Let's break down this problem involving percentages of students playing different games. We are given the percentage of students who play cricket, football, and both games. We need to find the percentage of students who play neither game.

We can solve this using the principle of inclusion-exclusion, which helps us find the number or percentage of elements in the union of two sets.

Here's what we know:

• Percentage of students playing cricket (C) = 50%
• Percentage of students playing football (F) = 40%
• Percentage of students playing both cricket and football (C & F) = 10%

To find the percentage of students who play at least one game (either cricket or football or both), we use the formula:

Percentage playing at least one game = % Playing Cricket + % Playing Football - % Playing Both

Using the given values:

Percentage playing at least one game = $\text{50\%} + \text{40\%} - \text{10\%}$

Percentage playing at least one game = $\text{90\%} - \text{10\%}$

Percentage playing at least one game = $\text{80\%}$

This means 80% of the students play at least one of the two games.

The total percentage of students in the school is 100%. Students either play at least one of these games or they play neither. Therefore, the percentage of students who play neither cricket nor football is the total percentage minus the percentage who play at least one game.

Percentage playing neither game = Total Percentage - Percentage playing at least one game

Percentage playing neither game = $\text{100\%} - \text{80\%}$

Percentage playing neither game = $\text{20\%}$

So, 20% of the students play neither cricket nor football.

Revision Table: School Sports Percentages

Let's quickly summarize the key percentages and the calculation process for finding students who play neither sport.

• Total students = 100%
• Students playing only cricket = % Cricket - % Both = 50% - 10% = 40%
• Students playing only football = % Football - % Both = 40% - 10% = 30%
• Students playing both = 10%
• Students playing at least one game = % Only Cricket + % Only Football + % Both = 40% + 30% + 10% = 80%
• Alternatively, Students playing at least one = % Cricket + % Football - % Both = 50% + 40% - 10% = 80%
• Students playing neither game = 100% - % Playing at least one = 100% - 80% = 20%

Additional Information: Set Theory and Percentages

This problem can be easily visualized using Venn diagrams, which are useful for understanding relationships between sets. In this case, we have two overlapping sets representing students who play cricket and students who play football.

• The overlap region represents students who play both.
• The areas within each circle but outside the overlap represent students who play only that specific sport.
• The area outside both circles represents students who play neither sport.

The formula $\text{P(A} \cup \text{B)} = \text{P(A)} + \text{P(B)} - \text{P(A} \cap \text{B)}$ is a fundamental concept in probability and set theory. It ensures that the elements counted twice (those in the intersection A & B) are subtracted once to get the accurate total in the union (A or B or both).

When dealing with percentages of a whole group (like students in a school), the total group is considered 100%. If we know the percentage of a group that satisfies a condition (e.g., playing at least one game), we can find the percentage that does not satisfy the condition by subtracting from 100%.

Understanding these basic principles allows us to solve various problems involving overlapping groups and percentages.