Therefore the number produced in 1995 cannot be determined. NOT SUFF (2) If x+y +xy/100 = 9.2, 10(x+y)+xy/10 =92, and the number produced in 1995 can be found. SUFF

89

An athlete runs R miles in H hours, then rides a bike Q miles in the same number of hours. Which of the following represents the average speed, in miles per hour, for these 2 activities combined? (R-Q)/H (R-Q)/2H 2(R+Q)/H 2(R+Q)/2H (R+Q)/2H (movement, easy) The average speed, in miles per hour, for these 2 activities combined, is the total distance covered in miles, R + Q , divided by the total number of hours, 2H.

(R-Q)/H (R-Q)/2H 2(R+Q)/H 2(R+Q)/2H (R+Q)/2H

90

An equilateral triangle ABC is inscribed in square ADEF, forming three right triangles: ADB, ACF and BEC. What is the ratio of the area of triangle BEC to that of triangle ADB?

4/3 3 2 5/2 5 (geometry, hard)

Since |AB|=|BC|, x2 + 2xy + y2 +x2 = 2y2, so 2x2 +2xy =2x(x+y)= y2

The ratio of the area of triangle BEC to that of triangle ADB is y2/x(x+y) =2

4/3 3 2 5/2 5

91

An integer greater than 1 that is not a prime number is called composite. If two-digit integer n is greater than 20, is n composite? (1) The tens digit of n is a factor of the units digit of n. (2) The tens digit of n is 2. (factors and multiples, hard) (1) Suppose the tens digit of n is x.The the units digit of n is kx, where k is a positive integer. Therefore n = 10x + kx = x (10 + k). Since x is at least two, x , an integer greater than 1 but less than n, is a divisor of n. Thus n is a not a prime number. SUFF (2) n may or may not be a prime number: n could be 22 ( a composite number) or 23 ( a prime number). NOT SUFF

92

An investment of d dollars at k percent simple annual interest yields $600 interest over a 2-year period. In terms of d, what dollar amount invested at the same rate will yield $2400 interest over a 3-year period? 2d / 3 3d /4 4d /3 3d /2 8d /3 (interest, ratios, medium) Remember that at r% simple interest over n year, an investment of I dollars yields I x r x n dollars. Let x be the amount invested over a 3-year period. 3xk=2400 But 2dk=600 Thus 3xk=4(2dk) and x=8d/3 Alternatively, to earn 4 times as much interest in 3/2 of the original time, one needs 4 2/3 = 8/3 times the original principal!

2d / 3 3d /4 4d /3 3d /2 8d /3

93

An investor opened a money market account with a single deposit of $6,000 on December 31, 2001. The interest earned on the account was calculated and reinvested quarterly. The compounded interest reported for the first three quarters was $125 , $130 , and $145, respectively. If the investor made no deposits or withdrawals during the year, approximately what annual rate of interest must the account earn for the fourth quarter in order for the total interest earned on the account for the year to be 10 percent of the inital deposit? 3.1% 9.3% 10.0% 10.5% 12.5% (interest, hard) 10 percent of the inital deposit is $600, of which $400 was earned in the first three quarters, and the balance at the end of the 3rd quarter is $6,400. Thus $200 must be earned in the 4th quarter, so the annual rate of interest r that the account must earn for the 4th quarter is given by the following: 6400 (1+r/4)=6600, so r = 4 (6600/6400 1) = 4 (33/32 1) > 4(0.03) =0.12

3.1% 9.3% 10.0% 10.5% 12.5%

94

An investor purchased 20 shares of a certain stock at a price of $45.75 per share. Late this investor purchased 30 more shares at a price of $46.25 per share. What is the average (arithmetic mean) price per share that this investor paid for the 50 shares? $45.80 $45.95 $46.00 $46.05 $46.20 (ratios, medium) The total sum of money paid is 20(45.75) + 30(46.25) = 50(45.75) + 30(0.50) Dividing this sum by 50, we get the average price per share: 45.75 + 0.3 = 46.05. This is just more than the average of the two prices, $46.00.

