Tennis and Golf Players
Find a percentage of a given quantity
Use Venn diagrams to represent intersecting sets.
Devise and use problem solving strategies (draw a diagram)
This problem involves reasoning with percentages. Percentages are fractions that are used to express the whole divided into one hundred equal parts. One percent means one-hundredth part of a whole or unit, and the notation used is 1%. In the extension to this problem the students need to find the lowest common multiples of the numbers 20, 30 and 50.
Probably the easiest way to do this problem is to draw a diagram, representing the problem with intersecting sets. We used the Venn Diagram below in solving this problem. Venn diagrams are sometimes difficult for students to understand and are more likely to be understood if they are built up one step at a time. Several examples should be done at each step.
- Identifying a subset eg blue shapes from all coloured shapes
- 2 non-intersecting sets eg triangles and circles
- Intersecting sets eg triangles and shapes with a right angle
In a class in the school down the road, everyone plays tennis or golf or both. In fact 80% play tennis and 70% play golf. What percentage plays both games?
- Introduce the problem to the class. As puzzles work well to motivate the students you could present the problem as a scenario in which they have been given 3 clues.
- Brainstorm for ways to solve the problem. The easiest way to solve this problem is to draw a diagram. You may need to check that the students understand that the students can play both golf and tennis.
- Circulate as the students work taking the opportunity for the students to share their understanding of percentages and their use of diagrams.
What does 80% of the class mean?
How do you know what percentage play both tennis and golf?
How has your diagram helped you to solve the problem?
What information did you need to include in your diagram?
- Share solutions
Extension to the problem
What is the smallest number of students in that class?
Other contexts for the problem
Change the sports involved.
Now if 80% of the class plays tennis, 20% don’t play tennis. This 20% must be part of the golf players. So they play golf but not tennis. This means that the remaining 50% play both tennis and golf.
Solution to the Extension:
As far as the extension goes, we want to find the smallest number so that the various percentages of the Venn Diagram are whole numbers. We therefore need 20%, 30% and 50% of the class to be whole numbers. Certainly this will be the case if we had 20, 30 and 50 in each group. But it would also be the case if we had 10, 15 and 25. So how do we get the smallest class?
What is the biggest number that goes into 20, 30 and 50. Once we’ve found that, if we divide 20, 30 and 50 by that amount we will have the smallest numbers we want. This biggest divisor is 10. So the separate amounts are 2, 3 and 5. The class has only 10 students.