# Exam Style Question

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

### Sets

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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