School Trip Activity Participation Probability

Let's read the problem carefully and represent it using a Venn diagram. We have participants doing hiking, canoeing, and swimming.

We are given that no students did fewer than two activities. This means the regions representing only one activity, as well as the outside region, are zero. Also, ten students did all three.

Let's label the remaining intersection regions as x, y, and z. If N is the total number of participants, we can write it as the sum of all these groups.

We know fifty percent participated in at least hiking and canoeing. This encompasses region x, along with the center ten.

Similarly, sixty percent participated in at least hiking and swimming. These are regions y and ten.

Now, let's substitute our expressions for x and y back into our equation for the total number of participants.

Replacing x and y gives us zero point five N minus ten, plus zero point six N minus ten, plus z plus ten.

Combining the N terms and the constants, we get one point one N minus ten plus z.

Solving for z, we find an elegant relationship: z equals ten minus zero point one N. Let's highlight this.

Since these regions represent groups of students, they must contain a non-negative whole number of people. Let's restrict N using this logic.

The first inequality tells us that zero point one N is at most ten, so N must be less than or equal to one hundred.

The second inequality implies that zero point five N is at least ten, which means N must be at least twenty.

Furthermore, for z to be a whole number, zero point one N must be an integer, which forces N to be a multiple of ten.