GMAT® Official Guide 2019 Quantitative Review
Adding equations (i) and (ii) gives
(17 + k)+(17 + k)+ (17 + k)+ ... +(17 + k) = 20n
n(17 + k) = 20n
17 +k= 20
k=3
Alternatively, because the numbers are
consecutive odd integers, they form a data
set that is symmetric about its average, and
so the average of the numbers is the average
of the least and greatest numbers. Therefore,
10 = k +^17 , from which a unique value
2
fork can be determined; SUFFICIENT.
The correct answer is D;
each statement alone is sufficient.
DS16044- If x, y, and z are positive numbers, is x > y > z?
(1) xz > yz
(2) yx > yzAlgebra Inequalities
( 1) Dividing both sides of the inequality by z
yields x > y. However, there is no information
relating z to either x or y; NOT sufficient.
(2) Dividing both sides of the inequality by
y yields only that x > z, with no further
information relatingy to either x or z; NOT
sufficient.
From (1) and (2) it can be determined that xis
greater than bothy and z. Since it still cannot be
determined which of y or z is the least, the correct
ordering of the three numbers also cannot be
determined.The correct answer is E;
both statements together are still not sufficient.
DS06644- K is a set of numbers such that
(i) if x is in K, then -x is in K, and
(ii) if each of x and y is in K, then xy is in K.Is 12 in K?(1) 2 is in K.
(2) 3 is in K.Arithmetic Properties of numbers
(1) Given that 2 is in K, it follows that K
could be the set of all real numbers,
which contains 12. However, if K is the set
[ ... , - 16, -8, -4, -2, 2, 4, 8, 16, ... }, then K
contains 2 and K satisfies both (i) and (ii), but
K does not contain 12. To see that K satisfies
(ii), note that K can be written as[ ... , -24,
-2^3 , -2^2 , -21, 21, 22 , 23, 24 , •.. }, and thus a
verification of (ii) can reduce to verifying that
the sum of two positive integer exponents is a
positive integer exponent; NOT sufficient.
(2) Given that 3 is in K, it follows that K could
be the set of all real numbers, which contains- However, if Kis the set[ ... ,-81, -27, -9,
-3, 3, 9, 27, 81, ... }, then K contains 3 and
K satisfies both (i) and (ii), but K does not
contain 12. To see that K satisfies (ii), note
that K can be written as[ ... ,-34,-33,-3^2 ,
-31, 31, 32 , 33, 34,. .. }, and thus a verification
of (ii) can reduce to verifying that the sum of
two positive integer exponents is a positive
integer exponent; NOT sufficient.
Given (1) and (2), it follows that both 2 and 3 are
in K. Thus, by (ii), (2)(3) = 6 is in K. Therefore,
by (ii), (2)(6) = 12 is in K.
The correct answer is C;
both statc!ments together are sufficient.
DS05637- If x^2 + y^2 = 29, what is the value of (x - y)^2?
(1) XY= 10
(2) X= ~>Algebra Simplifying algebraic expressions
Since (x --y)^2 = (x2 + y2) -2xy and it is given that
x2 + y2 = 29, it follows that (x -y)^2 = 29 - 2xy.
Therefore, the value of (x -y )^2 can be determined
if and only if the value of xy can be determined.(1) Since the value of xy is given, the value of
(x -y)^2 can be determined; SUFFICIENT.
(2) Given only that x = 5, it is not possible to
determine the value of xy. Therefore, the
value of (x -y)^2 cannot be determined;
NOT sufficient.
The correct answer is A;
statement 1 alone is sufficient.