Question

A certain high school with a total enrollment of 900 students held a science fair for three days last week. How many of the students enrolled in the high school attended the science fair on all three days?
(1) Of the students enrolled in the school, 30 percent attended the science fair on two or more days.
(2) Of the students enrolled in the school, 10 percent of those that attended the science fair on at least one day attended on all three days.

• 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.

• Let $a,b$, and $c$ be the numbers of students who attended on exactly one day, exactly two days, and exactly three days, respectively, and let $n$ be the number of students who did not attend on any of the three days. Given that $a+b+c+n=900$, we have to determine the value of $c$.
• Considering statement 1, it is given that $\left(30\%\right)\left(900\right)=b+c$, or $b+c=270$, more than one positive integer value for $c$ is possible. Thus, statement 1 is insufficient.
• From statement 2, it is given that $\left(10\%\right)(a+b+c)=c$, or $a+b+c=10c$, or $a+b=9c$, more than one positive integer value for $c$ is possible. Again statement 2 is insufficient.
• Considering data from both the statements is still insufficient as more than one positive integer value for $c$ is possible.