$45.80 $45.95 $46.00 $46.05 $46.20

95

Ann deposited money into two accounts, A and B. Account A earns 8% simple interest and B earns 5% simple interest. If there were no other transactions, A gained how much more interest in the first year than did B in the first year? (1) Ann invested $200 more in B than she did in A. (2) The total amount of interest gained by the two accounts in the first year was $120. (interest, hard) Suppose the amounts deposited are a and b. We are asked for the value of D= 0.08a 0.05b (1) b= 200 + a, so D = 0.03a 10 NOT SUFF (2) 0.08a + 0.05b = 120, so D= 120 0.1b NOT SUFF (T) As we have two linear equations involving a and b, we can solve for both variables and then answer the question. SUFF

96

{ - 10, -6, -5, -4, -2.5, -1, 0, 2.5, 4, 6, 7, 10} A number is to be selected at random from the set above. What is the probability that the number selected will be a solution of the equation (x 5)(x + 10)(2x 5) = 0 ? 1/12 1/6 1/4 1/3 1/2 (proabability,medium) The solutions of the equation above are x= 5, x= - 10, x= 2.5. As two of these values are elements of the set of numbers above, 2 of the 12 elements are solutions of the equation. Therefore, the probability that the number selected will be a solution of the equation (x 5)(x + 10)(2x 5) = 0 is 2/12 = 1/6.

1/12 1/6 1/4 1/3 1/2

97

A parent established a college fund for his daughter. Each year the parent made a contribution to the fund, and each year he increased his contribution by a constant amount. If he made a contribution of $800 in the first year, by what amount did the parent increase his contribution to the fund each year? (1) The parents contribution to the fund in the 18th year was $7600. (2) The parents contribution to the fund in the 7th year was twice what it was in the 3rd year. (sequences, hard) The contribution in the nth year is 800 + (n 1)x , where x is the amount by which his contribution increases each year. We are asked to find the value of x. (1) 800 + (18 1)x = 7600 SUFF (2) 800 + (7 1)x =2(800 + (3 -1)x) SUFF

98

A point is arbitrarily selected on a line segment, breaking it into two smaller segments.What is the probability that the bigger segment is at least twice as long as the smaller? 1/4 1/3 1/2 2/3 3/4 (algebra, hard) Think of the number line with the segment between 0 and 1, which point x chosen as cutting point. The left segment is at least twice as long as the right if x 2(1-x) i.e. x 2/3 Thus the probability that the left segment is at least twice as long as the right is 1/3, as is the probability that the right is at least twice as long as the left. So the required probability is 1/3 + 1/3 =2/3

1/4 1/3 1/2 2/3 3/4

99

A positive integer n is said to be prime-saturated if the product of all the different positive prime factors of n is less than the square root of n. What is the greatest two-digit prime-saturated integer? 99 98 97 96 95 (factors and multiples, hard) For a positive integer n slightly less than 100 to be prime saturated, its only prime factors should be 2 and 3, as if it has larger prime factors the product of the prime factors will be greater than 10. 96 = 24 4 = 3 25 is prime-saturated, but none of the other numbers are: 99 is a multiple of 11, 98 is a multiple of 7 and 2, 97 is a prime number, and 95 is a multiple of 19 and 5.

99 98 97 96 95

100

Are at least 10% of the people in Country X who are 65 years old or older employed? (1) In Country X, 11% of the population is 65 years old or older. (2) In Country X, of the population 65 years old or older, 20% of the men and 10% of the women are employed. (algebra, hard) (1) gives us no information about employment. NOT SUFF (2) tells us that the percentage of all people 65 years old or order that is employed is between 10% and 20%, and is thus at least 10%. SUFF

101

Are positive integers p and q both greater than n? (1) p q is greater than n. (2) p < q (inequalities, hard) (1) p > n + q, so p is greater than n. However, q may or may not be be greater than n. NOT SUFF (2) p q is negative, but nothing is known about n NOT SUFF (1) and (2): p q < 0 but p q > n, so n

102

A rectangular region has a fence along three sides and a wall along the fourth side. The fenced side opposite the wall is twice the length of each of the other two fenced sides. If the area of the rectangular region is 128 square feet, what is the total length of the fence in feet? 4 8 16 32 64 (algebra, easy) The lengths of the 3 sides are x, x and 2x, for a total of 4x , and the area of the

rectangular region is 2x2 = 128, so x=8 and 4x = 32.

