It is known that in a sports club, there are 1000 registered members. 60% of members play Tennis, 50% of members play Cricket, 70% of members play Football, 20% of members play Tennis and Cricket, 40% of members play Cricket and Football and 30% of members play Football and Tennis. If someone claimed that 20% of members play all the three sports, what is your opinion and why? [Use inclusion and exclusion principle to provide your opinion]
Manipal university
well, if x% play all three, then
60+50+70 - (20+40+30) + x = 100
so x = 10%
To determine whether the claim that 20% of members play all three sports is accurate, we can use the principle of inclusion and exclusion.
According to the principle of inclusion and exclusion, the total number of members who play at least one sport can be calculated by summing the number of members who play each sport separately and subtracting the intersections between the sports.
Let's break down the information given:
- Total registered members: 1000
- Members who play Tennis: 60% (600 members)
- Members who play Cricket: 50% (500 members)
- Members who play Football: 70% (700 members)
- Members who play Tennis and Cricket: 20% (200 members)
- Members who play Cricket and Football: 40% (400 members)
- Members who play Football and Tennis: 30% (300 members)
Using these numbers, we can calculate the total number of members who play at least one sport:
Total = Members who play Tennis + Members who play Cricket + Members who play Football - (Members who play Tennis and Cricket + Members who play Cricket and Football + Members who play Football and Tennis)
Total = 600 + 500 + 700 - (200 + 400 + 300)
= 1600 - 900
= 700 members
The calculated value of 700 members is less than the total registered members of 1000, which indicates that there is a discrepancy. According to the inclusion and exclusion principle, if the total number of members who play at least one sport is less than the total number of members, it means that there is an overlap or inconsistency in the information provided.
In this case, the claim that 20% of members play all three sports cannot be true since we calculated that there are only 700 members who play at least one sport. This means that there is an inconsistency in the data provided.
To determine whether the claim that 20% of members play all three sports is accurate, we can use the principle of inclusion and exclusion. This principle allows us to calculate the total number of members who play at least one of the three sports by considering the individual percentages and their intersections.
Let's break down the given information:
- Total number of registered members: 1000
- Percentage of members who play Tennis: 60%
- Percentage of members who play Cricket: 50%
- Percentage of members who play Football: 70%
- Percentage of members who play Tennis and Cricket: 20%
- Percentage of members who play Cricket and Football: 40%
- Percentage of members who play Football and Tennis: 30%
To find the total number of members who play at least one of the sports, we can add the percentages of each sport individually:
Total = Tennis + Cricket + Football
But we also need to take into account the overlaps between the sports. To do this, we subtract the percentages of the intersections:
Total = Tennis + Cricket + Football - (Tennis and Cricket) - (Cricket and Football) - (Football and Tennis)
Using the given percentages:
Total = 60% + 50% + 70% - 20% - 40% - 30%
Total = 60% + 50% + 70% - 20% - 40% - 30%
= 70%
Therefore, according to the principle of inclusion and exclusion, the total number of members who play at least one of the three sports is 70% of 1000 (total members) which equals 700.
Now, let's evaluate the claim that 20% of members play all three sports. If this claim is true, it implies that there are 20% of 1000 (total members) who play all three sports, which equals 200 members.
However, since the total number of members who play at least one of the three sports is only 700, it is impossible for 200 members (20%) to play all three sports. Therefore, the claim that 20% of members play all three sports is inaccurate.