If y is the circumference,
then y = 2 PI r. If x is the diameter, x is only 2 r,
without the PI. Since PI is a positive number
(approximately 3.14), x < y.18.Is figure A a cube?
(1)The volume of the figure is 9 cubic units
(2)The height of 1/2
the diameter, the radius is 9 and the area is 81 PI.17.Is x < y?
(1)y is the circumference of circle O
(2)x is the diameter of circle O
(A) (B) (C) (D) (E)17. (C) You must know the relative values of x and y to
answer the question.16. (D) If the field is circular, its area is
PI r squared and its circumference is 2 PI r. If the
circumference is 18 PI, the radius must be 9 and
the area 81 PI. Statement two tells you the diameter
and that the field is a circle; since the radius is ual 180, and angles are
supplementary when they sum up to 180 degrees.16.What is the area of the athletic field?
(1)The field is circular and has a circumference
of 18 PI
(2)The diameter of the circular field is 18
(A) (B) (C) (D) (E) (B) (C) (D) (E)15. (B) Statement one only gives you the ratio of x
and y; x is twice as large as y. They could be 120
degrees and 60 degrees, and be supplementary, but you
don't know for certain. Statement two is sufficient
since x plus y would eq5,260 degrees
(B)1,260 degrees
(C)900 degrees
(D)90 degrees
(E)Cannot be determined
olygon has in order to determine the number
of triangles you may make from the figure; knowing the
number of triangles will allow you to determine the
total interior angle measure.10.What is the total interior angle measure of a
regular nonagon?
(A)easure of a polygon
of side 3?
(A)300 degrees
(B)90 degrees
(C)30 degrees
(D)6 degrees
(E)Cannot be determined9. (E) The interior ANGLE measure is not affected by the
LENGTH of a side of a polygon. You must know how MANY
sides the p answers. Point B is directly below point X, just
two units away; point C is above point X and slightly
farther away, since it is located diagonally across a
1 x 2 unit rectangle. Therefore, point B is the closest.9.What is the total interior angle mint X?
(A)(1,2)
(B)(3,1)
(C)(2,3)
(D)(2,1)
(E)(5,5)8. (B) Sketch the graph, including each of the five
points offered as answer choices. Points A, D, and E are
all quite far from point X, leaving only points B and C as
possible pair of
parallel lines. Angle 8 = Angle 6, since they are
vertical angles. Therefore, Angle 1 (155 degrees)
 Angle 8 (25 degrees) = 130 degrees.8.On a coordinate graph, if point X is located at
(3,1), which of the following points is closest to
po5 degrees
(D)55 degrees
(E)0 degrees7. (B) Since Angles 5 and 6 are along a straight line,
they are supplementary and must add up to 180 degrees,
making Angle 5 = 155 degrees. Angle 1 = Angle 5 since
they are in corresponding positions around a10.1 / 2
/L1
4 / 3
L1  L2/
5 / 6
/L2
8 / 7
7.In the figure above, if Angle 6 = 25 degrees, then
Angle 1  Angle 8 =
(A)155 degrees
(B)130 degrees
(C)12 5,
since it is parallel to and equal to AD. Since BE is 5
and BC is 13, you have a right triangle that must have the
ratio of 5:12:13, making EC 12. Since EC = 12, DC = 48
(add 36 + 12). The two bases together are 84; 1/2 that
is 42. Multiply 42*5 = 2s the area of trapezoid ABCD?
(A)210 square units
(B)180 square units
(C)164 square units
(D)148 square units
(E)Cannot be determined6. (A) The formula for the area of a trapezoid is
(1/2)(base 1 + base 2)*height. The height, BE, isits area. Without this area, you cannot find the volume
of the cylinder.A36B
++\
\
\
5\13
\
90 deg
\
++
DE C
6.What i,
then y = 2 PI r. If x is the diameter, x is only 2 r,
without the PI. Since PI is a positive number
(approximately 3.14), x < y.18.Is figure A a cube?
(1)The volume of the figure is 9 cubic units
(2)The height of the figure is 2 units
(A)s is 9 and the area is 81 PI.17.Is x < y?
(1)y is the circumference of circle O
(2)x is the diameter of circle O
(A) (B) (C) (D) (E)17. (C) You must know the relative values of x and y to
answer the question. If y is the circumferencercular, its area is
PI r squared and its circumference is 2 PI r. If the
circumference is 18 PI, the radius must be 9 and
the area 81 PI. Statement two tells you the diameter
and that the field is a circle; since the radius is 1/2
the diameter, the radiunce the measures of the interior angles of any
triangle add up to 180 degrees, multiply the number of
triangles by 180 degrees.ant), use the
formula (s  2)*180 degrees, where s represents the number
of sides of the figure. A nonagon has 9 sides; 9  2 = 7.
7*180 degrees = 1,260 degrees. Note that the number of
sides minus two equals the number of TRIANGLES within the
figure; si5,260 degrees
(B)1,260 degrees
(C)900 degrees
(D)90 degrees
(E)Cannot be determined
10. (B) To find the total interior angle measure of ANY
polygon (whether or not it is regular  that is, having
all equal sides and angles  is irrelevolygon has in order to determine the number
of triangles you may make from the figure; knowing the
number of triangles will allow you to determine the
total interior angle measure.10.What is the total interior angle measure of a
regular nonagon?
(A)easure of a polygon
of side 3?
(A)300 degrees
(B)90 degrees
(C)30 degrees
(D)6 degrees
(E)Cannot be determined9. (E) The interior ANGLE measure is not affected by the
LENGTH of a side of a polygon. You must know how MANY
sides the p answers. Point B is directly below point X, just
two units away; point C is above point X and slightly
farther away, since it is located diagonally across a
1 x 2 unit rectangle. Therefore, point B is the closest.9.What is the total interior angle mint X?
(A)(1,2)
(B)(3,1)
(C)(2,3)
(D)(2,1)
(E)(5,5)8. (B) Sketch the graph, including each of the five
points offered as answer choices. Points A, D, and E are
all quite far from point X, leaving only points B and C as
possible pair of
parallel lines. Angle 8 = Angle 6, since they are
vertical angles. Therefore, Angle 1 (155 degrees)
 Angle 8 (25 degrees) = 130 degrees.8.On a coordinate graph, if point X is located at
(3,1), which of the following points is closest to
po5 degrees
(D)55 degrees
(E)0 degrees7. (B) Since Angles 5 and 6 are along a straight line,
they are supplementary and must add up to 180 degrees,
making Angle 5 = 155 degrees. Angle 1 = Angle 5 since
they are in corresponding positions around a10.1 / 2
/L1
4 / 3
L1  L2/
5 / 6
/L2
8 / 7
7.In the figure above, if Angle 6 = 25 degrees, then
Angle 1  Angle 8 =
(A)155 degrees
(B)130 degrees
(C)12 5,
since it is parallel to and equal to AD. Since BE is 5
and BC is 13, you have a right triangle that must have the
ratio of 5:12:13, making EC 12. Since EC = 12, DC = 48
(add 36 + 12). The two bases together are 84; 1/2 that
is 42. Multiply 42*5 = 2s the area of trapezoid ABCD?
(A)210 square units
(B)180 square units
(C)164 square units
(D)148 square units
(E)Cannot be determined6. (A) The formula for the area of a trapezoid is
(1/2)(base 1 + base 2)*height. The height, BE, isits area. Without this area, you cannot find the volume
of the cylinder.A36B
++\
\
\
5\13
\
90 deg
\
++
DE C
6.What irrect.11.6a + 3b = 9.8a + 11b = 19.Solve for (a + b).
(A)19
(B)9
(C)4
(D)2
(E)Cannot be determined11. (D) One way to solve simultaneous equations is to
combine the equations. Line the equations up vertically
and add like ter(C)III only
(D)II or III
(E)I or II or III or IV or V10. (D) When the product of two numbers is 0, AT LEAST
one of those numbers must be 0. x could be 9, such that
9 + 9 = 0. x could be +3, such that +3  3 = 0. Either
value would be co Since x is
the same as x to the first, you are adding a 1 and a 2
(x squared is x to the second power), to get x to the
third, or x cubed.10.(x + 9)*(x  3) = 0.x could be
I.9
II.9
III.3
IV.3
V.0
(A)I only
(B)I or II
he information frequently.9.6x*4(x squared) =
(A)6x + 4(x squared)
(B)10x squared
(C)24x squared
(D)24x cubed
(E)Cannot be determined9. (D) Multiply 6*4 to get 24. When multiplying like
bases (the x's), simply add the exponents.ed  b squared8. (E) Take one term at a time and multiply through.
a*a = a squared; a*b = ab; b*a = ab; b*b = b squared.
The +ab and ab cancel one another out, leaving you with
a squared  b squared. You should MEMORIZE this problem;
you may use tred =
a squared  2ab + b squared; you are likely to use the
information frequently.8.(a + b)*(a  b) =
(A)a squared + 2ab + b squared
(B)a squared  2ab + b squared
(C)a squared + 2ab  b squared
(D)a squared + b squared
(E)a squar(a  b)*(a  b).
Take one term at a time and multiply through.
a*a = a squared; a*b = ab; b*a = ba;
b*b = +b squared. Since ab and ba are the same,
combine them to get 2ab. You should MEMORIZE this
problem so that you instantly know (a  b) squainformation frequently.7.(a  b) squared =
(A)a squared  2ab  b squared
(B)a squared  2ab + b squared
(C)a squared + 2ab  b squared
(D)a squared + 2ab + b squared
(E)a squared + b squared7. (B) (a  b) squared is the same as squared; a*b = ab; b*a = ba; b*b = b squared.
Since ab and ba are the same, combine them to get 2ab.
You should MEMORIZE this problem, so that you can
instantly look at (a + b) squared and know it is
a squared + 2ab + b squared; you are likely to use the
(A)a squared + b squared
(B)a squared + ab + b squared
(C)a squared + 2ab + b squared
(D)a squared + 4ab + b squared
(E)a cubed + b cubed6. (C) (a + b) squared is (a + b)*(a + b). Take one
term at a time and multiply through.
a*a = agure) has INTERIOR angles summing to
360 degrees. The interior angles of the figure sum to 360
degrees
(A) (B) (C) (D) (E)20. (B) The exterior angles of ANY polygon sum to 360
degrees, such that statement one is not sufficient.
Statement two is sufficient; only a quadrilateral
(a foursided fiuld measure 60 degrees, such that two would
sum to 120 degrees), it does not HAVE to be (one angle
could be 100 degrees and the other 20 degrees).20.How many sides does figure 1 have?
(1)The exterior angles of the figure sum to 360
degrees
(2) (B) (C) (D) (E)19. (E) The interior angles of ANY type of triangle,
scalene (no equal angles), isosceles (two equal angles)
or equilateral (three equal angles) is 180 degrees. In
statement two, while the triangle COULD be equilateral
(each angle wome of 2*2*2 = 8. You need both
pieces of information, the total volume and the length
of one edge.19.Is triangle ABC equilateral?
(1)The sum of the interior angles is 180 degrees
(2)The sum of two of the interior angles is 120
degrees
(A) (B) (C) (D) (E)18. (C) The volume of a cube is found by using the
formula e cubed, where e stands for an edge of the
figure. If the height of the figure is 2, the figure
CANNOT be a cube, since if it were, each edge would
measure 2, giving a volucal and background
knowledge, determine whether the data are
SUFFICIENT for answering the question.Choose:
(A) if statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statement (2) alone is sufficient but
stating information may be useful:
SYMBOLS
*multiplied by parallel to
>is greater than<is less than
DIRECTIONS:Each problem consists of a question
and two statements.Using the data given in the
problem plus your own mathemati25 =
(A)3.25
(B)3.00
(C)2.75
(D)2.00
(E)1.958. (B) This is a simple problem. When dividing decimals,
move the decimal points to the right to make the numbers
whole. This gives you 975.0 divided by 325.0 which
equals 3.The followd by 5/9 =
(A)1
(B)54/55
(C)30/18
(D)11/18
(E)Cannot be determined7. (B) When dividing fractions, INVERT (turn upside
down) the number by which you are dividing and multiply
the numbers. (6/11)*(9/5) = 54/55.8.9.75 divided by 3.t be greater
than 21. Since 5*5 = 25, your answer must end in a 5.
Only the correct answer has both of these qualifications.
Look BEFORE you begin doing the problem to see whether you
can estimate rather than doing the actual mathematics.7.6/11 dividelying the
denominators (3*4*8 = 96) and automatically looking for
an answer with that result. (A) is the trap answer here.6.7.65*3.25 =
(A)18.6565
(B)21.8950
(C)24.8625
(D)28.3644
(E)28.39006. (C) Since 7*3 = 21, your answer mus (C) When multiplying fractions, simply multiply the
numerators, then the denominators. You can make it even
easier by canceling; here cancel the 3's so that your
numerators are 1*1*7 = 7. Your denominators are
1*4*8 = 32. Do NOT be tricked into multip.115 = .945. Note that your answer must be LESS THAN 1
because of the decimals, and must end in a 5 since you are
subtracting 5 from "0" which you convert to a "10."5.(1/3)*(3/4)*(7/8) =
(A)37/96
(B)11/20
(C)7/32
(D)3/8
(E)1/35./3
will be positive, so that the answer will be 4 plus some
additional fraction. Only one answer fits this
description.4.4.06  3.115 =
(A)1.045
(B).945
(C).650
(D).550
(E).5064. (B) 4.06 is the same as 4.060; subtract 4.060
 3nominator is 15. The problem
can be restated as 6 12/15  2 5/15 = 4 7/15. It may also
be possible to estimate and therefore bypass the actual
calculation. You know that 6  2 = 4. If you realize
that 4/5 is greater than 1/3, then you know that 4/5  1st be GREATER than 22; only the
correct choice is so. Do NOT take the time to do all the
actual addition; look at the answer key first.3.6 4/5  2 1/3 =
(A)6 7/15
(B)5 3/5
(C)4 7/15
(D)3 6/15
(E)2 11/153. (C) The least common de + 16b = 19b; 27 + 11 = 38.
19a + 19b = 38, or in other words, 19(a + b) = 38;
divide both sides through by 19 to find that a + b = 2.r equation
to get the value of x, you may stop  you know that
either statement is sufficient.)20.What is (a + b)?
(1)11a + 3b = 27
(2)8a + 16b = 11
(A) (B) (C) (D) (E)20. (C) Combine the statements; add like terms.
11a + 8a = 19a; 3b(A)39
(B)27
(C)9
(D)6
(E)3
(C)6
(D)3
(E)Cannot be determined12. (D) Multiply both sides by 4 to get rid of the
fraction. This gives you 11x + 3 = 36. Subtract 3 from
both sides. Your equation is now 11x = 33; x = 3.13.4x + 5y = 39.x  y = 3.Solve for x.
ms. 6a + 8a = 14a; 3b + 11b = 14b;
9 + 19 = 28. You have 14a + 14b = 28 which can be
rewritten as 14(a + b) = 28. Since you want to find
(a + b), divide both sides by 14. This gives you
a + b = 2.12.(11x + 3)/4 = 9.Solve for x.