4 8 16 32 64

103

Are x and y both positive? (1) 2x -2y = 1 (2) x/y > 1 (inequalities, hard) (1) tells us that x= y + 1/2, so x and y can be both positive or both negative. Also y could be negative and x positive. NOT SUFF (2) x/y > 1 tells us that x > y if y>0 and x y if and only if y >0. Thus x and y are both positive. SUFF

104

A saleswomans monthly income has two components, a fixed component of $1000, and a variable component, which is $C for each set of encyclopaedias that she sells in that month over a sales target of n sets, where n > 1. How much did she earn in March? (1) If she had sold three fewer sets than she actually did, her March income

would have been $600 less. (2) If she had sold 8 sets of encyclopaedias, her income in March would

have been over $4000. (inequalities, hard) We see that the saleswomans income in a given month on sales for that month of x sets is (in dollars) 1000 if xn. (1) 600 is equal to either C, 2C or 3C. If 600=C or 600=2C. her income for that month was $1600. However, if 600=3C, her income could be $1600 or more. NOT SUFF (2) Clearly, n4000 and thus C > 3000/(8-n). We do not know, however, the exact value of

C. Nor do we know how many sets she actually sold. NOT SUFF (T) From (2), we see that C> 3000/7 > 300, so (1) tells us that

C=600 and her income for the month was $1,600. SUFF

105

A scientist recorded the number of eggs in each of 10 birds nests. What was the standard deviation of the numbers of eggs in the 10 nests? (1) The average (arithmetic mean) number of eggs for the 10 nests was 4. (2) Each of the 10 nests contained the same number of eggs. (statistics, medium) (1) Remember that standard deviation of a set of numbers is the average distance between each element. No information is given in (1) about how far apart the elements are. NOT SUFF (2) Since each element is equal to the mean number of eggs, the standard deviation is 0. SUFF

106

A set of 15 different integers has a median of 25 and a range of 25. What is the largest possible integer that could be in the set? 32 37 40 43 50 (statistics, hard) In a nutshell, since the range is 25, the greater the value of the smallest integer in the set, the greater the value of the largest integer in the set. If we look at the integers in increasing order, x1

107

A set of data consists of the following 5 numbers: 0,2,4,6, and 8. Which two numbers, if added to create a set of 7 numbers, will result in a new standard deviation that is closest to the standard deviation for the 5 original numbers? -1 and 9 4 and 4 3 and 5 2 and 6 0 and 8 (statistics, hard) Note that the terms of the original set are on average 2 units from the arithemetic mean, 4. Thus the addition of elements that are 2 units from 4 (i.e. 2 and 6) will result in a larger set with a very similar standard deviation.

-1 and 9 4 and 4 3 and 5 2 and 6 0 and 8

108

A state legislature had a total of 96 members. The members who did not vote on a certain bill consisted of 25 who were absent and 3 who abstained. How many of those voting voted for the bill? (1) Exactly 1/3 of the total membership of the legislature voted against the bill. (2) The number of legislators who voted for the bill was 8 more than the total number who were absent or abstained. We need to know how many of the remaining 96 -28 legislators voted for the bill. Each of these legislators voted either for the bill or against it. (1) If 1/3 of the 96 voted against the bill, the number who voted for it is 96 28 1/3(96) SUFF (2) The number required is 8 more than 25 + 3 SUFF

109

A store purchased 20 coats that each cost an equal amount and then sold each of the 20 coats at an equal price. What was the store's gross profit on the 20 coats? (1) If the selling price per coat had been twice as much, the store's gross

profit on the 20 coats would have been $2,440. (2) If the selling price per coat had been $2 more, the store's gross profit

on the 20 coats would have been $440. (algebra, hard) Suppose that the cost of each coat is c dollars and the selling price for each

coat is s > c. We are asked for the value of 20(s - c). (1) 20 (2s c) = 2400, so 2s c = 200 and c = 200 2s. Thus the stores

gross profit is 20(200 s). We cannot answer the question without the value of s. If fact, if s=200, the gross profit would have been 0!

