GMAT® Official Guide 2019 Quantitative Review
DS09385- For all x, the expression x is defined to be ax+ a,
where a is a constant. What is the value of 2 ?
(1) 3*=2
(2) 5*=3Algebra
Determine the value of 2• = (a)(2) +a= 3a, or
equivalently, determine the value of a.(1) Given that 3* = 2, it follows that (a)(3) +a=
2, or 4a = 2, or a= l; SUFFICIENT.
2
(2) Given that 5* = 3, it follows that (a)(5) + a=
3, or 6a = 3, or a= l; SUFFICIENT.
2
The correct answer is D;
each statement alone is sufficient.
DS09260- Is k + m < 0?
(l) k < 0
(2) km> 0Arithmetic
(1) Given that k is negative, it is not possible
to determine whether k +mis negative. For
example, if k = -2 and m = 1, then k +mis
negative. However, if k = -2 and m = 3, then
k +mis not negative; NOT sufficient.
(2) Given that km is positive, it is not possible
to determine whether k +mis negative. For
example, if k = - 2 and m = -1, then km is
positive and k + m is negative. However,
if k = 2 and m = 1, then km is positive and
k +mis not negative; NOT sufficient.
Taking (1) and (2) together, k is negative and km is
positive, it follows that m is negative. Therefore, both
k and m are negative, and hence k + m is negative.The correct answer is C;
both statements together are sufficient.
DS08352- The symbol 6. represents which one of the following
operations: addition, subtraction, or multiplication?
(1) a 6. (b 6. c) * a 6. (c 6. b) for some numbers a, b,
and c.
(2) a 6. (b 6. c) * (a 6. b) 6. c for some numbers a, b,
and c.Arithmetic
Can we determine which of the operations-
addition, subtraction, or multiplication- is the
operation ~?(1) Given the condition that~ has the property
that, for some numbers a, b, and c, a~ (b ~ c) "#
a~ (c ~ b), we can infer that, for some
numbers b and c, b ~ c-:/:-c ~ b. Both addition
and multiplication have the commutative
property, whereby, for any numbers x and y,
x + y = y + x and x X y = y x x. For example,
7 + 2 = 9 = 2 + 7, and 7 x 2 = 14 = 2 x 7.
We thus see that, for all numbers x, y, and z,
both of the statements x + (y + z) = x + (z +
y) and x x (y x z) =xx (z x y) are true. The
operation ~ therefore cannot be addition or
multiplication.
Subtraction, on the other hand, lacks the
commutative property; for example, 7 - 2
= 5 and 2 - 7 = -5. The operation ~ could
therefore be subtraction. Subtr2 ction is
therefore the one operation among addition,
subtraction, and multiplication that satisfies
statement 1; SUFFICIENT.(2) The reasoning in this case is similar to the
reasoning for statement 1, but concerning
the associative property rather than the
commutative property. Both addition and
multiplication have this property. For any
numbers x, y, and z, the statements x + (y +
z) = (x + y) + z and x x (y x z) =(xx y) x z
are always true. For example, in the case of
multiplication, 2 x (3 x 5) = 2 x 15 = 30 =
6 x 5 = (2 x 3) x 5. However, in contrast to
addition and multiplication, the operation
of subtraction does not have the associative
property. For example, for the numbers 2,
3, and 5,2 -(3 -5) = 2 - (-2) = 4, whereas
(2 - 3) - 4 = -1 - 4 = -5. Subtraction is
therefore the one operation among addition,
subtraction, and multiplication that satisfies
statement 2; SUFFICIENT.
The correct answer is D;
each stat1ement alone is sufficient.