(A)14
(B)9ave sufficient information to do so
(you would need either the number of kittens or puppies,
or the total number of both) and cannot answer the
question. 7.For every 2 complaints that Chef Charles gets about
his meat loaf, he gets 9 compliments.If the 3:7.How many more
puppies are there than kittens?
(A)4
(B)7
(C)10
(D)37
(E)Cannot be determined6. (E) In order to find out the difference, you must find
out the total number of kittens and the total number of
puppies. You do not hmber
that follows.
DIRECTIONS:Each problem consists of a question
and two statements.Using the data given in the
problem plus your own mathematical and background
knowledge, determine whether the data are
SUFFICIENT for answering the quest stop at 19 because
20*20 = 400.The following information may be useful:
SYMBOLS
*multiplied by parallel to
>is greater than<is less than
NOTE:The word "radical" in these questions
indicates the positive square root of the nu3 and 399?
(A)12
(B)18
(C)21
(D)59
(E)1008. (B) A perfect square is the product of a number
multiplied by itself. The perfect squares between 3 and
399 are the products of numbers from 2 (2*2 = 4) through
19 (19*19 = 361). You must(E)2897. (B) A perfect square is the product obtained when
an integer is multiplied by itself. 4 is a perfect
square since 2*2 = 4. 144 is a perfect square:
12*12 = 144. 256 = 16*16 and 289 = 17*17.8.How many perfect squares are there between al 169 = 13, radical 170 is slightly more than 13.
Since radical 625 = 25, radical 626 is slightly more
than 25. The difference is approximately 13  25 = 12.7.Which of the following is not a perfect square?
(A)4
(B)44
(C)144
(D)256
6.Which of the following is closest in value to radical
170  radical 626?
(A)38
(B)12
(C)12
(D)38
(E)radical 4566. (C) You cannot simply subtract radicals; you must
approximate the square roots and subtract those. Since
radiche thousandths place).11.What fraction of the pies were sold at the bazaar?
(1).75% of the pies were sold
(2)There were 1,000 pies available for sale
(A) (B) (C) (D) (E)ls to the right of the
decimal point (add the number of numerals to the right of
the decimal point in each term). Statement two answers
the question because the thousandths place is 3 numerals
to the right of the decimal point (e.g., in 123.456, 6 is
in t
10. (D) In statement one, there are two numerals to the
right of the decimal point in one of the terms being
multiplied; there is one numeral to the right of the
decimal point in the other term. The product of the terms
will therefore have three numeraanswer is C.10.What is the largest number of nonzero numerals that
may appear to the right of the decimal point in x?
(1)x is the product of 3.16 and 2.4
(2)The last nonzero numeral in x is in the
thousandths place
(A) (B) (C) (D) (E)e is not sufficient.
Statement two alone tells you nothing about y, and so it
is not sufficient. But combined with statement one, it
allows you to infer that both x and y are positive
numbers, with y > x. In that case, x/y is less than 1.
So the correct ominator is greater than its numerator, the value of
the fraction is usually less than 1 (1/2, for example).
However, if both numbers are negative, the reverse is
true: for instance, 2 > 3, but 3/2 = 3/2, which is
greater than 1. So statement one aloner statement alone nor both statements
together are sufficient to answer the question.9.Is x/y a fraction less than 1?
(1)y > x
(2)x is a positive, whole number
(A) (B) (C) (D) (E)9. (C) Start with statement one. If a fraction's
denement (1) alone is not sufficient;
(C) if both statements (1) and (2) together are
sufficient to answer the question, but neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
answer the question;
(E) if neithTo find the percent increase, take the number
increase (81  18 = 63) and put it over what you are
increasing from, the original whole. 63/18 = 3.5 or 350%.10.Approximately what percent of the area of a square
of side 14 is a triangle with a base of 9k an exam and answered 18 questions correctly.
She took the same exam the next day and answered 81
questions correctly.By what percent did she increase the
number of her correct answers?
(A)350%
(B)250%
(C)63%
(D)35%
(E)3.5%9. (A) not be determined8. (C) To find the percent increase, take the number
increase and put it over the original whole. Here,
$1,500  $700 = $800. Put that over what you are
increasing from, the $700. $800/$700 = approximately
1.14 or 114%. 9.Enid toow the area of the
rectangle, you can't answer the question.8.Tasha made $700 from gambling last year and
$1,500 from gambling this year.By approximately what
percent did her winnings increase?
(A)800%
(B)700%
(C)114%
(D)98%
(E)Can area
of the rectangle you need to know the length and the
width. Knowing the perimeter is not sufficient; the
length and width could be 8 and 10 or 7 and 11 or any
other combination for which the sum of the four sides
would equal 36. Since you don't knoof side 16 is what percent of
the area of a rectangle of perimeter 36?
(A)100%
(B)90%
(C)75%
(D)60%
(E)Cannot be determined7. (E) You can find the area of the square by multiplying
side times side: 16*16 = 256. However, to find theto pistols?
(1)There are 3 times as many pistols as there
are rifles
(2)There are 18 pistols
(A) (B) (C) (D) (E)but neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
answer the question;
(E) if neither statement alone nor both statements
together are sufficient to answer the question.10.What is the ratio of rifles statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statement (2) alone is sufficient but
statement (1) alone is not sufficient;
(C) if both statements (1) and (2) together are
sufficient to answer the question, DIRECTIONS:Each problem consists of a question
and two statements.Using the data given in the
problem plus your own mathematical and background
knowledge, determine whether the data are
SUFFICIENT for answering the question.Choose:
(A) if here are 3 sets of 5; therefore, there must be 3
sets of 7, or 21. Check your work by seeing that 15:21
reduces to 5:7.The following information may be useful:
SYMBOLS
*multiplied by parallel to
>is greater than<is less than
what is b?
(A)21
(B)22
(C)23
(D)24
(E)Cannot be determined9. (A) The ratio of 25:35 may be reduced (by a factor
of 5) to the ratio of 5:7. This means that for every set
of 5 a's, there is a corresponding set of 7 b's. If a is
15, t OR BETTER and B OR BETTER. Therefore, you cannot
determine how many, or what proportion, of the students
got C's and how many got B's. (Some of the students
could have gotten A's.)9.The ratio of a to b is the same as the ratio of 25:35.
If a is 15, of B or better.What is the ratio of those
getting semester averages of C's to those getting semester
averages of B's?
(A)3:7
(B)7:3
(C)1:10
(D)10:1
(E)Cannot be determined8. (E) The key is noting that the students got averages
of Cf 11; therefore,
there must be 6 SETS of complaints and 6 SETS of
compliments. With 2 complaints to a set, there are
12 complaints.
8.At the University of Miami, 70% of the students get a
semester average of C or better, and 30% get a semester
average re are 66
comments about his meat loaf, all of which are either
complaints or compliments, how many are complaints?
(A)2
(B)12
(C)22
(D)32
(E)397. (B) The sum of the numbers in the ratio is 11
(2 + 9 = 11). In 66, there are 6 SETS o a total of $2.70. The difference in their
amounts is $3.15; since there are 20 nickels to a dollar,
this amount could be changed to 63 nickels.The following information may be useful:
SYMBOLS
*multiplied by parallel to
>is greater th(D)42
(E)169. (C) Since a quarter is 25 cents, 18 quarters = $4.50;
a dime is 10 cents, 12 dimes = $1.20; a nickel is 5 cents,
3 nickels = 15 cents. $4.50 + $1.20 + $0.15 = $5.85.
Benedict's sister has 9 quarters ($2.25) and 9 nickels
($0.45) for, and 3 nickels.
His sister has half as many quarters and three times as
many nickels.She has no dimes.If Benedict puts the
money he has in excess of the money his sister has into
nickels, how many nickels will he have?
(A)75
(B)69
(C)63
ces in
16 lbs.Solve for x.
(A)16
(B)160
(C)256
(D)356
(E)Cannot be determined8. (A) There are 16 ounces in a pound; in 16 pounds,
there are 16*16 or 256 ounces. The square root of
256 is 16.
9.Benedict has 18 quarters, 12 dimes,005
(E)1,1527. (E) There are 36 inches in a yard, and 36*36 or 1,296
square inches in a square yard. There are 12 inches in a
foot and 12*12 or 144 square inches in a square foot.
1,296  144 = 1,152.8.x is the square root of the number of oun35 pounds a trip, it would take 9,000/335 or
approximately 26.8 trips. Since the truck cannot make a
partial trip, it would make 27 trips.7.How many more square inches are in a square yard than
in a square foot?
(A)24
(B)36
(C)568
(D)1), but only for the percentage of
incorrect answers. Out of 225
questions, he therefore answered 38 questions incorrectly.
Find the percentage by dividing 38 by 225. Note that the
question does not ask for his SCORE (which you could not
determine without knowing how unanswered questions were
accounted fored was 9
(A) (B) (C) (D) (E)
20. (C) You need to know the total number of questions;
statement 1 gives this. You need to determine the number
he answered incorrectly; if he answered 178 correctly, and
left 9 blank, that accounts for 187 questions.t
it is negative.20.Dith answered 178 questions correctly on his English
exam.What percent of all the questions on that exam did
he answer incorrectly?
(1)The number of questions on the exam was 225
(2)The number of questions he left unansweres.
Two complementary angles sum up to 90 degrees. Since 180
is TWICE AS MUCH AS 90, it is 100% MORE THAN 90. Do not
be tricked into thinking that if a number is twice as
much, it is 200% GREATER; it is just 200% OF, not
MORE THAN. .002 = b.
500a = b. 500 = 50,000%.13.The sum of two supplementary angles is what percent
greater than the sum of two complementary angles?
(A)200%
(B)150%
(C)100%
(D)75%
(E)50%13. (C) Two supplementary angles sum up to 180 degre)*(1/2) = 1/30 which is
approximately .0333 or 31/3%.12.a = .1% of 2b.What percent of a is b?
(A)50,000%
(B)5,000%
(C)500%
(D)5%
(E) .5%12. (A) .1% is the same as the decimal .001. If
a = .001*2b, then a = .002b. Therefore, a/blue eyes.Approximately what percent of all the people
in Burnsville are blueeyed men over 6' tall?
(A)1/3 %
(B)331/3%
(C)30%
(D)31/3%
(E)Cannot be determined11. (D) Convert all the numbers to fractions and
multiply: (1/3)*(1/5 answer (A) is the approximate percentage
that the area of the square is of the area of the
triangle; the question asks for the opposite. 11. Onethird of the people of Burnsville are men; 20% of
the men are over 6' tall; 50% of the men over 6' tall have
and a height
of 4?
(A)1,089%
(B)990%
(C)36%
(D)18%
(E)9%10. (E) The area of the square is side times side,
or 196. The area of the triangle is 1/2 base times
height, or (1/2)(9*4) = 18. 18/196 = approximately .09
or 9%. Note that = $22.50. Add the numbers
to get a total of $87.50; divide by 17 to get about 5.
Try to do these types of problems in your mind; the
answers are far enough apart to allow you to
estimate easily.
11.Three children want to buy a book priced at $4.90.
Eaach, and 3 records at $7.59 each.To the nearest
dollar, what was the average price he paid per record?
(A)$7.00
(B)$6.00
(C)$5.00
(D)$4.00
(E)$3.0010. (C) You should have estimated; 8 * $4.00 = $32.00;
6 * $5.50 = $33.00; 3 * $7.50ou should have ESTIMATED
the answer to this problem: since 18,500 divided by 10
equals 1,850, the same number divided by 16 must equal
less than that. Only one answer is less than 1,850.10.Humphrey bought 8 records at $3.99 each, 6 records at
$5.49 e
hauled per trip?
(A)12,000
(B)11,560
(C)6,678.50
(D)1,156.25
(E)Cannot be determined9. (D) To find the average, divide the total number of
pounds hauled by the number of trips it took to haul them.
18,500/16 = 1,156.25. Note that y  3 + 5 + 6 = 10) to have a total of 14x + 10.
Divide by the number of terms, 4, to have (14x + 10)/4.
Reduce by a factor of 2 to get (7x + 5)/2.9.Eunice hauled 18,500 pounds of turnips in 16 trips.
What was the average number of pounds of turnips she 25 + 29 = 54; 54/2 = 27.8.What is the average of (3x + 2), (2x  3), (16x + 5)
and (6  7x)?
(A)(7x + 5)/2
(B)(7x  5)/2
(C)(11x + 4)/4
(D)(11x  4)/4
(E)(14x + 5)/48. (A) Add all the like terms: (3x + 2x + 16x  7x =
14x), (2ch that the
series is 1317212529333741. With an even number
of evenly spaced numbers, you find the average by adding
the first and last numbers and dividing by 2:
13 + 41 = 54; 54/2 = 27. You could also add the central
two numbers and divide by 2:.7.What is the average of a series of 8 evenly spaced
integers whose first two terms are 13 and 17?
(A)18
(B)22
(C)23
(D)27
(E)297. (D) Since the first two terms are 4 units apart, all
the numbers must be spaced 4 units apart, sue determined6. (A) Since the series begins with terms that are 3
units apart, all the terms must be evenly spaced by 3's,
such that the series is 36912151821. With an odd
number of evenly spaced terms, the number in the
middle, 12, is the averagefind Ronnie's age in years. old is Ronnie?
(1)Ronnie is exactly 5 years older than his sister
(2)Ronnie's sister is 11,323 days old
(A) (B) (C) (D) (E)11. (C) Statement two could be converted into years
(31 years; 11,323/365 = just over 31). Add 5 years to
that to Y container will hold twice as many pints as
quarts. Statement two allows you to solve the problem;
since 1 gallon = 16 cups, 14 gallons = 14*16 cups. Do
NOT take the time to do the multiplication; simply
determine that the data are sufficient.11.Howw many cups of water will it take to fill the
barrel?
(1)The barrel will hold twice as many pints as
quarts of water
(2)The barrel holds 14 gallons of water
(A) (B) (C) (D) (E)10. (B) Statement one is useless; since 2 pints =
1 quart, ANt to answer the question, but neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
answer the question;
(E) if neither statement alone nor both statements
together are sufficient to answer the question.10.Hostion.Choose:
(A) if statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statement (2) alone is sufficient but
statement (1) alone is not sufficient;
(C) if both statements (1) and (2) together are
sufficienan<is less than
DIRECTIONS:Each problem consists of a question
and two statements.Using the data given in the
problem plus your own mathematical and background
knowledge, determine whether the data are
SUFFICIENT for answering the queph for 60 miles; he took
1.5 hours to go the distance, or 90 minutes. If Richard
traveled 5/8 as fast, he went 25 mph for the same
distance, 60 miles, taking 2.4 hours, or 144 minutes.
144  90 = 54 minutes that David will have to wait.7.How much inteance.
How long does David have to wait at the final destination
for Richard?