NOT SUFF (2) 20( s+2 c) = 440, so 20(s c) =440 40 = 400 SUFF

110

A researcher computed the mean, the median, and the standard deviation for a set of performance scores. If 5 were to be added to each score, which of these statistics would change? mean only median only standard deviation only mean and median mean and standard deviation (statistics, medium) Remember that mean and median are measurements of the central tendancy of the score. If 5 were to be added to each score, the sum of the scores would increase, and thus the mean would increase as well. If 5 were added to each score, each score will rise, including the one score or two scores used to compute the median. The standard deviation, in contrast, is a measurement of the average distance from each score to the mean of the scores. If 5 were added to each score, the mean would increase by 5 and thus the distance between any score and the mean would not be changed. Therefore, the standard deviation would not change. mean only median only standard deviation only

mean and median mean and standard deviation

111

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

112

A straight line in the xy-plane has a slope of 2 and a y-intercept of 2. On this line, what is the x-coordinate of the point whose y-coordinate is 500 ? 249 498 676 823 1,002 (coordinate geometry, medium) Remember that the equation of a line with slope m and y-intercept b is y=mx+ b. Thus the equation of the line is question is y= 2x + 2. If y=500, x = 249

249 498 676 823 1,002

113

A student worked for 20 days. For each of the amounts shown in the first row of the table, the second row gives the number of days that the student earned that amount. What is the median amount of money that the student earned per day for the 20 days? 96 84 80 70 48 (statistics, medium) As 20 is an even number, to find the median amount of money, we must find the mean of 10th and 11th largest of the 20 amounts shown. Each of these is $84, so that is the median amount of money that the student earned.

96 84 80 70 48

114

A store purchased a Brand C computer for the same amount that it paid for a Brand D computer and then sold them both at higher prices. The stores gross profit on the Brand C computer was what percent greater than its gross profit on the Brand D computer? (1) The price at which the store sold the Brand C computer was 15 percent greater than the price at which the store sold the Brand D computer. (2) The stores gross profit on the Brand D computer was $300. (percents, algebra, hard) Let p be the price paid for each computer and c and d be the selling price of the Brand C and Brand D computers respectively. We are asked for the value of

)1(100

pd

pc. In other words can we find the ratio c p : d p ?

(1) c=1.15d. NOT SUFF. A couple of examples: if p=90 and d=100, c=115 and (c - p)/(d - p)=25/10=2.5. However, if p=10 and d=100, c=115 and (c - p)/(d - p)=95/80. (2) d-p=300. NOT SUFF, as no information is given about c-p. (T) Given that d=p+300 and c=1.15d c-p=1.15p + 1.15(300) - p =0.15p +1.15(300). Clearly, the numerator depends on the value of p, whereas the denominator is 300, so the ratio will depend on the value of p. NOT SUFF

115

At a certain bakery, each roll costs r cents and each doughnut costs d cents. if Alfredo bought rolls and doughnuts at the bakery, how many cents did he pay for each roll?

(1) Alfredo paid $5 for 8 rolls and 6 doughnuts. (2) Alfredo would have paid $10 if he had bought 16 rolls and 12 doughnuts.

(algebra, medium) We are asked for the value of r. (1) 8r + 6d = 500 4r + 3d = 250 r = (250 -3d)/ 4 NOT SUFF (2) Equivalent to (1) NOT SUFF (T) NOT SUFF

116

At a certain college there are twice as many English majors as history majors and three times as many English majors as mathematics majors. What is the ratio of the number of history majors to the number of mathematics majors? 6 to 1 3 to 2 2 to 3 1 to 5 1 to 6 (ratios, medium) We are given that e = 2h=3m, so h/m= 3/2

6 to 1 3 to 2 2 to 3 1 to 5 1 to 6

117

At a certain company, each employee has a salary grade s that is at least 1 and at most 5. Each employee receives an hour wage, p , in dollars, determined by the formula p= 9.50 + 0.25 ( s 1) . An employee with a salary grade of 5 receives how many more dollars per hour than an employee with a salary grade of 1 ? $0.50 $1.00 $1.25 $1.50 $1.75 Note that there is a linear relationship between p and s, and if s and p were plotted and the x and y axes of the xy-coordinate plane, the slope would be $0.25. Thus as 5 is 4 higher than 1, p(5) is $0.25(4)= $1.00 higher than p(1).

