3. Out of 99 fifth-grade students who watched the film screening, 76 bought juice, and 59 bought popcorn. One-ninth of the total number of students did not buy either juice or popcorn. Calculate how many students bought:

a) only juice

b) only popcorn

c) both juice and popcorn.
3. Out of 99 fifth-grade students who watched a film screening, 76 bought juice, and 59 bought popcorn. One-ninth of the total number of students did not buy either juice or popcorn. Calculate how many students bought:

a) only juice

b) only popcorn

c) both juice and popcorn.

## Solution.

The problem can be solved using a Venn diagram.

Let U be the set of all students who watched the film, S be the set of all students who bought juice, and K be the set of all students who bought popcorn.

One-ninth of the total number of students is $99 : 9 = 11$ who did not buy either juice or popcorn.

99 students watched the film, and 11 did not buy either juice or popcorn, so we conclude that $99 - 11 = 88$ bought something (either only juice, only popcorn, or both juice and popcorn).

The total number of juices and popcorns bought is $76 + 59 = 135$.

$135 - 88 = 47$ students bought both juice and popcorn (the intersection of sets S and K is not an empty set).

76 students bought juice, and 47 bought both juice and popcorn, so we conclude that $76 - 47 = 29$ students bought only juice.

59 students bought popcorn, and 47 bought both juice and popcorn, so we conclude that $59 - 47 = 12$ students bought only popcorn.

![](https://cdn.mathpix.com/cropped/2024_05_30_1ae5e90785d026149547g-08.jpg?height=234&width=394&top_left_y=1345&top_left_x=411)

Note: If a student presents the solution using a Venn diagram (image) and clearly indicates which elements belong to each set, the student can achieve all 6 POINTS.

If the sets are not defined, the student should achieve 3 POINTS for the Venn diagram (image), and the remaining 3 POINTS should be distributed among the required answers.