(A)54 minutes
(B)45 minutes
(C)38 minutes
(D)24 minutes
(E)13 minutes6. (A) Use the formula RT = D (Rate times Time equals
Distance). David's rate was 40 mrofit). The cost was $36.00. The profit was
25% of $36.00, or $9.00. $36.00 + $9.00 = $45.00.
6.David and Richard start driving from the same point
at the same time.David drives 40 mph for 60 miles.
Richard drives 5/8 as fast and goes the same distal number of points is. You can find that half the
players scored 520,000 points (13 players at 40,000 points
each), but you have no idea how many points each of the
remaining 13 players scored.rs scored 40,000 points each
(2)There were 26 players
(A) (B) (C) (D) (E)20. (E) To find the average number of points, find the
total points and divide that by the number of players.
You have the number of players, but have no idea what the
tot third
highest tests: 194 (97 + 97). Add the three lowest tests
to get a total of 264 points; the remaining or highest
test must have had a score of 100 points.20.What is the average number of points scored on the
video game?
(1)50% of the player.
(2)The average of the second and third highest
scores was 97.
(A) (B) (C) (D) (E)19. (C) Find the total points scored on all four exams;
if the average exam was 91, the total was 4*91, or 364.
Find the total points scored on the second andre could be any number of terms in the series
since you are not told that the series features
consecutive or evenly spaced terms.19.Scott had an average score of 91 on four exams.What
was the highest of his four scores?
(1)The lowest score was 70annot find the average for one student.18.How many terms are in a series?
(1)The average of the series is exactly three times
the first number of the series
(2)The sum of the numbers in the series is 48
(A) (B) (C) (D) (E)18. (E) Theudents received 90's
(2)5/8 of the students received 85's
(A) (B) (C) (D) (E)17. (E) Since 35% is NOT QUITE 3/8 (3/8 = 37.5%), the
total using both statements does not sum up to 100%.
Since you don't know the scores for all the students,
you cives you basically the
same information; if the last term of 7 consecutive
numbers is 25, those numbers must be 25242322212019.
The central term, x, is 22.17.What was the average score per student on the
Anatomy test?
(1)Exactly 35% of the sts whose last number is 25
(A) (B) (C) (D) (E)16. (D) You could find x in the first statement; if the
integers are consecutive, they follow one another and are
19202122232425. The number in the middle, 22, is
the average, x. Statement two g him.How much does the least
generous student contribute?
(A)$50.00
(B)$40.00
(C)$37.25
(D)$31.40
(E)$16.00 the total number
of children, 8, you would have about $0.61, BUT THE
QUESTION WAS NOT ASKING FOR THIS INFORMATION. 12.Four students contribute an average of $60.00 each
to charity.Each student contributes exactly twice as
much as the student prior to11. (B) If each of the first group of children has
contributed $0.65, the total so far is $1.95, which leaves
$2.95 to be contributed by the 5 new children.
$2.95/5 = $0.59 contribution PER ADDITIONAL CHILD. Note
that if you divided the total, $4.90, bych child has $0.65.If five more children each
contribute an amount that makes the total of all
contributions enough to buy the book, how much on the
average must each new child contribute?
(A)$0.61
(B)$0.59
(C)$0.55
(D)$0.43
(E)$0.3330.5.An item which cost $36.00 to manufacture was then
sold for a 25% profit.What was the selling price of
that item?
(A)$64.00
(B)$56.00
(C)$51.00
(D)$45.00
(E)$41.005. (D) Use the formula: S = C + P (Selling Price =
Cost + P5:30
(D)5:45
(E)6:004. (C) He made 80 toys in 2 hours, making his rate 40
toys an hour. If he wants to make 100 toys, it will take
him 2 1/2 hours (40 * 2 1/2 = 100). Add 2 1/2 hours to
the starting time of 3:00 to get a finishing time of 5:s in a dollar, 100. 4.George begins assembling toys at 12:30 and has 80 done
by 2:30.The next day, he begins at 3:00.If he works at
the same rate as he did the previous day, at what time
will he have assembled 100 toys?
(A)5:00
(B)5:15
(C))(b/a)/100
(B)(a/b)/100
(C)(b/a)*100
(D)100 (ab)
(E)(a + b)/1003. (C) To find an average, divide the total by the
number of items; this gives you b/a. Since everything is
in terms of dollars, you want to multiply by the number of
cent; the wholesale price
is $8.00. The agent gets 20%, or 1/5, of the author's 5%.
He therefore gets 1% of the wholesale price; 1% of
$8.00 = 8 cents.3.Frances bought "a" items for a total of "b" dollars.
What is the average price per item in cents?
(A2. (E) You may recognize that 12.5% is the same as 1/8,
and therefore 112.5% is the same as 1 1/8, or 9/8. (If
you did not know this, you could have converted 112.5%
to the fraction 1125/1000 and reduced it to get 9/8.)
$9.00 = 9/8 of the wholesale pricer $9.00 which is 112.5% of
the wholesale price.If the author gets a 5% royalty of
the wholesale price and her agent gets 20% of her
royalties, how much money does her agent make per book?
(A)$3.00
(B)$1.25
(C)$0.90
(D)$0.50
(E)$0.08(B)$2.50
(C)$1.75
(D)$1.50
(E)$1.251. (A) Work backwards. At the fifth stop, she spent
25 cents, meaning that she spent 50 cents at stop four,
$1.00 at stop three, $2.00 at stop two, and $4.00 at
stop one. 2.A book sells at retail foJessica stops at 5 concession stands while she is
enjoying the carnival.At each stop she spends 50% as
much as she spent at the previous stand.At the last
stand, she spends 25 cents.How much money did Jessica
spend at the first stand?
(A)$4.00
l indicates addition or subtraction
(4 + 3 = 7).If the sign appears without a space
(e.g. 4), the symbol indicates whether the number
following it is positive or negative.Press^Wto
enter theSCRATCHPADand^StoSEE QUESTION
again. **
1.ates otherwise.All numbers used in
this section are real numbers.
NOTE:The word "radical" in these questions
indicates the positive square root of the number
that follows.
** A space appears on both sides of a + or  sign
when the symbois greater than<is less than
NOTE:Although the figures that accompany the
problems in this section are drawn as accurately
as possible, they are NOT drawn exactly to scale.
All figures lie in a plane unless the problem
states or indicPress the RIGHT
arrow key>to begin and to continue to the next
question.Press the LEFT arrow key<if you wish
to review previous questions. **
The following information may be useful:
SYMBOLS
*multiplied by
is parallel to
>+7[0W\+7]Pt^+7
TOPIC 9:WORD PROBLEMS
Problem Solving and Data Sufficiency
DIRECTIONS:Read and solve each problem carefully.
Choose the best answer and type the letter
corresponding to your choice.** d060 P
0N0
@] PUP;R*6
TPT+7UgV+7W X+7YPZ
f just one individual item is irrelevant to finding
out the average price per item.16.Solve for x.
(1)x is the average of a series of 7 consecutive
integers that begins with 19
(2)x is the central number of a series of 7
consecutive integerr allows you to solve
the problem.
solve the problem.
2.READ each statement (statement one and
statement two) INDIVIDUALLY, determining whether
each one ALONE has sufficient data to solve the
problem.
3.COMBINE the two statements and determine
whether using them togethee
whether or not the data are sufficient to do so.
HOW TO APPROACH THE QUESTION:
1.READ the question stem and PREDICT what
information you will need to solve the problem or
answer the question.Do NOT begin doing any
calculations or try tore
two numbered statements.You are to determine
whether either statement alone, both statements
together or neither statement gives you sufficient
data to solve the problem.You are NOT required
actually to solve the problem; you only determin expensive item was $1.20.
(2)There were 10 items.
(A) (B) (C) (D) (E)15. (B) To find the average, you need the total (which
is given to you in the question) and the number of items
(which is given to you in the second statement). The
price o and that the series must be 3579111315. With
an odd number of evenly spaced integers, the integer in
the middle is the average. You need both pieces
of information.15.What was the average cost per item in a $40.00
grocery bill?
(1)The most(B) (C) (D) (E)14. (C) If you know that the integers are evenly spaced,
you know that they are an equal distance apart from one
another, like 246 or 369. If you know that the first
two terms are 3 and 5, you know that the terms are spaced
by 2'snswer the question;
(E) if neither statement alone nor both statements
together are sufficient to answer the question.14.What is the average of seven integers?
(1)The integers are evenly spaced
(2)The first two integers are 3 and 5
(A) (2) alone is sufficient but
statement (1) alone is not sufficient;
(C) if both statements (1) and (2) together are
sufficient to answer the question, but neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
ae
problem plus your own mathematical and background
knowledge, determine whether the data are
SUFFICIENT for answering the question.Choose:
(A) if statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statementadditional
9 friends.The following information may be useful:
SYMBOLS
*multiplied by parallel to
>is greater than<is less than
DIRECTIONS:Each problem consists of a question
and two statements.Using the data given in th contribute the same amount in order to
have the party?
(A)13
(B)10
(C)9
(D)8
(E)613. (C) You want a total of $650.00, with each person
spending only $50.00. $650.00/$50.00 = 13. Since there
are already 4 friends, there must be an 2x, 4x, and 8x. Therefore,
15x = $240.00; $240.00/15 = $16.00. x (the contribution
of the least generous student) = $16.00. 13.Four friends want to throw a party that will cost
$650.00.They want to spend only $50.00 each.How many
more friends must~;12. (E) If the average contribution per student is
$60.00 and there are 4 students, the total contribution
is $240.00. Let x be the amount contributed by the least
generous student; since each contribution doubles the
previous one, the series is x,
u
cannot determine the central term (or any term other than
the first) and cannot solve the problem.6.What is the average of a series of 7 evenly spaced
integers whose first two terms are 3 and 6?
(A)12
(B)13
(C)14
(D)15
(E)Cannot benly spaced numbers, the number
in the middle is the average. However, since you do not
know what the spacing is between the numbers (they could
be 2 units apart and be 1012141618, or 3 units apart
and be 1013161921, or have any other spacing), yoset of 5 evenly spaced
numbers whose first term is 10?
(A)5
(B)10
(C)15
(D)25
(E)Cannot be determined 5. (E) Evenly spaced numbers are an equal number of
units apart, such as 5101520 or 36912 or 2468.
With an odd number of ev11. With an EVEN
number of consecutive integers, add the first and last
terms and divide by 2 to find the average:
6 + 11 = 17; 17/2 = 8.5. You could also add the two
central terms and divide by 2: 8 + 9 = 17; 17/2 = 8.5.5.What is the average of a o add the numbers and divide by 7.4.What is the average of 6 consecutive integers of which
the smallest is 6?
(A)36
(B)12
(C)8.5
(D)6.5
(E)5.54. (C) Since the integers are consecutive, they
are right after one another, 678910ers are integers that follow one
another directly, like 234 or 111213. With an odd
number of such integers, the one in the middle is the
average. The series here is 11121314151617; 14 is
in the middle and is the average. Do NOT take the time
t + 10/4 = 16/4; divide by the number of
terms. 16/4 divided by 4 = 4/4 or 1.3.What is the average of seven consecutive integers of
which the smallest is 11?
(A)16
(B)14
(C)12
(D)11
(E)Cannot be determined3. (B) Consecutive integf those numbers by the number of numbers. The six
numbers sum to 300; 300/6 = 50.2.What is the average of 1/4, 1/2, 3/4, and 2 1/2?
(A)2
(B)1 3/4
(C)1
(D)3/4
(E)1/22. (C) Convert all the terms to fourths and add:
1/4 + 2/4 + 3/4itive or negative.Press^Wto
enter theSCRATCHPADand^StoSEE QUESTION
again. **1.What is the average of 37, 19, 45, 81, 63, and 55?
(A)300
(B)100
(C)75
(D)50
(E)471. (D) To find the average of numbers, divide the sum
onumber
that follows.
** A space appears on both sides of a + or  sign
when the symbol indicates addition or subtraction
(4 + 3 = 7).If the sign appears without a space
(e.g. 4), the symbol indicates whether the number
following it is posOT drawn exactly to scale.
All figures lie in a plane unless the problem
states or indicates otherwise.All numbers used in
this section are real numbers.
NOTE:The word "radical" in these questions
indicates the positive square root of the following information may be useful:
SYMBOLS
*multiplied by
is parallel to
>is greater than<is less than
NOTE:Although the figures that accompany the
problems in this section are drawn as accurately
as possible, they are Nm carefully.
Choose the best answer and type the letter
corresponding to your choice.** Press the RIGHT
arrow key>to begin and to continue to the next
question.Press the LEFT arrow key<if you wish
to review previous questions. **
TheMUX0JY+7MUZ [+7MU\@k]+7MU^P_+7MU_`P*a+7MUa0b+7MUcPd+7
TOPIC 8:AVERAGES
Problem Solving and Data Sufficiency
DIRECTIONS:Read and solve each proble ~6W@O0J
0
Pva @@f0 pP;RS0T
ement (1) alone is not sufficient;
(C) if both statements (1) and (2) together are
sufficient to answer the question, but neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
answer the question;
(E) if neithcal and background
knowledge, determine whether the data are
SUFFICIENT for answering the question.Choose:
(A) if statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statement (2) alone is sufficient but
statlowing information may be useful:
SYMBOLS
*multiplied by parallel to
>is greater than<is less than
DIRECTIONS:Each problem consists of a question
and two statements.Using the data given in the
problem plus your own mathematie interest rate = $12,000; therefore, divide
both sides by $80,000 to solve for the interest rate.
$12,000/$80,000 = .15 or 15%. (You can check your work
by working backwards: multiply $80,000 by .15 to see
that it does in fact equal $12,000.)The folest in the first quarter year of
investment on a principal investment of $80,000?
(A)331/3%
(B)30%
(C)27.5%
(D)25%
(E)15%8. (E) If a quarteryear's interest is $3,000, the
annual interest must be four times that, $12,000.
$80,000*thuals Interest. Multiply the
$15,000 principal by the 5% rate to get annual interest
earnings of $750; multiply that by the time, 1 year, to
have a final answer of $750.8.What is the annual rate of interest on an account that
earned $3,000 simple interrest will an investor earn in one year
on a principal investment of $15,000 that earns 5%
simple interest annually?
(A)$1,500
(B)$750
(C)$550
(D)$500
(E)$507. (B) The formula to use here is PRT = I, Principal
times Rate times Time eq2
inches to a foot, there are 24 inches in 2 feet; 24
inches is the length. Multiply 24*16 to get the number
of square inches in the area of the table top.tar table top is how many
square inches?
(1)The length of the table is 2 feet
(2)The width of the table is 16 inches
(A) (B) (C) (D) (E)15. (C) To find an area of a rectangle, you need both
length and width. Since you know that there are 1that there must be 6 nickels ($0.30), 6 dimes ($0.60) and
6 quarters ($1.50) for a total of $2.40. Note that you
did not need to find the exact answer, only to determine
that the data were sufficient to answer the question.15.The area of the rectangul
(A) (B) (C) (D) (E)14. (C) Statement one only gives you the ratio of the
number of coins. Alone it is not sufficient. Statement
two tells you how many there are of one type of coin, and
ALONE is not sufficient. Using both statements, you know
ime it is presently, so that you can
find out how many minutes past 10:00 it is.14.How much money is in Dion's piggy bank?
