## 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.

The worked solutions to these exam-style questions are only available to those who have a Transum Subscription. Subscribers can drag down the panel to reveal the solution line by line. This is a very helpful strategy for the student who does not know how to do the question but given a clue, a peep at the beginnings of a method, they may be able to make progress themselves. This could be a great resource for a teacher using a projector or for a parent helping their child work through the solution to this question. The worked solutions also contain screen shots (where needed) of the step by step calculator procedures. A subscription also opens up the answers to all of the other online exercises, puzzles and lesson starters on Transum Mathematics and provides an ad-free browsing experience. |

Drag this panel down to reveal the solution

The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

The solutions to the questions on this website are only available to those who have a Transum Subscription.

Exam-Style Questions Main Page

To search the **entire Transum website** use the search box in the grey area below.

Do you have any comments about these exam-style questions? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.

©1997 - 2023 Transum Mathematics :: For more exam-style questions and worked solutions go to Transum.org/Maths/Exam/