At the start of the school year, students in Mr. Pi’s class were asked the
following question: “Do you love Math?” They were only allowed to answer “yes”
or “no”, and everyone had to answer. Of the 30 students in the class, 21 answered
“yes” and 9 answered “no”.
That day, with every student present, the probability of randomly selecting a
student who answered the question “yes” was 21
30 =
7
10 and the probability of
randomly selecting a student who answered the question “no” was 9
30 =
3
10.
However, on one particular morning later in the year, the following information
was known about the class:
• at least one of the students who had answered “yes” was absent and at least
one of the students who had answered “no” was absent;
• more than half of the class was present; and
• the probability of randomly selecting a student who had answered the
question “yes” was 3
4
.

Is there enough information to determine how many students were absent that
particular morning? If yes, how many students were absent? If no, explain why

The probability of randomly selecting a student who answered the question “yes” is the number of students that responded yes divided by the number of students present. Thus, it was:

21/30 = 7/10

The probability of randomly selecting a student who answered the question “no” is the number of students who answered "no" divided by the number of students present. Thus, it was:

9/30 = 3/10

On one particular morning later in the year, the probability of randomly selecting a student who had answered the question “yes” was 3/4

The equivalent ratios to 3/4 will tell you the possible combinations of students that answered yes and students present on that particular morning.

For instance: 3/4 = 6/8

6/8 would mean that there were 6 students that answered yes and 8 total students on that particular morning.

Since more than half of class was present, the total number of students was greater than 15. Then, you must find a ratio with a denominator greater than 15 (and less than 30).

The other equivalent ratios to 3/4 are: 9/12, 12/16, 15/20, 18/24, 21/28, 24/32, ...

That means that the only possibilities with more than 15 and less than 30 present students are 12/16, 15/20, 18/24, and 21/28.

The total number of absents might have been:

30 - 28 = 2
30 - 24 = 6
30 - 20 = 10
30 - 16 = 14

Since at least one of the students who had answered “yes” was absent and at least one of the students who had answered “no” was absent, all the above possibilities are valid: there is not enough information to determine how many students were absent that particular morning; they could be 2, 6, 10, or 14.