(1)The bank contains equal numbers of nickels,
dimes, and quarters, and no other coins
(2)There are 6 dimes in the bankShe was supposed to arrive at 10:00
(A) (B) (C) (D) (E)13. (A) Since an hour is 60 minutes, 1/3 of an hour
is 20 minutes. In 20 minutes she will be 45 minutes late;
she must be 25 minutes late now. Statement two is useless
unless you know what tfind the number of laps in a
mile. Since there are 3 feet to a yard, 1,760 yards are
the same as 1 mile; statement two tells you that
1 mile = 211.20 laps.13.How many minutes late is Lorna?
(1)In 1/3 of an hour, she will be 45 minutes late
(2);12.How many laps of the pool will Julia have to swim
to go a mile?
(1)Each lap of the pool is 25 feet
(2)1,760 yards = 211.20 laps
(A) (B) (C) (D) (E)12. (D) Since 1 mile = 5,280 feet, and if 1 lap is 25
feet, divide 5,280 by 25 to
truck can haul 335 pounds of trash per trip, how
many trips must it make to haul 4.5 tons?
(A)18
(B)24
(C)26
(D)27
(E)286. (D) There are 2,000 pounds in a ton; 4.5*2,000 =
9,000 pounds of trash to be hauled. If the truck can take
3
(B)1,760
(C)1,173 1/3
(D)1,O49 1/2
(E)8805. (E) There are 5,280 feet in a mile; since there are 3
feet to 1 yard, there are 1,760 yards in a mile (5,280/3 =
1,760), and 1,320 yards in 3/4 mile. Divide 1,320 by 1.5
to get 880.6.If a14 days; add the 3 additional days to get 17 days. There
are 24 hours in a day; 24*17 = 408 hours. Since there
are 60 minutes in an hour, 15 minutes is 1/4 or .25 hour.5.Into how many 1.5 yard segments can a 3/4mile
sidewalk be divided?
(A)5,280e to go through
the problem one step at a time.4.If a job takes 2 weeks, 3 days and 15 minutes, how
many hours does it take?
(A)390.25
(B)408.25
(C)415.25
(D)500.50
(E)512.754. (B) There are 7 days in a week, such that 2 weeks =
number of feet in a yard is what
fraction of a yard?
(A)1/3
(B)1/6
(C)2/3
(D)1/9
(E)2/93. (A) There are 3 feet in a yard, such that 1/3 the
number is 1 foot, which is 1/3 of a yard. Don't
automatically choose (D), 1/9; take the timyear:
July 31; August 31; September 30; October 31; November 30
and December 31 for a total of 184 (you could simply
subtract 181 from the number of days in a year, 365).
If the problem discussed a leap year, February would have
29 days.3.Onethird the(A)3
(B)4
(C)9
(D)12
(E)13 2. (A) The numbers of days per month in the first six
months of the year are as follows: January 31;
February 28; March 31; April 30; May 31 and June 30 for a
total of 181. In the second six months of the are 6 pints in 3 quarts and 12 cups in 6 pints.
There are 4 quarts, or 16 cups, to a gallon such that
there are 32 cups in 2 gallons. 8 + 12 + 32 = 52.2.How many more days are in the second six months of a
nonleap year than in the first six months?
**1.What is the sum of the number of cups in 4 pints,
3 quarts, and 2 gallons?
(A)49
(B)50
(C)51
(D)52
(E)531. (D) There are 2 cups to a pint, such that there are 8
cups in 4 pints. There are 2 pints to a quart, such that
theree symbol indicates addition or subtraction
(4 + 3 = 7).If the sign appears without a space
(e.g. 4), the symbol indicates whether the number
following it is positive or negative.Press^Wto
enter theSCRATCHPADand^StoSEE QUESTION
again.r indicates otherwise.All numbers used in
this section are real numbers.
NOTE:The word "radical" in these questions
indicates the positive square root of the number
that follows.
** A space appears on both sides of a + or  sign
when tho
>is greater than<is less than
NOTE:Although the figures that accompany the
problems in this section are drawn as accurately
as possible, they are NOT drawn exactly to scale.
All figures lie in a plane unless the problem
states oe.** Press the RIGHT
arrow key>to begin and to continue to the next
question.Press the LEFT arrow key<if you wish
to review previous questions. **
The following information may be useful:
SYMBOLS
*multiplied by
is parallel t+7U0RV+7X0X+7
TOPIC 7:UNITS OF MEASUREMENT
Problem Solving and Data Sufficiency
DIRECTIONS:Read and solve each problem carefully.
Choose the best answer and type the letter
corresponding to your choic 06@+ i
, P7
@P`0 +70+7;R@R+7SuT
ss of which is the value of x,
you can answer the question. Since 100% of any number is
that number itself, x = 1. Note that you do NOT have to
have BOTH statements together, since you are not solving
for the precise value of x but only for whether or no be given a lot of information, but none of it is
sufficient to answer the question.19.Is x negative?
(1)x squared = 100% x
(2)100% x = 1
(A) (B) (C) (D) (E)19. (D) Only 0 or 1 remain 100% of themselves when
squared, such that regardle(A) (B) (C) (D) (E)18. (E) Squaring a number means multiplying it by
itself; the percent that the square is of the square root
depends on the number itself. While 2 squared = 4, which
is 200% of 2, 4 squared = 16, which is 400% of 4. You
seem to, then x = (4/3)y, or 1331/3%
OF y, or 331/3% GREATER THAN y. Statement two is
insufficient; the percentages depend entirely on the
values of y and x.18.What percent of x is x squared?
(1)x > 1
(2)x is an even whole number greater than zero
ld is
the original price, $15.99. Statement two is unnecessary;
alone it is not sufficient as you are missing the original
price.17.What percent greater than y is x?
(1)y = (3/4)x
(2)y + 25 = x
(A) (B) (C) (D) (E)17. (A) If y = (3/4)x regular price
(A) (B) (C) (D) (E)16. (A) You need the dollar discount (found by
subtracting the discounted price from the original price)
and the original price. Statement one allows you to find
the dollar discount; place it over what you are toas the sides sum up to 29, the
bases and height could be any size.16.What percent discount is there on the computer disks?
(1)The price of the disks was dropped from $15.99
to $13.99
(2)The discount price of the disks is $2.00 less
than thee sides of
the triangle are 4, 4, 4 radical 2, such that the area
is 4*4, or 16. However, statement one is not sufficient
to find the area of the trapezoid, which requires you to
know both the bases (so that you may average them) and
the height. As long g statement two alone.15.What percent of x is y?
(1)x is the area of a trapezoid of perimeter 29
(2)y is the area of an isosceles right triangle
of hypotenuse 4 radical 2
(A) (B) (C) (D) (E)15. (E) Statement two lets you know that ththe portion or part. Statement two is sufficient; to
have twice as many students pass as flunk means the ratio
is 2:1, and the percentages are 662/3% passing and
331/3% flunking. Note that you could not determine the
NUMBER of students who flunked usinnked French class?
(1)There were 150 students in the class
(2)Exactly twice as many students passed as flunked
(A) (B) (C) (D) (E)14. (B) Statement one alone gives you the total; in
order to find a percentage, you need BOTH the total and
neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
answer the question;
(E) if neither statement alone nor both statements
together are sufficient to answer the question.14.What percent of the students fluement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statement (2) alone is sufficient but
statement (1) alone is not sufficient;
(C) if both statements (1) and (2) together are
sufficient to answer the question, but CTIONS:Each problem consists of a question
and two statements.Using the data given in the
problem plus your own mathematical and background
knowledge, determine whether the data are
SUFFICIENT for answering the question.Choose:
(A) if stat:1 The following information may be useful:
SYMBOLS
*multiplied by parallel to
>is greater than<is less than
NOTE:The word "radical" in these questions
indicates the positive square root of the number
that follows.
DIRE
(A)3000%
(B)300%
(C)30%
(D)3%
(E)None of the above6. (B) Start with the smallest term and plug in
hypothetical values. Say that c = 10. Then b = 15 and
a = 30. Since a is three times c, it is 300% of c.7.The area of a square must have slightly more than that. The
closest answer is (C). Note that you should try to
estimate; do NOT take the time to work out the answer
precisely unless the answer choices are very close.6.If a = 200% b and b = 150% c, a is what percent of c?eek after he
has paid rent and bought groceries?
(A)$192.05
(B)$175.60
(C)$154.96
(D)$130.83
(E)$86.705. (C) If he spent 48% on rent and groceries, then he
DIDN'T spend 52%, or slightly MORE than half. Since 1/2
of $298 = $149, he s and multiply.
331/3% = 1/3; 62.5% = 5/8; 50% = 1/2. (1/3)*(5/8)*(1/2) =
5/48. Since 5/48 is ABOUT 5/50, or 1/10, the closest
approximation is 10%.5.Curtis makes $298 a week and spends 48% of it on rent
and groceries.How much money is left every w 9a = 1b. Therefore, b is 9 times as great
as a, or 900% of a.4.331/3% of 62.5% of 50% of x is closest in value to
(A)40% of x
(B)10% of x
(C)4% of x
(D)1% of x
(E)Cannot be determined4. (B) Convert the percentages to fractioned, both those
lost and those won: 65 + 85 = 150. 65/150 =
approximately .4333 or 431/3%.3.If 3a = (1/3)b, what percent of a is b?
(A)1/9 %
(B)9%
(C)11%
(D)90%
(E)900%3. (E) Multiply both sides by 3 to get rid of the
fraction:t
percent of the games did it lose?
(A)85%
(B)65%
(C)431/3%
(D)25%
(E)162/3%2. (C) To find a percent, put the part over the whole.
Here, the part is the number of games the team lost, 65.
The whole is the total number of games playbe determined1. (A) To find a percentage, divide the part by the whole.
Here, the part is the number of high notes she made, 32.
The whole is the number of high notes she tried, 160.
32/160 = 1/5 = 20%.2.A softball team won 85 games and lost 65.Whasitive or negative.Press^Wto
enter theSCRATCHPADand^StoSEE QUESTION
again. **1.A singer made 32 out of every 160 high notes she
tried.What percent of the high notes did she make?
(A)20%
(B)12%
(C)10%
(D)9%
(E)Cannot number
that follows.
** A space appears on both sides of a + or  sign
when the symbol indicates addition or subtraction
(4 + 3 = 7).If the sign appears without a space
(e.g. 4), the symbol indicates whether the number
following it is poNOT drawn exactly to scale.
All figures lie in a plane unless the problem
states or indicates otherwise.All numbers used in
this section are real numbers.
NOTE:The word "radical" in these questions
indicates the positive square root of theThe following information may be useful:
SYMBOLS
*multiplied by
is parallel to
>is greater than<is less than
NOTE:Although the figures that accompany the
problems in this section are drawn as accurately
as possible, they are blem carefully.
Choose the best answer and type the letter
corresponding to your choice.** Press the RIGHT
arrow key>to begin and to continue to the next
question.Press the LEFT arrow key<if you wish
to review previous questions. **
;RV V+7;RNXP
Y+7;RzZ`[+7;R\ ]+7;R]PQ^+7;R_@_+7;RIa0Zb+7
TOPIC 6:PERCENTAGES
Problem Solving and Data Sufficiency
DIRECTIONS:Read and solve each pro 6X
0_Y P =
0w
DP80PJ@3 0
the 1984 sales are 115% of the 1983's sales.
Although you should not take the time to do so, you COULD
determine the exact number of sales in 1983. Statement
two is not sufficient; you would have to know the average
profit per widget and divide it into $1ure at a time
(2)Katherine transports a total of 51 pieces of
furniture
(A) (B) (C) (D) (E) er statement alone nor both statements
together are sufficient to answer the question.9.How many trips does Katherine make to transport
furniture by truck from Store X to Store Y?
(1)Katherine's truck can carry a maximum of six
pieces of furniteople who finish AT LEAST two years in fact go on to
graduate or not; the same is true of those who finish AT
LEAST three years. You do not have enough information
to answer the question. (1)40% of the people finish at least two years
of a fouryear college program
(2)60% of the people finish at least three years
of a fouryear college program
(A) (B) (C) (D) (E)15. (E) The key is AT LEAST. You do not know whether
the parea of
the triangle is 1/2 the product of those numbers, while
the area of the rectangle is the whole product of those
numbers. You do not need to know the actual numbers to
find the ratio.15.What is the ratio of college graduates to
nongraduates?
e inscribed in a rectangle has an
area half that of the rectangle, such that the ratio of
their areas is 1:2. This is true because the triangle and
rectangle have the same base and height (in the rectangle
these measures are called length and width); the er the question.14.What is the ratio of the area of the triangle to
the area of the rectangle?
(1)The rectangle has a length of 8 and a width of 6
(2)The triangle is inscribed within the rectangle
(A) (B) (C) (D) (E) 14. (B) Any triangla to b?
(1)a = 12.5% of b
(2)b = 800% of a
(A) (B) (C) (D) (E)13. (D) If a is 12.5% or 1/8 of b, then a:b is 1:8.
If b is 800% of, or 8 times, a, then b:a is 8:1, and
a:b would be that reversed, 1:8. Either statement
is sufficient to answ every set of 7 items
not returned, there are 8 items returned. Since 7 + 8 =
15, and there are 20 SETS of 15 in 300 (300/15 = 20),
there must be 20 SETS of items purchased and returned.
With 8 items per set, that is 160 items.13.What is the ratio of total number of items purchased
and returned?
(1)There are 300 items purchased
(2)The ratio of items purchased and not returned
to the number of items purchased and returned
is 7:8
(A) (B) (C) (D) (E)12. (C) With the ratio of 7:8, for dancers as well. 34 (the number
of sets) * 17 (the number of dancers per set) = 578
dancers; 578 + 34 = 612 people at the ball. Note that you
did not have to work this problem out; only see that you
do have enough information to do so.12.What is thee ratio of dancers to nondancers is 17:1
(A) (B) (C) (D) (E)11. (C) The ratio of 17:1 means that for every set of 17
dancers, there is a set of 1 nondancer. With 34
nondancers, there are 34 sets (because 34/1 = 34) and
there must be 34 sets ofis being asked for.
Statement two would help you find the number of rifles if
you combined it with statement one; however, it is not
necessary for finding the ratio.11.What is the number of dancers at the ball?
(1)There are 34 nondancers
(2)Th}:10. (A) If there are 3 times as many pistols as rifles,
the ratio of pistols to rifles is 1:3 and the ratio of
rifles to pistols is that reversed, 3:1. It makes no
difference how MANY rifles or pistols there are; the
ratio or proportion is all that
z #?б®àеРиà
и®à йàй®àбàб®à( бà д®à
"еàе®à
жàж®à
зàз®à бàб®àвàв®àгàг®à дàemployee turnover for
any year and thus cannot make a comparison.'DIRECTIONS:Each problem consists of a question
and two statements.Using the data given in the
problem plus your own mathematical and background
knowledge, determine whether the ndidate A was 2,000 (10% of the 20,000).