$0.50 $1.00 $1.25 $1.50 $1.75

118

At a certain food stand, the price of each apple is 40 and the price of each orange is 60. Mary selects a total of 10 apples and oranges from the food stand, and the average (arithmetic mean) price of the 10 pieces of fruit is 56. How many oranges must Mary put back so that the average price of the pieces of fruit that she keeps is 52? 1 2 3 4 5 (algebra, hard) If she puts back x oranges, the price of the 10 x pieces of fruit she keeps is 560 60x cents. If the average price of the fruit she keeps is 52,

584052520605605210

60560

xxxx

x

x

1 2 3 4 5

119

At a certain refreshment stand, all hot dogs have the same price and all sodas have the same price. What is the total price of 3 hot dogs and 2 sodas at the refreshment stand? (1) The total price of 5 sodas at the stand is less than the total price of 2 hot dogs. (2) The total price of 9 hot dogs and 6 sodas at the stand is $21. (algebra, medium) We need to find the value of 3h + 2s , where h is the price of each hot dog and s is the price of each soda. (1) 5s < 2h NOT SUFF (2) 9h + 6s = 21. Since 9h + 6s = 3(3h + 2s), we can find the value of 3h + 2s. SUFF

120

At a certain store, each notepad costs $x and each marker $y. $10 is enough to buy 5 notepads and 3 markers. Is $10 enough to buy 4 notepads and 4 markers? (1) Each notepad costs less than $1. (2) $10 is enough to buy 11 notepads. (inequalities, hard) We know that $10 is enough to buy 5 notepads and 3 markers. We know that $10 will be enough for 4 notepads and 4 markers if the price of a marker is no higher than that of a notepad. If this is not the case, $10 may or may not be enough. Alternatively, one could rephrase the question as follows: Is the price of one marker plus the price of one notepad no more than $2.50? (T) Nothing is said about the price of markers. If the price of a marker is $2 and that of a notepad is $0.90, $10 will not be enough. If the price of a marker is $1 instead, $10 will be enough. NOT SUFF

121

At a certain supplier, a machine of type A costs $20,000 and a machine of type B costs $50,000. Each machine can be purchased by making a 20 percent down payment and repaying the remainder of the cost and the finance charges over a period of time. If the finance charges are equal to 40 percent of the remainder of the cost, how much less would 2 machines of type A cost than 1 machine of type B under this arrangement? $10,000 $11,200 $12,000 $12,800 $13,200 (percents, medium) Note that under this arrangement, an article costing p dollars requires a down payment of 0.2p and subsequent payments totaling 1.4(0.8p) = 1.12p, for a total payment of 1.32p. Since 2 As cost $10,000 less than 1 B, under this arrangement, they could cost $13,200 less.

$10,000 $11,200 $12,000 $12,800 $13,200

122

At a certain university, the ratio of teachers to students for every course is always greater than 3:80. At this university, what is the maximum number of students possible in a course that has 5 teachers? 130 131 132 133 134 (inequalities, medium) If there are t teachers and s students, we are told that t/s > 3/80, so s< 80t/3. If t=5, s

123

At a constant rate of flow, it takes 20 minutes to fill a swimming pool if a large hose is used and 30 minutes if a small hose is used. At these constant rates, how many minutes will it take to fill the pool when both hoses are used simultaneously?

10 12 15 25 50

(combined work, medium) In one hour, the large and small hose would fill 3 and 2 pools

respectively, for a total of 5 pools. Thus in 1/5 of an hour, i.e. 12 minutes, one pool could be filled by the two hoses.

Alternatively, in one minute, 1/20 + 1/30 = 5/60=1/12 of the pool is

filled, so in 12 minutes, one pool could be filled.

10 12 15 25 50

124

At a dinner party, 5 people are to be seated around a circular table. Two seating arrangements are considered different only when the positions of the people are different relative to one another. What is the total number of different seating arrangements for the group? 5 10 24 36 120 (combinatronics, hard)

There is a one-to-one correspondence between each different seating arrangement and a seating arrangement in which person A is seated in position *. Thus we simply need to count the number of ways of arranging the four remaining people in the four remaining seats, which is 4!=24. Alternatively, one could suppose for a moment that the seats were distinguishable. In that case, there would be 5! ways to seat the 5 people. However, we can rotate the table 4 times and we get a different arrangement in which the positions of the people are the same relative to one another. Thus 5 absolute arra