-
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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%
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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
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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%
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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