Since 1,000 votes were cast by men, men cast 1/2 or 50%
of all the votes for Candidate A.&A, the remaining 5,500 voters must have been
women.8.What percent of all voters aged 5165 who voted for
Candidate A are men?
(A)1,000%
(B)100%
(C)50%
(D)20%
(E)10%8. (C) The TOTAL number of people in this age group
voting for CaCandidate A fell
into this age group, find 35% of 20,000. .35*20,000 =
7,000. A TOTAL of 7,000 age 2230 voters, men and women,
voted for Candidate A. Since the righthand column of
the graph tells you that 1,500 men, aged 22 to 30, voted
for Candidate find 13% of the total number of votes cast for
Candidate A. .13*20,000 = 2,600.7.How many women aged 2230 voted for Candidate A?
(A)7,000
(B)6,800
(C)5,900
(D)5,500
(E)Cannot be determined7. (D) Since 35% of all the voters for ALL CANDIDATES:100,000
TOTAL VOTES CAST FOR CANDIDATE A:20,0006.How many people over 65 voted for Candidate A?
(A)13,000
(B)11,000
(C)3,600
(D)2,600
(E)1,1106. (D) Since 13% of those who voted for Candidate A were
over 65, HPADand^StoSEE QUESTIONagain. **
BREAKDOWN OF VOTES CAST FOR CANDIDATE A
% 0F ALL
AGE A'SVOTESVOTES CAST BY MEN
1821171,600
2230351,500
3150252,500
5165101,000
Over 65
131,110
TOTAL VOTES CAST FOR idget, then multiply it by the
number of widgets to get the total dollar amount
of profit.DIRECTIONS:Read and solve the following problem.
Choose the best answer and type the letter
corresponding to your choice.** Press^Wto enter
theSCRATC (E)5. (C) You must have both statements. The graph tells
you how many widgets were sold altogether. If the selling
price were $11.00 and 12.5% of that was profit, you could
(don't actually take the time to do so) determine the
amount of profit per w,400,000 to find
the total number of widgets.5.What was the dollar amount of profit Company D made on
its widgets in 1984?
(1)It made 12.5% profit per widget
(2)The average price per widget for all widgets
sold was $11.00
(A) (B) (C) (D)data are
SUFFICIENT for answering the question.Choose:
(A) if statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statement (2) alone is sufficient but
statement (1) alone is not sufficient;
(C) if both statey was spent. Statement
two tells you the same thing; you can divide the total,
$12.50, by the number of prints, 5, to find that the cost
per print was $2.50. You still do not know the number of
prints and thus cannot find the total money spent.td have cost an
additional $12.50
(A) (B) (C) (D) (E)15. (E) This problem gives you the same information
twice. Statement one alone is not sufficient; you would
have to know HOW MANY prints, at $2.50 each, were
purchased to find out how much mone been 300 miles as well) and that the total time was
11 hours. Divide the total distance by the total time to
find the average rate of speed.15.How much was spent on color prints?
(1)The cost per print was $2.50
(2)An additional 5 prints woulhours
(A) (B) (C) (D) (E)14. (C) You must know the total distance traveled and
the total time taken. Using BOTH statements, you can
determine that the total distance was 600 miles (since
the "remaining half" was 300 miles, the first half must
havnow the principal, the
interest rate, and the time the money remained in the
account to find the amount of interest earned.14.What was the average speed for the trip?
(1)The first half of the trip took 6 hours
(2)The remaining 300 miles took 5 that paid
12.5% annually
(2)He left the money in that account for two years
(A) (B) (C) (D) (E)13. (E) You are missing one of the three vital pieces of
information, the principal or original amount of money
placed in the account. You must ktement two to find the temperature at 2:00
(45 degrees). Statement one alone is insufficient; it
would only be useful if you knew what the temperature
was at noon.13.How much interest did Chris earn on his money?
(1)He had the money in an account than it was at noon
(2)The temperature at the airport was 25 degrees at
10:00 a.m. and rose 5 degrees every hour
(A) (B) (C) (D) (E)12. (B) Since there are four hours between 10:00 a.m.
and 2:00 p.m. you can use the information given in
stasufficient; you must
have some specific numbers for the speed of either the
first third or second twothirds of the trip.12.What was the temperature at the airport at 2:00 p.m.?
(1)The temperature at the airport at 2:00 p.m. was
10 degrees higher remaining twothirds
(A) (B) (C) (D) (E)11. (A) To find the average rate of speed, divide the
total distance traveled by the total time taken to travel
that distance. Statement one gives you both vital pieces
of information. Statement two is inone gives you the first; neither
statement gives you the second. You do not have enough
information.11.What is Norbert's average rate of speed?
(1)He drives 273 miles in 4.2 hours
(2)He drives the first third of the trip twice as
fast as theitem?
(1)The item originally sold for $49.00
(2)The new profit on the item is 10%
(A) (B) (C) (D) (E)10. (E) In order to find the percent of discount, you
must know the original selling price and the new, reduced
selling price. Statement er of pieces transported
(statement two) to determine the number of trips. This
is because you have no way of knowing whether or not the
truck is fully loaded on each trip. It's not safe to
assume that it is.10.What was the percent of discount on an d:
9. (E) You cannot answer this question even through the
use of both statements. Statement one tells you that
Katherine's truck can carry a maximum of six pieces of
furniture at once. However, you cannot simply divide
this number into the total numb
mpany A sell in 1983?
(1)Company A's 1984 sales represented a 15% increase
over its 1983 sales
(2)Company A earned $1,400,000 from the sales of
its widgets
(A) (B) (C) (D) (E)4. (A) Statement one alone is sufficient; it tells you
that =====
20 +======
======
10 +======
======
+
ABCDEF4.How many widgets did Coo
answer the question;
(E) if neither statement alone nor both statements
together are sufficient to answer the question.
SALES OF WIDGETS, 1984
(in thousands)

50 +

40 +=
==
30 +====
ement (2) alone is sufficient but
statement (1) alone is not sufficient;
(C) if both statements (1) and (2) together are
sufficient to answer the question, but neither
statement alone is sufficient;
(D) if either statement alone is sufficient tin the
problem plus your own mathematical and background
knowledge, determine whether the data are
SUFFICIENT for answering the question.Choose:
(A) if statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if stat(C)$120,000
(D)$80,000
(E)$30,0003. (B) Company E sold 30,000 widgets; if it made a profit
of $6.00 per widget, it made 30,000 * $6.00 = $180,000.'DIRECTIONS:Each problem consists of a question
and two statements.Using the data given t be greater than 1, and greater than
100%, allowing you to eliminate answers (D) and
(E) immediately.3.If Company E made a profit of $6.00 per widget sold,
what was its total profit from the sales of widgets
in 1984?
(A)$200,000
(B)$180,000
mula "is" over
"of" (or in this case, "were" over "of"). Here, that
gives you 40,000/25,000 or 40/25. Reduce by a factor of 5
to get 8/5, which is the same as 1 3/5, or 160%. Note
that since the numerator is larger than the denominator,
the fraction mussold. Company C sold
30,000. Company A sold 20,000. 30,000  20,000 = 10,000.2.In 1984, what percent of Company F's sales were
Company B's sales?
(A)200%
(B)160%
(C)150%
(D)40%
(E)25%2. (B) To find percent, you can use the for
ABCDEF
1.Company C sold how many more widgets than did Company
A in 1984?
(A)30,000
(B)20,000
(C)10,000
(D)20
(E)101. (C) Look at the title to see that the graph
represents THOUSANDS of widgets

40 +=
==
30 +====
=====
20 +======
======
10 +======
======
+CTIONS:Read and solve the following problem.
Choose the best answer and type the letter
corresponding to your choice.** Press^Wto enter
theSCRATCHPADand^StoSEE QUESTIONagain. **
SALES OF WIDGETS, 1984
(in thousands)

50 + carefully.
Choose the best answer and type the letter
corresponding to your choice.** Press the RIGHT
arrow key>to begin and to continue to the next
question.Press the LEFT arrow key<if you wish
to review previous questions. **DIREya+ee+7yag0wh+7imk0kil Tmico pieqPqWs;w0w+7
TOPIC 10:GRAPHS
Problem Solving and Data Sufficiency
DIRECTIONS:Read and solve each problem 6'0M c (
*6 *0*6WA@WM@W00;R:VW*6;RIX@IY+78[\@^]8[ ^P^8[_P7`
13. (E) This was a tricky question. You CAN determine
in which year the PERCENT of employee turnover was the
greatest (1981 with 20%); however, since you do not know
the total number of employees in any given year, you
CANNOT find the numerical rate of (in 1982, the percentage would be 15%) to find a
specific number. You do not have enough information.13.In which year did the greatest number of employees
leave Althaus Co.?
(A)1980
(B)1981
(C)1982
(D)1983
(E)Cannot be determined982?
(A)15,000
(B)1,500
(C)150
(D)100
(E)Cannot be determined12. (E) Nowhere are you given a TOTAL number which the
graph (or any portion thereof) represents. Since you do
not have a total, you cannot take a percentage of that
total10%
(C)15%
(D)20%
(E)25%11. (D) This is a very simple question. Look at the
graph above the 1981; check the lefthand axis to see
that in that year, there was a turnover of
approximately 20%.12.How many employees left Althaus Co. in 1
20 +x
ANNUAL
15 + x x
PERCENT
10 +xx

5 +

+
1979198019811982198311.What was the approximate percent of employee turnover
at Althaus Co. in 1981?
(A)5%
(B)DIRECTIONS:Read and solve the following problem.
Choose the best answer and type the letter
corresponding to your choice.** Press^Wto enter
theSCRATCHPADand^StoSEE QUESTIONagain. **
EMPLOYEE TURNOVER,
ALTHAUS CO.

25 +
number who were male from the total number).
You could compare the numbers and determine which sex was
in the majority. Remember not to waste your time
performing calculations on problems of this sort; simply
determine that you COULD answer the question.B) (C) (D) (E) 10. (D) Either statement alone is sufficient. Although
you should NOT take the time actually to do so, you could
figure out the total number of voters for Candidate A in
each age group, then determine how many were female
(subtract theate needs to win.10.Is Candidate A a man?
(1)Candidate A is the same sex as the majority of
those aged 2230 who voted for Candidate A
(2)Candidate A is the opposite sex of the minority
of those aged 1821 who voted for Candidate A
(A) (t Candidate
A only got 20,000 votes out of a total of 100,000 cast.
You could use this information to answer the question.
Statement two is not sufficient; you do not care how many
people were in the race, only the percent of the total
vote that one candid009.Did Candidate A win the election?
(1)A candidate must receive at least 51% of the
total vote to win
(2)There were three candidates besides Candidate A
in the race
(A) (B) (C) (D) (E)9. (A) The note under the graph tells you tha FOR CANDIDATE A
% 0F ALL
AGE A'SVOTESVOTES CAST BY MEN
1821171,600
2230351,500
3150252,500
5165101,000
Over 65
131,110
TOTAL VOTES CAST FOR ALL CANDIDATES:100,000
TOTAL VOTES CAST FOR CANDIDATE A:20,0stion, but neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
answer the question;
(E) if neither statement alone nor both statements
together are sufficient to answer the question.
BREAKDOWN OF VOTES CAST(A) if statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statement (2) alone is sufficient but
statement (1) alone is not sufficient;
(C) if both statements (1) and (2) together are
sufficient to answer the que8&&DIRECTIONS:Each problem consists of a question
and two statements.Using the data given in the
problem plus your own mathematical and background
knowledge, determine whether the data are
SUFFICIENT for answering the question.Choose:
re $2.00 each, for a total of $8.00. Pounds
510 are $1.00 each for a total of $5.00. Pounds 1120
are $1.00 each for a total of $10.00. The total cost is
$2.00 + $8.00 + $5.00 + $10.00 = $25.00. Divide the total
by the number of pounds: $25.00 / 20 pthe first one at $6.00 and one more
at $5.00.19.What is the average cost per pound to send 20 pounds
nonpriority?
(A)$25.00
(B)$16.50
(C)$9.75
(D)$2.25
(E)$1.2519. (E) Find the total cost. The first pound is $2.00.
Pounds 25 aotal cost of nonpriority
delivery of 6 pounds. The first pound costs $2.00.
Pounds 25 cost $2.00 each, for a total of $8.00.
Pound 6 costs $1.00. The total cost is $2.00 + $8.00 +
$1.00 = $11.00. Using priority delivery, you could
send only 2 pounds, The difference
is $35.00  $13.00 = $22.00.18.Sending 6 pounds nonpriority costs the same as
sending how many pounds by priority delivery?
(A)1 pound
(B)2 pounds
(C)3 pounds
(D)4 pounds
(E)5 pounds18. (B) First, figure out the t= $35.00. With nonpriority delivery, the first
pound costs $2.00. Pounds 25 cost $2.00 each, for a
total of $8.00. Pounds 68 cost $1.00 each, for a total
of $3.00. To send the item nonpriority would cost a
total of $2.00 + $8.00 + $3.00 = $13.00.
(E)Cannot be determined
17. (B) With priority delivery, the first pound costs
$6.00. Pounds 25 cost $5.00 each, for a total of $20.00.
Pounds 68 cost $3.00 each for a total of $9.00. The
total cost of priority delivery is $6.00 + $20.00 +
$9.00 ords, the remaining 4 lbs.) cost $5.00 each, for a
total of $20.00. $6.00 + $20.00 = $26.00.17.How much does a shipper save sending an 8lb.
package by nonpriority rather than priority delivery?
(A)$35.00
(B)$22.00
(C)$13.00
(D)$11.006.003.00 1.00
10.1  25 lbs.3.002.00 1.0016.How much does it cost to send a 5lb. package by
priority mail?
(A)$42.00
(B)$36.00
(C)$26.00
(D)$12.00
(E)$11.0016. (C) The first pound costs $6.00. Pounds 25 (in
other w
theSCRATCHPADand^StoSEE QUESTIONagain. **
DELIVERY CHARGES
WEIGHT 1DAY AIRPRIORITYNONPRIORITY
Up to 1 lb.$10.00$6.00$2.00
Each additional pound or fraction thereof:
1.1  5 lbs.8.005.00 2.00
5.1  10 lbs.ployee turnover in 1980;
combine it with statement one to find the percentage that
one is of the other.DIRECTIONS:Read and solve the following problem.
Choose the best answer and type the letter
corresponding to your choice.** Press^Wto enters a turnover in 1980 of 200 employees
(A) (B) (C) (D) (E)15. (C) You need both statements. Statement one allows
you to use the total to find the numerical rate of
employee turnover (20% of 1,500). Statement two gives
you the numerical rate of emnt at the firm from year to year,
the information is useless in helping you answer this
particular question.15.The numerical amount of employee turnover in 1980
was what percent of that in 1981?
