GMAT® Official Guide 2019 Quantitative ReviewD$04468- Is x > y?
(1) x +y>x-y
(2) 3x> 2yAlgebra
(1) Given that x + y > x -y, it follows that
y > -y, or 2y > 0, or y > 0. However, nothing
is known about the value of x. If x = 2 and
y = l, then x + y > x -y and the answer to
the question is yes. However, if x = l and
y = l, then x + y > x -y and the answer to
the question is no; NOT sufficient.
(2) Given that 3x > 2y, then x = 2 and y = l is
possible and the answer to the question is
yes. However, if 3x > 2y, then x = l and
y = l is also possible and the answer to
the question is no; NOT sufficient.
Taking (1) and (2) together is of no more help
than either (1) or (2) taken separately because
the same examples used to show that (1) is not
sufficient also show that (2) is not sufficient.The correct answer is E;
both statements together are still not sufficient.
ADDS 17588- In the figure above, AB, which has length z cm, is
tangent to the circle at point A, and BO, which has
lengthy cm, intersects the circle at point C. If
BC = x cm and z = Jxy, what is the value of x?
(1) CD=x cm
(2) Z= 5'12_Geometry
(1) Given that CD= x cm, it is not possible
to determine the value of x because all the
given information continues to hold whenall the parts of the figure increase in length
by any given nonzero factor; NOT sufficient.
(2) Given that z = 5 ✓2, the value of x will vary
when the radius of the circle varies and CD
is a diameter and thus passes through the
center of the circle. To see this, let r be the
radius, in centimeters, of the circle and let 0
be the center of the circle, as shown in the
figure below. Then, because CD is a
diameter, it follows that CD = 2r and y =
x + CD= x + 2r. Also, ~ OAB is a right triangle
and the Pythagorean theorem gives ( OA)^2 +
(AB)^2 = (OB)2, or-?+ (5✓2)2 = (x + r)2, or-?
+ 50 = x2 + 2xr + -?,or x(x + 2r) = 50, which
implies that xy = z2 and z = J;ry,
since y = x + 2r and z = 5 ✓2. Therefore, if
z = 5 ✓ 2 and CD is a diameter, then z = fry
holds, and the value of x can vary.---~A
This can be seen by considering the equation
x(x + 2r) = 50, or x = ___iQ__ If the value of r
x +2r
changes slightly to a new value R, then the
value of x must also change. Otherwise, there
would be two different numbers, namely ___iQ__
50 x +2r
and -----''----'--, equal to each other, which is a
x +2R
contradiction; NOT sufficient.Taking (l) and (2) together,y = x +CD= x + x =
2x and z = 5 ✓2, so z = fry becomes
5 ✓ 2 = ✓x(2x), or (5✓2)^2 = (✓x(2x))^2 , or
50 = x(2x), or x2 = 25, or x = 5.The correct answer is C;
both statements together are sufficient.