## Exam-Style Question on Sets## A mathematics exam-style question with a worked solution that can be revealed gradually |

Question id: 24. This question is similar to one that appeared on an IB Studies paper in 2012. The use of a calculator is allowed.

900 professional footballers were surveyed with the following results

- 200 have a swimming pool
- 305 have a second home
- 120 have an boat
- 45 have a boat and a second home
- 30 have a swimming pool and a second home
- 32 have a boat and a swimming pool
- 16 have all three.

(a) Draw a Venn diagram to show this information. Use P to represent the set of footballers who have a swimming pool, H the set of footballers who have a second home and B the set of footballers who have a boat.

(b) Write down the number of footballers that have a swimming pool only;

(c) Write down the number of footballers that have a swimming pool and a boat but no second home.

(d) Write down \(n[B\cap (H\cup P)']\).

(e) Calculate the number of footballers who have none of the three.

Two footballers are chosen at random from the 900 footballers. Calculate the probability that:

(f) neither footballer has a swimming pool;

(g) only one of the footballers has a swimming pool.

The footballers are asked to collect money for charity. In the first month, the footballers collect \(x\) pounds and then they collect \(y\) pounds in each subsequent month.

In the first 6 months, they collect 15700 pounds. This can be represented by the equation \(x + 5y = 15700\).

In the first 10 months they collect 25700 pounds.

(h) Write down a second equation in \(x\) and \(y\) to represent this information.

(i) Write down the value of \(x\) and of \(y\).

(j) Calculate the number of months that it will take them to collect at least 50000 pounds.

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