(1)There were 1,500 employees in 1981
(2)There wa
the turnover percentage by the total. Statement one alone
is sufficient. Statement two is insufficient. You could
use the information to find the total number of employees
in 1982, but since you cannot assume that the number of
employees remains constany employees left in 1979?
(1)The total number of employees in 1979 was 1,000
(2)In 1982, 200 employees left
(A) (B) (C) (D) (E) 14. (A) Since you know the total number of employees in
1979, you could find the number who left by multiplyingsufficient to answer the question.
EMPLOYEE TURNOVER,
ALTHAUS CO.

25 +

20 +x
ANNUAL
15 + x x
PERCENT
10 +xx

5 +

+
1979198019811982198314.How maements (1) and (2) together are
sufficient to answer the question, but neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
answer the question;
(E) if neither statement alone nor both statements
together are A) The ! means "factorial," which tells you to begin
with the number before the ! and multiply down to 1. For
example, 5! = 5*4*3*2*1 = 120. Here, you can do much
canceling.
(5*4*3*2*1) (3*2*1)

which is 24) you
know is less than 27 because it is must be less than 25
(you should have memorized that 25*25 = 625).
1 to any power is 1; any number to the first power
is itself.2.What is (5!3!)/4!?
(A) 30
(B) 32
(C) 41
(D) 90
(E) 3212. ((A)8 PI
(B)radical 576
(C)3 cubed
(D)1 to the 24th power
(E)24 to the first power1. (C) 3 cubed is 3*3*3 = 27. Since PI is approximately
3.14, you can estimate that 8 PI is larger than 24, but
less than 27. The square root of 576 (btraction
(4 + 3 = 7).If the sign appears without a space
(e.g. 4), the symbol indicates whether the number
following it is positive or negative.Press^Wto
enter theSCRATCHPADand^StoSEE QUESTION
again. **1.Which is the greatest?
bers used in
this section are real numbers.
NOTE:The word "radical" in these questions
indicates the positive square root of the number
that follows.
** A space appears on both sides of a + or  sign
when the symbol indicates addition or sulue
of x.
NOTE:Although the figures that accompany the
problems in this section are drawn as accurately
as possible, they are NOT drawn exactly to scale.
All figures lie in a plane unless the problem
states or indicates otherwise.All numto the next
question.Press the LEFT arrow key<if you wish
to review previous questions. **
The following information may be useful:
SYMBOLS
*multiplied by
is parallel to
>is greater than<is less than
x means absolute va
TOPIC 11:MISCELLANEOUS MATH
Problem Solving
DIRECTIONS:Read and solve each problem carefully.
Choose the best answer and type the letter
corresponding to your choice.** Press the RIGHT
arrow key>to begin and to continue q
6`0a``i P `
03`; `@`M0`0``j@L``i /`;R@S
ne the cost per pound for the first pound, then
pounds 25, 610, etc. using 1day air delivery, and
determine how many pounds it would take to add up to
$72.00.(.0020.How many lbs. did Kit send?
(A)She used 1day air delivery
(B)She spent a total of $72.00
(A) (B) (C) (D) (E)20. (C) You need both pieces of information. Although
you should NOT take the time actually to do so, you could
determi
DELIVERY CHARGES
WEIGHT 1DAY AIRPRIORITYNONPRIORITY
Up to 1 lb.$10.00$6.00$2.00
Each additional pound or fraction thereof:
1.1  5 lbs.8.005.00 2.00
5.1  10 lbs.6.003.00 1.00
10.1  25 lbs.3.002.00 1sufficient to answer the question, but neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
answer the question;
(E) if neither statement alone nor both statements
together are sufficient to answer the question.g the question.Choose:
(A) if statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statement (2) alone is sufficient but
statement (1) alone is not sufficient;
(C) if both statements (1) and (2) together are
ounds = 5/4 or
$1.25 per pound.&DIRECTIONS:Each problem consists of a question
and two statements.Using the data given in the
problem plus your own mathematical and background
knowledge, determine whether the data are
SUFFICIENT for answerinxcept
(A)3
(B)4
(C)5
(D)6
(E)712. (B) All of the numbers divide evenly into 630 except
for 4.s
135. If the highest term is 17, work backwards to find
the lowest term: 1715131197. 17  7 = 10.12.If 630 boxes of cookies are to be divided evenly
among x Girl Scout troops, there could be any of the
following number of Girl Scout troops e 17.What is the difference between the
highest and lowest terms in the series?
(A)10
(B)11
(C)12
(D)13
(E)Cannot be determined11. (A) Consecutive integers come right after one
another; consecutive odd integers are two apart, such a many as Alice, 12, in 2/3 the time, 30 minutes.
At a rate of 12 posters per 30 minutes, he can do 36
posters in 90 minutes. Working together, they do
12 + 36 = 48 posters in 90 minutes.11.A series of 6 consecutive odd integers has as its
highest termthey decorate
in 1 1/2 hours?
(A)90
(B)60
(C)54
(D)48
(E)3610. (D) 1 1/2 hours is 90 minutes, or twice 45 minutes.
If Alice can decorate 6 posters in 45 minutes, she can
decorate twice as many, 12, in 90 minutes. Steve does
twice as (A) If 560 = 1/8 capacity, full capacity is 8*560,
or 4,480 gallons. A third of that is 1,493 1/3 gallons. 10.Alice can decorate 6 posters in 45 minutes.Steve
can decorate twice as many in 2/3 of the time.Working
together, how many posters could A rectangular pool filled to 1/8 capacity holds 560
gallons.How much will the tank hold when it is filled
to 1/3 capacity?
(A)1,493 1/3 gallons
(B)2,803 1/3 gallons
(C)2,999 2/5 gallons
(D)3,570 2/5 gallons
(E)Cannot be determined9. 5 shirts.
How many different combinations of sweater, shirt and
pants can he make?
(A)13
(B)17
(C)60
(D)180
(E)2708. (C) To find the total number of combinations possible,
multiply the individual numbers of items. 6*2*5 = 60.9.re CONSECUTIVE, they are only
one unit apart. You can quickly find the numbers by
dividing 127 by 2; the quotient is 63.5, which means 1
number is 63 and 1 is 64. The square root of the even
number, 64, is 8.8.Ace has 6 pairs of pants, 2 sweaters andp. Since there are 6 terms, there is a difference of
1 + 1 + 1 + 1 + 1 + 1 = 6.7.The sum of two consecutive integers is 127.The
square root of the even number of the pair is
(A)10
(B)9
(C)8
(D)7
(E)67. (C) Since the integers a the answer is 3.6.(18 + 20 + 22 + 24 + 26 + 28) 
(17 + 19 + 21 + 23 + 25 + 27) =
(A)138
(B)106
(C)24
(D)6
(E)16. (D) There is a difference of 1 between each term in
the first group and the corresponding term in the second
grouabsolute value of a number is its magnitude, and is
always positive. The absolute value of 3 is 3. However,
you must then find the negative of that 3 because of the
negative sign OUTSIDE the absolute value lines. The
negative of a positive is negative;All of these are prime
factors; there are three distinct ones.5.3 =
(A)3
(B)3
(C)1
(D)1
(E)Cannot be determined5. (B) The vertical lines indicate that you are to find
the ABSOLUTE VALUE of the number within the lines. The
in 1,690?
(A)1
(B)2
(C)3
(D)4
(E)54. (C) A prime number is any number that can be divided
evenly only by itself and 1. Examples of primes are
2,3,5,7 and 11. You can factor 1,690 down into 169*10,
and further down into 13*13*2*5. stitute the
numbers for the symbols. Everywhere you see an "a," place
a 2. Everywhere you see a "b," place a 3. Everywhere you
see a "c," place a 4. This gives you (2*4) + 3*(2 + 4),
which is 8 + 18 = 26.4.How many distinct prime factors are there (4*3*2*1)
Cancel, leaving a numerator of 5*3*2*1 and a denominator
of 1. Your final answer is 30/1 or 30.3.a # b # c = (ac) + b(a + c).Solve for 2 # 3 # 4.
(A)86
(B)84
(C)62
(D)39
(E)263. (E) This is a SYMBOLISM problem; subt be greater
than 21. Since 5*5 = 25, your answer must end in a 5.
Only the correct answer has both of these qualifications.
Look BEFORE you begin doing the problem to see whether you
can estimate rather than doing the actual mathematics.7.6/11 dividelying the
denominators (3*4*8 = 96) and automatically looking for
an answer with that result. (A) is the trap answer here.6.7.65*3.25 =
(A)18.6565
(B)21.8950
(C)24.8625
(D)28.3644
(E)28.39006. (C) Since 7*3 = 21, your answer mus (C) When multiplying fractions, simply multiply the
numerators, then the denominators. You can make it even
easier by canceling; here cancel the 3's so that your
numerators are 1*1*7 = 7. Your denominators are
1*4*8 = 32. Do NOT be tricked into multip.115 = .945. Note that your answer must be LESS THAN 1
because of the decimals, and must end in a 5 since you are
subtracting 5 from "0" which you convert to a "10."5.(1/3)*(3/4)*(7/8) =
(A)37/96
(B)11/20
(C)7/32
(D)3/8
(E)1/35./3
will be positive, so that the answer will be 4 plus some
additional fraction. Only one answer fits this
description.4.4.06  3.115 =
(A)1.045
(B).945
(C).650
(D).550
(E).5064. (B) 4.06 is the same as 4.060; subtract 4.060
 3.11.What is the value of x?
(1)x is equal to the sum of radical 9 and
radical 144
(2)x > radical 400
(A) (B) (C) (D) (E)d the square root of x is 27 (since
27*27 = 729). Note that you personally should not take
the time to find the square root of 531,441; simply
recognize that the data are sufficient to find it, such
that you can use that information to answer the questiony itself
three times, gives that product. If the cube root of x
is 9, then x = 729 (since 9*9*9 = 729) and the square root
of x is 27 (since 27*27 = 729). If the perfect square of
x is 531,441 and x is positive, then x is 729 (since
729*729 = 531,441) anWhat is the square root of x?
(1)The cube root of x is 9
(2)x is positive and the perfect square of
x is 531,441
(A) (B) (C) (D) (E)10. (D) Either statement is sufficient. The cube root
of a number is the number which, when multiplied bis the value of x?
(1)x is radical 625
(2)x is the perfect square of 5
(A) (B) (C) (D) (E)9. (D) Either statement tells you that x = 25. 25 is
the square root of 625 since 25*25 = 625. 25 is the
perfect square of 5 since 5*5 = 25.10.to answer the question, but neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
answer the question;
(E) if neither statement alone nor both statements
together are sufficient to answer the question.9.What ion.Choose:
(A) if statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statement (2) alone is sufficient but
statement (1) alone is not sufficient;
(C) if both statements (1) and (2) together are
sufficient a role in determining which product
will be greatest. Once these are eliminated, it's easy
to compare the five pairs of numbers that remain. Of the
five, the (9) and (17) in set D will produce by far the
largest product.nmultiply all the sets of
numbers. Instead, look for ways to simplify the
comparison process. Note that three factors appear in
ALL the sets: (8), (13), and (16). These can be
eliminated from the reckoning; since they are common to
all, they cannot play q813.Which of the following sets of numbers has the
greatest product?
(A)(13) (8) (16) (3) (7)
(B)(10) (16) (8) (7) (13)
(C)(5) (8) (16) (7) (13)
(D)(8) (9) (16) (13) (17)
(E)(9) (16) (13) (10) (8)
13. (D) Don't take the time to
mation refer to the MATH RULES and SPECIAL
STRATEGIES:DATA SUFFICIENCY portions of the
printed booklet.
NUMBER OF QUESTIONS ON THE GMAT:25
QUESTION FORMAT:You are given a math problem or a
question, called a "stem."Following the stem athe one closest to
your estimate.Unless all the choices are very
close together, an estimate will be sufficient.
"DATA SUFFICIENCY
Use this AID! SCREEN to remind yourself of the best
approach to this question type.For additional
infor ANSWER CHOICES carefully.
Immediately eliminate any that are obviously
incorrect, such as those that have the wrong units
(for example, square instead of cubic).
3.PREDICT an answer.Make a rough estimate; look
at the answer choices to find QUESTION:
1.READ the question ALL THE WAY THROUGH before
beginning any calculations; you may be able
immediately to determine that the problem cannot be
solved and thus prevent yourself from wasting time
on any calculations.
2.LOOK at theMBER OF QUESTIONS ON THE GMAT:20
QUESTION FORMAT:These are standard multiple
choice questions.A question will be followed by
five answer choices; select the best one.Figures
or diagrams may accompany the problem.
HOW TO APPROACH THE }$ !PROBLEM SOLVING
Use this AID! SCREEN to remind yourself of the best
approach to this question type.For additional
information refer to the MATH RULES and SPECIAL
STRATEGIES:PROBLEM SOLVING portions of the
printed materials.
NU
3 =
(A)23.255
(B)21.995
(C)21.005
(D)20.315
(E)19.2652. (A) You should have ESTIMATED the answer to this
problem; begin by adding only the whole numbers:
19 + 1 + 2 = 22. Since there are also values after the
decimals, your answer muing fractions, find the least common
denominator. Here it is 30 (since all the denominators
divide into 30 evenly). Convert all numbers using the
least denominator: 10/30 + 6/30 + 21/30 = 37/30, which
is the same as 1 7/30.2.1.35 + 2.005 + .60 + 19.he symbol indicates whether the number
following it is positive or negative.Press^Wto
enter theSCRATCHPADand^StoSEE QUESTION
again.**1.1/3 + 1/5 + 7/10 =
(A)29/30
(B)1
(C)1 1/30
(D)1 7/30
(E)1 3/41. (D) When addates or indicates otherwise.All numbers used in
this section are real numbers.
** A space appears on both sides of a + or  sign
when the symbol indicates addition or subtraction
(4 + 3 = 7).If the sign appears without a space
(e.g. 4), tlel to
>is greater than<is less than
NOTE:Although the figures that accompany the
problems in this section are drawn as accurately
as possible, they are NOT drawn exactly to scale.
All figures lie in a plane unless the problem
stchoice.** Press the RIGHT
arrow key>to begin and to continue to the next
question.Press the LEFT arrow key<if you wish
to review previous questions. **
The following information may be useful:
SYMBOLS
*multiplied by
is paral+7bW@W+7X3Y+7
TOPIC 3:FRACTIONS AND DECIMALS
Problem Solving and Data Sufficiency
DIRECTIONS:Read and solve each problem carefully.
Choose the best answer and type the letter
corresponding to your 06@F0u
0x
0
S =U0*6 @+7;R+7S@S+7ZUPV
(E)Cannot be determined5. (E) To find volume, you need to multiply the height
by the area of the base. In a cylinder, the base is a
circle. Since you are not given the radius or diameter
or any other information about the circle, you cannot find
re is always in CUBIC
units. Since a cube has all equal sides (called edges),
multiply 6*6*6 to get 216 CUBIC units.5.What is the volume of a cylinder of height 9?
