7. The set S of numbers has the following properties:
I) If x is in S, then 1/x is in S.
II) If both x and y are in S, then so is x + y.
Is 3 in S?
(1) 1/3 is in S.
(2) 1 is in S.
Consider (1) alone. Since 1/3 is in S, we know from Property I that 1/(1/3)
= 3 is in S. Hence, (1) is sufficient.
Consider (2) alone. Since 1 is in S, we know from Property II that 1 + 1 = 2
(Note, nothing in Property II prevents x and y from standing for the same n
umber. In this case both stand for 1.) is in S. Applying Property II again s
hows that 1 + 2 = 3 is in S. Hence, (2) is also sufficient. The answer is D.
8. What is the area of the triangle above?
(1) a = x, b = 2x, and c = 3x.
(2) The side opposite a is 4 and the side opposite b is 3.
From (1) we can determine the measures of the angles: a + b + c = x + 2x + 3
x = 6x = 180
Dividing the last equation by 6 gives: x = 30
Hence, a = 30, b = 60, and c = 90. However, different size triangles can hav
e these angle measures, as the diagram below illus