Question

In College X the number of students enrolled in both a chemistry course and a biology course is how much less than the number of students enrolled in neither?
(1) In College X there are 60 students enrolled in a chemistry course.
(2) In College X there are 85 students enrolled in a biology course.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is E Statements (1) and (2) together are not sufficient.

Consider the Venn diagram above, in which x represents the number of students in chemistry only, y represents the number of students in both chemistry and biology, z represents the number of students in biology only, and w represents the number of students in neither chemistry nor biology. Find the value for $w-y$.

(1) Since there are 60 students enrolled in chemistry, $x+y=60$, but there is no way to determine the value of $y$. Also, no information is given for determining w. For example, if $x=y=30$ and $w=30$, then $w-y=0$. However, if $x=y=30$ and $w=40$, then $w-y=10$; NOT sufficient.

(2) Since there are 85 students enrolled in biology, $y+z=85$, but there is no way to determine the value of $y$. Also, no information is given for determining w. For example, if $x=y=30,z=55$, and $w=30$, then $w-y=0$. However, if $x=y=30,z=55$, and $w=40$, then $w-y=10$; NOT sufficient.

Taking (1) and (2) together and subtracting the equation in (1) from the equation in (2) gives $z-x=25$. Then, adding the equations gives $x+2y+z=145$, but neither gives information for finding the value of $w$. For example, if $x=Y=30,z=55$, and $w=30$, then $w-y=0$. However, if $x=Y=30,z=55$, and $w=40$, then $w-y=10$.

The correct answer is E;

both statements together are still not sufficient.