(A)9 cubic units
(B)18 cubic units
(C)81 cubic units
(D)729 cubic units arc is 1/6 (60/360) of the circumference.
(1/6)*6 PI = PI.4.What is the volume of a cube of edge 6?
(A)6 cubic units
(B)36 square units
(C)216 square units
(D)216 cubic units
(E)Cannot be determined4. (D) The volume of any figuI r, or 6 PI in this problem. Next, find
the degree measure of the arc. It has the same degree
measure as the central angle that intercepts it: 60
degrees. Since the total degree measure of the
circumference of the circle is 360 degrees, the 60
degree the radius of circle O is 3, what is the
length of intercepted arc AB?
(A)60 PI
(B)30 PI
(C)10 PI
(D)6 PI
(E)PI 3. (E) To find the measure of an arc, first find the
circumference of the circle. The circumference of any
circle is 2 Pnscribed in a square has
a diameter that is the same length as a side of the
square, 14. The radius is 1/2 the diameter, 7. The area
of the circle is PI radius squared, or 7*7*PI, 49 PI.3.Given circle O with central Angle AOB measuring 60
degrees, ifelf.2.A circle is inscribed inside a square of area 196.
What is the area of the circle?
(A)196 PI
(B)49 PI
(C)13 PI
(D)7 PI
(E)Cannot be determined2. (B) A square of area 196 must have sides of 14,
since 14*14 = 196. A circle ixy%
(E)Cannot be determined1. (B) The area of a parallelogram is base times height,
or y*x. The area of a rectangle is length times width or
y*x. Regardless of the values of y and x, the areas of
the figures are the same. A number is 100% of its or negative.Press^Wto
enter theSCRATCHPADand^StoSEE QUESTION
again.**1.The area of a parallelogram with height x and base y
is what percent of the area of a rectangle with length y
and width x?
(A)200%
(B)100%
(C)50%
(D)ber
that follows.
** A space appears on both sides of a + or  sign
when the symbol indicates addition or subtraction
(4 + 3 = 7).If the sign appears without a space
(e.g. 4), the symbol indicates whether the number
following it is positivedrawn exactly to scale.
All figures lie in a plane unless the problem
states or indicates otherwise.All numbers used in
this section are real numbers.
NOTE:The word "radical" in these questions
indicates the positive square root of the numfollowing information may be useful:
SYMBOLS
*multiplied by
is parallel to
>is greater than<is less than
NOTE:Although the figures that accompany the
problems in this section are drawn as accurately
as possible, they are NOT carefully.
Choose the best answer and type the letter
corresponding to your choice.** Press the RIGHT
arrow key>to begin and to continue to the next
question.Press the LEFT arrow key<if you wish
to review previous questions. **
The Y{\0]+7Y^ 0_+7Yc`@a+7YUb0b+7Yc0Ld+7YePKf+7Yg h+7 TOPIC 1:GEOMETRY
Problem Solving and Data Sufficiency
DIRECTIONS:Read and solve each problem 6S P
9@r
PF
)
IP a;R0)SAU VV W
plementary when they sum up to 180 degrees.16.What is the area of the athletic field?
(1)The field is circular and has a circumference
of 18 PI
(2)The diameter of the circular field is 18
(A) (B) (C) (D) (E)16. (D) If the field is ci) Statement one only gives you the ratio of x
and y; x is twice as large as y. They could be 120
degrees and 60 degrees, and be supplementary, but you
don't know for certain. Statement two is sufficient
since x plus y would equal 180, and angles are
supthe area without knowing what type of figure
it is. Combining the statements lets you know that the
figure is a square of side 12; the area is 144.15.Are Angles x and y supplementary?
(1)x = 2y
(2)180  y = x
(A) (B) (C) (D) (E)15. (B not sufficient; it tells
you that all sides are equal, but does not give you
information on the measure of any one side. Statement
two alone is not sufficient; you have the total sum of
the lengths of the outside lines of the figure, but you
cannot find r both statements
together are sufficient to answer the question.14.What is the area of the quadrilateral?
(1)The quadrilateral is a square
(2)The perimeter of the quadrilateral is 48
(A) (B) (C) (D) (E)14. (C) Statement one alone ist sufficient;
(C) if both statements (1) and (2) together are
sufficient to answer the question, but neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
answer the question;
(E) if neither statement alone noknowledge, determine whether the data are
SUFFICIENT for answering the question.Choose:
(A) if statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statement (2) alone is sufficient but
statement (1) alone is nouseful:
SYMBOLS
*multiplied by
is parallel to
>is greater than<is less than
DIRECTIONS:Each problem consists of a question
and two statements.Using the data given in the
problem plus your own mathematical and background
d that the sides are in a 6:6 radical 3:12
ratio, which may be reduced to 1:1 radical 3:2. Any
triangle with sides in this ratio is a 30:60:90 triangle.
The angle opposite the (side*radical 3) side measures
60 degrees.
The following information may be degx\
+
6
13.In the figure above, what is the measure of Angle x?
(A)90 degrees
(B)60 degrees
(C)30 degrees
(D)15 degrees
(E)Cannot be determined13. (B) You are told that this is a right triangle; you
are also toly polygon sum up to
360 degrees. Since a hexagon has 6 sides, its interior
angles are (6  2)*180, or 4*180 = 720 degrees. 720
degrees is twice (or 200% of) 360 degrees.\
 \
\
6(radical 3)\12
\
\
\
\
\
90 3 ratio).12.The measure of the sum of the interior angles in a
hexagon is what percent of the sum of the exterior angles
of that same figure?
(A)400%
(B)200%
(C)100%
(D)50%
(E)Cannot be determined12. (B) The EXTERIOR angles of ana line from point
A to the bottom of the line you just dropped from point B
and see that it is 5 units long. You now have two sides
of a triangle, 5 and 12. The line connecting A and B
must be the hypotenuse of the triangle and be 13
(remember the 5:12:1 the graph for yourself, noting that point A
is to the left of the point of origin, and on the Xaxis.
Point B is above and to the right of the point of origin.
Drop a line from point B to the horizontal axis, and see
that the line is 12 units long. Draw ;.11.On a coordinate graph, point A is located at (3,0)
and point B is located at (2,12).What is the measure
of a straight line connecting the two points?
(A)30 units
(B)17 units
(C)13 units
(D)12 units
(E)8 units
11. (C) Draw
(C)12(s squared)  18s + 12
(D)7s + 4
(E)Cannot be determined5. (A) Take one term at a time and multiply it through.
3s*4s = 12(s squared); 3s*2 = 6s; 6*4s = 24s;
6*2 = 12. Combine like terms: 6s + 24s = 18s.6.(a + b) squared =
xy
(E)3(x squared) + 3(y squared)4. (D) Multiply x by each term in the binomial
separately: x*3x = 3(x squared). x*y = xy. Add
the terms: 3(x squared) + xy.5.(3s + 6)*(4s  2) =
(A)12(s squared) + 18s  12
(B)12(s squared)  18s  12is 4*25 = 100. You
could use paper and divide both sides by 3/4 which is the
same as multiplying 75*(4/3). Cancel the 3 into the 75
to get 25*4 = 100.4.x*(3x + y) =
(A)4x + y
(B)4(x squared) + xy
(C)3(x squared) + y
(D)3(x squared) + First add 11 to both sides; then you have (1/4)x = 75 
(1/2)x. Now add (1/2)x to both sides; then you have
(3/4)x = 75, since (1/4)x + (1/2)x = (1/4)x + (2/4)x
= (3/4)x. Now, if THREEfourths is 75, ONEfourth is a
third of that, or 25, and the total d 100, to get 25*5 = 125.3.(1/4)x  11 = 64  (1/2)x.Solve for x.
(A)500
(B)300
(C)100
(D)75
(E)Cannot be determined
3. (C) Rearrange so that multiples of x are on one side
of the equals sign, and numerical values on the other.
mind, think that since
FOURfifths equals 100, ONEfifth must be onefourth of
that, or 25. If ONEfifth is 25, the total is five times
that, or 125. If you do this on paper, divide both sides
by 4/5, which means to multiply 100 by 5/4. Cancel the 4
an MORE than 140, you could have eliminated
answers (D) and (E) immediately.2.(4/5)x  16 = 84.Solve for x.
(A)250
(B)225
(C)200
(D)150
(E)1252. (E) Add 16 to both sides; then you have (4/5)x = 100.
If you do this problem in yourn ONEthird is half that,
or 70. Since 70 is ONETHIRD, multiply it by 3 to get a
total of 210. If you do it on paper, divide both sides
by 2/3, which is the same as multiplying 140*(3/2).
Cancel the 2 and the 140 to get 70*3 = 210. Note that
since x isve or negative.Press^Wto
enter theSCRATCHPADand^StoSEE QUESTION
again.**1.(2/3)x = 140.Solve for x.
(A)210
(B)200
(C)180
(D)120
(E)701. (A) You could do this in your mind quickly: if
TWOthirds of x is 140, theber
that follows.
** A space appears on both sides of a + or  sign
when the symbol indicates addition or subtraction
(4 + 3 = 7).If the sign appears without a space
(e.g. 4), the symbol indicates whether the number
following it is positidrawn exactly to scale.
All figures lie in a plane unless the problem
states or indicates otherwise.All numbers used in
this section are real numbers.
NOTE:The word "radical" in these questions
indicates the positive square root of the numfollowing information may be useful:
SYMBOLS
*multiplied by
is parallel to
>is greater than<is less than
NOTE:Although the figures that accompany the
problems in this section are drawn as accurately
as possible, they are NOT carefully.
Choose the best answer and type the letter
corresponding to your choice.** Press the RIGHT
arrow key>to begin and to continue to the next
question.Press the LEFT arrow key<if you wish
to review previous questions. **
The SW0zW+7StX0X+7S+[0[+7S.] ]+7S@^P^+7S_@#`+7Sa0a+7
TOPIC 2:ALGEBRA
Problem Solving and Data Sufficiency
DIRECTIONS:Read and solve each problem )6UGP
0
@"
G0 QP@@>
@{@+@;R
d 3x to both sides in statement one;
you have 16 = 7 + 3x. Now you can subtract 7 from both
sides; you have 9 = 3x, and so x = 3. Statement two tells
you that x must be the cube root of 27; since 3*3*3 = 27,
x = 3. (Once you see that you CAN solve eitherms COULD both
be 0, or x could equal 30 while y equals 27. Since there
is more than one possible solution, you cannot answer
the question.19.What is the value of x?
(1)16  3x = 7
(2)x cubed = 27
(A) (B) (C) (D) (E)19. (D) You can ad 1 = 0.18.What is the value of x?
(1)(3/5)x = (2/3)y
(2)y = (9/10)x
(A) (B) (C) (D) (E)18. (E) Either statement alone or both together only
allow you to find the ratio of the numbers but not the
numerical value of either term. The tet is the value of y?
(1)y squared = y
(2)y  1 = 0
(A) (B) (C) (D) (E)17. (B) Statement one alone is not sufficient; y could
be EITHER 0 or 1, since the square of either is itself.
Statement two alone tells you that y must be +1, since
+1 of the numbers is; you cannot determine their
difference. Statement two gives you only one number; you
cannot find the difference. Combining the two statements,
you find that if x + y = 12 and y = 8, x must equal 4.
4  8 = 4; 4 squared = 16.17.Whas the value of (x  y)squared?(x and y are
positive numbers.)
(1)(x + y)squared = 144
(2)y = 8
(A) (B) (C) (D) (E)16. (C) If (x + y)squared = 144, (x + y) must be the
square root of 144, or 12. Statement one tells you only
what the sum of x. Once you see that you can do
this, you can stop  you know that both statements
together are sufficient to answer the question. If you
took the time to solve the problem, you would have
18  3y + 9y = 42; 18 + 6y = 42; 6y = 24; x = 2.16.What iplace x with
6  y in statement two to get 3*(6  y) + 9y = 42. Since
you could then combine like terms in statement two, you
see that you could find the value of y; then you could
insert this value in place of y in statement one, and
thus find the value ; therefore, 10 must be 2y,
and y must be 5. x = 3y, or 15.15.What is the value of x?
(1)x + y = 6
(2)3x + 9y = 42
(A) (B) (C) (D) (E)15. (C) The first statement allows you to determine only
that x = 6  y. Knowing this, you could ref x?
(1)x = 300% y
(2)x  y = 10
(A) (B) (C) (D) (E)14. (C) Statement one tells you only that x is 3
times y. Statement two tells you only that x and y are
10 units apart. Combining the statements allows you to
determine that 3y  y = 10uestion, but neither
statement alone is sufficient;
(D) if either statement alone is sufficient to
answer the question;
(E) if neither statement alone nor both statements
together are sufficient to answer the question.14.What is the value o(A) if statement (1) alone is sufficient but
statement (2) alone is not sufficient;
(B) if statement (2) alone is sufficient but
statement (1) alone is not sufficient;
(C) if both statements (1) and (2) together are
sufficient to answer the qthan
DIRECTIONS:Each problem consists of a question
and two statements.Using the data given in the
problem plus your own mathematical and background
knowledge, determine whether the data are
SUFFICIENT for answering the question.Choose:
Subtract 12 from both sides.
You now have 9y = 27; y = 3. BUT the question wants to
know the value of x; since x  y = 3, x = 6.The following information may be useful:
SYMBOLS
*multiplied by parallel to
>is greater than<is less );13. (D) Use the second equation to find that x = 3 + y
(find this by adding y to both sides). Substitute 3 + y
for x in the first equation, giving you 4*(3 + y) +
5y = 39. Multiply through to get 12 + 4y + 5y = 39. Add
like terms: 12 + 9y = 39.
t be greater
than 21. Since 5*5 = 25, your answer must end in a 5.
Only the correct answer has both of these qualifications.
Look BEFORE you begin doing the problem to see whether you
can estimate rather than doing the actual mathematics.7.6/11 dividelying the
denominators (3*4*8 = 96) and automatically looking for
an answer with that result. (A) is the trap answer here.6.7.65*3.25 =
(A)18.6565
(B)21.8950
(C)24.8625
(D)28.3644
(E)28.39006. (C) Since 7*3 = 21, your answer mus (C) When multiplying fractions, simply multiply the
numerators, then the denominators. You can make it even
easier by canceling; here cancel the 3's so that your
numerators are 1*1*7 = 7. Your denominators are
1*4*8 = 32. Do NOT be tricked into multip.115 = .945. Note that your answer must be LESS THAN 1
because of the decimals, and must end in a 5 since you are
subtracting 5 from "0" which you convert to a "10."5.(1/3)*(3/4)*(7/8) =
(A)37/96
(B)11/20
(C)7/32
(D)3/8
(E)1/35./3
will be positive, so that the answer will be 4 plus some
additional fraction. Only one answer fits this
description.4.4.06  3.115 =
(A)1.045
(B).945
(C).650
(D).550
(E).5064. (B) 4.06 is the same as 4.060; subtract 4.060
 3nominator is 15. The problem
can be restated as 6 12/15  2 5/15 = 4 7/15. It may also
be possible to estimate and therefore bypass the actual
calculation. You know that 6  2 = 4. If you realize
that 4/5 is greater than 1/3, then you know that 4/5  1st be GREATER than 22; only the
correct choice is so. Do NOT take the time to do all the
actual addition; look at the answer key first.3.6 4/5  2 1/3 =
(A)6 7/15
(B)5 3/5
(C)4 7/15
(D)3 6/15
(E)2 11/153. (C) The least common de (A) If 3y = x, then y is 1/3 of x. The fact that
x = 15 is irrelevant and alone is not enough to find the
fractional relationship between x and y. inator. (3/5)*(1/3) = 3/15 or 1/5. 1 divided
by 5 = .2. A percentage may be converted to a decimal
by placing the decimal point two places to the
left: 20% = .20.15.What fraction of x is y?
(1)3y = x
(2)x = 15
(A) (B) (C) (D) (E)15.got right and cannot answer this question.14.What is the decimal representation of x?
(1)x = (3/5)*(1/3)
(2)x = 20%
(A) (B) (C) (D) (E)14. (D) A fraction may be converted to decimal
representation by dividing the numerator by the
denomw many questions Ginger DID
NOT ANSWER AT ALL. You know the total and the number she
answered incorrectly; however, you cannot simply subtract
the two numbers. Unless you are told that she answered
all of the questions, you cannot determine how many she
1. The reciprocal of 4/3 is 3/4.13.What fraction of questions did Ginger answer
correctly on the exam?
(1)There were 200 questions on the exam
(2)Ginger answered 100 questions incorrectly
(A) (B) (C) (D) (E)13. (E) You do not know hoction requires inverting and multiplying, statement one
means that x*(5/3) = 5/4. You should not take the time to
do so, but you could solve for x (x = 3/4). Statement two
alone is also sufficient; a fraction must be multiplied by
its reciprocal to equal many were sold in order to put
the part (the number sold) over the whole (the number
available).12.What is the value of x?
(1)x/(3/5) = 1 1/4
(2)x*(4/3) = 1
(A) (B) (C) (D) (E)12. (D) 1 1/4 is the same as 5/4. Since division by a
fra:11. (A) Since .75% is the same as .0075, which is the
same as 75/10,000, you can determine the fraction of pies
sold. Do NOT take the time to reduce this fraction.
Statement two alone does not give enough information; you
would also have to know how
square root: since radical 25 = 5,
radical 24 must be just slightly less than 5. Since
radical 36 = 6, radical 37 must be slightly more than 6.
Since radical 144 = 12, radical 145 must be slightly more
than 12. The sum is approximately 5 + 6 + 12 = 23.adical 37 + radical 145 is closest in
value to which of the following?
(A)23
(B)30
(C)41
(D)206
(E)radical 2065. (A) To add radicals, you CANNOT simply add the
numbers in the radical signs. You must approximate the
answers to each lf three times, will produce the
original number. Since 5*5*5 = 125, the cube root of 125
is 5. You do NOT have to know how to find a cube root;
you could have answered this problem by the process of
elimination, using the answer key.5.Radical 24 + ry radical 8 must equal
radical 8 (the same as "since 2*3 = 6, 6/3 = 2").4.What is the cube root of 125?
(A)25
(B)125/3
(C)3
(D)5
(E)Cannot be determined4. (D) The cube root of a number is the number which,
when multiplied by itseives you
a numerator of 8 (radical 8) and a denominator of 8
(since radical 8*radical 8 = radical 64, or 8). Cancel
the 8's to get 1 (radical 8), or just plain radical 8.
You could have reasoned this out as well; since radical
8*radical 8 = 8, 8 divided bcal 6 = radical 36 or 6). Cancel the 6
into the 18 to get 3*(radical 6).3.8/radical 8 =
(A)8
(B)radical 8
(C)1/8
(D).8
(E)8.83. (B) Multiply both numerator and denominator through
by 1 in the form of radical 8/radical 8. This gical 3)
(D)9 (radical 2)
(E)2 (radical 9)2. (B) Get rid of the radical in the denominator by
multiplying through by 1 in the form of radical
6/radical 6. That gives you a numerator of 18
(radical 6), and a denominator of 6 (since
radical 6*radio get a numerator of radical 264. To divide
radicals, divide the numbers inside the radicals:
264/88 = 3. Replace the radical sign after the division,
to have an answer of radical 3.2.18/radical 6 =
(A)radical 6
(B)3 (radical 6)
(C)6 (radN
again. **1.(radical 11 * radical 24)/radical 88 =
(A)radical 5
(B)radical 3
(C)radical 2
(D)3
(E)21. (B) To multiply radicals, first ignore the radical
sign and multiply the numbers: 11*24 = 264, then reinsert
the radical tn
when the symbol indicates addition or subtraction
(4 + 3 = 7).If the sign appears without a space
(e.g. 4), the symbol indicates whether the number
following it is positive or negative.Press^Wto
enter theSCRATCHPADand^StoSEE QUESTIO
states or indicates otherwise.All numbers used in
this section are real numbers.
NOTE:The word "radical" in these questions
indicates the positive square root of the number
that follows.
** A space appears on both sides of a + or  sig parallel to
>is greater than<is less than
NOTE:Although the figures that accompany the
problems in this section are drawn as accurately
as possible, they are NOT drawn exactly to scale.
All figures lie in a plane unless the problem your choice.** Press the RIGHT
arrow key>to begin and to continue to the next
question.Press the LEFT arrow key<if you wish
to review previous questions. **
The following information may be useful:
SYMBOLS
*multiplied by
is+7%W W+7X0bY+7
TOPIC 4:PERFECT SQUARES AND SQUARE ROOTS
Problem Solving and Data Sufficiency
DIRECTIONS:Read and solve each problem carefully.
Choose the best answer and type the letter
corresponding to 06  .F E@
0 :@e*6@+7;R+7S@ 1. cube root of 125
(A) (B) (C) (D) (E)15. (C) You need to know the value of x AND the value of
y to determine whether x/y > 1. Therefore, both
statements are necessary to answer the question. Note
that it is not necessary to solve for x and y. As is not sufficient.
Statement two alone IS sufficient, since if the square
root of a number is an even whole number (such as 4 or 6)
the square of that number is a perfect square (such as 16
or 36).15.Is x/y > 1?
(1)radical x = 13
(2)y is the ct square?
(1)x is an even whole number
(2)Radical x is an even whole number
(A) (B) (C) (D) (E)14. (B) There are even whole numbers that are perfect
squares (e.g., 144) and that are not perfect squares
(e.g., 146), so statement one alone 5 = radical 5; 1 divided by radical 1 =
radical 1). Statement one alone tells you that x
could be, but doesn't have to be, 1. Statement two
gives no information; ANY number divided by itself
equals 1. Neither statement is sufficient.14.Is x a perfes therefore greater
than 7.13.Is x = 1?
(1)x/radical x = radical x
(2)radical x/radical x = 1
(A) (B) (C) (D) (E)13. (E) ANY number when divided by the square root of
itself has a quotient of the square root (e.g., 5 divided
by radical (E)12. (D) The only whole number between 2 and 4 is 3;
its perfect square is 9. If x = 9, it is greater than 7.
For statement two, 7 squared = 49. The square root of
any number greater than 49 must be greater than 7.
x > radical 55 > radical 49 and i statements just because
statement two is true; if one alone is sufficient,
choose (A).12.Is x less than 7?
(1)x is the perfect square of a whole number
between 2 and 4
(2)x is the square root of a number greater than 55
(A) (B) (C) (D);^11. (A) Radical 9 = 3; radical 144 = 12; 9 + 12 = 21.
Alone, statement one is sufficient; statement two merely
tells you that x must be greater than 20. It does not
tell you precisely what the value of x is. Do not choose
(C) thinking you need both
u
cannot determine the central term (or any term other than
the first) and cannot solve the problem.6.What is the average of a series of 7 evenly spaced
integers whose first two terms are 3 and 6?
(A)12
(B)13
(C)14
(D)15
(E)Cannot benly spaced numbers, the number
in the middle is the average. However, since you do not
know what the spacing is between the numbers (they could
be 2 units apart and be 1012141618, or 3 units apart
and be 1013161921, or have any other spacing), yoset of 5 evenly spaced
numbers whose first term is 10?
(A)5
(B)10
(C)15
(D)25
(E)Cannot be determined 5. (E) Evenly spaced numbers are an equal number of
units apart, such as 5101520 or 36912 or 2468.
With an odd number of ev11. With an EVEN
number of consecutive integers, add the first and last
terms and divide by 2 to find the average:
6 + 11 = 17; 17/2 = 8.5. You could also add the two
central terms and divide by 2: 8 + 9 = 17; 17/2 = 8.5.5.What is the average of a o add the numbers and divide by 7.4.What is the average of 6 consecutive integers of which
the smallest is 6?
(A)36
(B)12
(C)8.5
(D)6.5
(E)5.54. (C) Since the integers are consecutive, they
are right after one another, 678910ers are integers that follow one
another directly, like 234 or 111213. With an odd
number of such integers, the one in the middle is the
average. The series here is 11121314151617; 14 is
in the middle and is the average. Do NOT take the time
t it
happens, x = 169 (since 13*13 = 169), y = 5 (since
5*5*5 = 125). Therefore, x/y = 169/5 > 1.1cube root of 125
(A) (B) (C) (D) (E)15. (C) You need to know the value of x AND the value of
y to determine whether x/y > 1. Therefore, both
statements are necessary to answer the question. Note
that it is not necessary to solve for x and y. As is not sufficient.
Statement two alone IS sufficient, since if the square
root of a number is an even whole number (such as 4 or 6)
the square of that number is a perfect square (such as 16
or 36).15.Is x/y > 1?
(1)radical x = 13
(2)y is the ct square?
(1)x is an even whole number
(2)Radical x is an even whole number
(A) (B) (C) (D) (E)14. (B) There are even whole numbers that are perfect
squares (e.g., 144) and that are not perfect squares
(e.g., 146), so statement one alone 5 = radical 5; 1 divided by radical 1 =
radical 1). Statement one alone tells you that x
could be, but doesn't have to be, 1. Statement two
gives no information; ANY number divided by itself
equals 1. Neither statement is sufficient.14.Is x a perfes therefore greater
than 7.13.Is x = 1?
(1)x/radical x = radical x
(2)radical x/radical x = 1
(A) (B) (C) (D) (E)13. (E) ANY number when divided by the square root of
itself has a quotient of the square root (e.g., 5 divided
by radical (E)12. (D) The only whole number between 2 and 4 is 3;
its perfect square is 9. If x = 9, it is greater than 7.
For statement two, 7 squared = 49. The square root of
any number greater than 49 must be greater than 7.
x > radical 55 > radical 49 and i statements just because
statement two is true; if one alone is sufficient,
choose (A).12.Is x less than 7?
(1)x is the perfect square of a whole number
between 2 and 4
(2)x is the square root of a number greater than 55
(A) (B) (C) (D);^11. (A) Radical 9 = 3; radical 144 = 12; 9 + 12 = 21.
Alone, statement one is sufficient; statement two merely
tells you that x must be greater than 20. It does not
tell you precisely what the value of x is. Do not choose
(C) thinking you need both!!
!!!
! !!!he " + 6," the ratio
could be changed. There could be a ratio of 5:1 (with 15
men and 3 women) or 9:2 (18 men and 4 women) or any other
ratio where the number of men is six more than three times
the number of women.6.The ratio of kittens to puppies is the ratio of men to women?There are 6 more
than 3 times as many men as there are women.
(A)2:3
(B)3:5
(C)1:6
(D)6:1
(E)Cannot be determined5. (E) If there were 3 times as many men as women, the
ratio would be 3:1. However, with ts in the ratio. 30 is a multiple of 1 (1*30),
2 (2*15), 3 (3*10), 5 (5*6), 6 (6*5), 10 (10*3),
15 (15*2), and 30 (30*1). The sum of the numbers in the
ratio must therefore equal 1,2,3,5,6,10,15 or 30. Only
choice (A) has numbers that do so.5.What isf there are 30 people, each of whom is either a
rancher or a farmer, which of the following could be the
ratio of ranchers to farmers?
(A)1:4
(B)1:7
(C)2:7
(D)3:5
(E)4:54. (A) The total must be a multiple of the sum of the
two number(E)Cannot be determined3. (D) For every SET of 4 boys, there is a SET of 5
girls. Since there are 30 girls, there are 6 SETS of
girls (because 30/5 = 6) and there must be 6 SETS of boys.
With 4 boys to a set, there are 24 boys. 24 + 30 = 54.4.Il must be a multiple of 7,
such as 7,14,21,35, etc. All the answers except (B), 31,
are multiples of 7.3.The ratio of boys to girls is 4:5.If there are 30
girls, how many boys and girls are there altogether?
(A)24
(B)30
(C)45
(D)54
of the
following could not be the total number of dogs and cats?
(A)21
(B)31
(C)56
(D)70
(E)842. (B) The total must be a multiple of the sum of the
numbers in the ratio. Since the ratio is 3:4, add 3 + 4
to get a sum of 7. The tota is 7:9, there is a SET of 7 men for every SET of 9
women. If there are 35 men, there are 5 SETS of men
(because 35/7 = 5) so there must be 5 SETS of women. With
9 women to a set, there are 5*9 = 45 women.2.If the ratio of dogs to cats is 3:4, which of men to women in the class is 7:9.If
there are 35 men, how many women are there?
(A)35
(B)40
(C)45
(D)63
(E)791. (C) To find the numbers using a ratio, think of the
ratio as defining the number of items in one SET. If the
ratio addition or subtraction
(4 + 3 = 7).If the sign appears without a space
(e.g. 4), the symbol indicates whether the number
following it is positive or negative.Press^Wto
enter theSCRATCHPADand^StoSEE QUESTION
again. **1.The ratiorwise.All numbers used in
this section are real numbers.
NOTE:The word "radical" in these questions
indicates the positive square root of the number
that follows.
** A space appears on both sides of a + or  sign
when the symbol indicatester than<is less than
NOTE:Although the figures that accompany the
problems in this section are drawn as accurately
as possible, they are NOT drawn exactly to scale.
All figures lie in a plane unless the problem
states or indicates othehe RIGHT
arrow key>to begin and to continue to the next
question.Press the LEFT arrow key<if you wish
to review previous questions. **
The following information may be useful:
SYMBOLS
*multiplied by
is parallel to
>is grea+7uHY "Z+7u[P\+7
TOPIC 5:RATIOS
Problem Solving and Data Sufficiency
DIRECTIONS:Read and solve each problem carefully.
Choose the best answer and type the letter
corresponding to your choice.** Press t }060 oe @
P
Ps P Ku;R+7uS0{T+7u&V0W+7uX@X""
"""
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