In a class of 120 students, 100 students participate in either Golf or Skating or both. Among them, total 60 students participate in Golf. A total of 56 students participate in Skating. How many students participate only in Skating?
40
Understanding the Problem: Golf and Skating Students
This problem involves analyzing data about students participating in two activities: Golf and Skating. We are given the total number of students in the class, the number of students involved in at least one activity (Golf or Skating or both), the total number of students participating in Golf, and the total number of students participating in Skating. Our goal is to find out how many students participate only in Skating.
Key Information Given
- Total students in the class: 120
- Students participating in Golf or Skating or both: 100
- Total students participating in Golf: 60
- Total students participating in Skating: 56
Applying Set Theory and Venn Diagrams
We can use set theory concepts, often visualized with Venn diagrams, to solve this type of problem. Let G represent the set of students participating in Golf and S represent the set of students participating in Skating.
- $|G \cup S|$ represents the number of students participating in Golf or Skating or both. We are given $|G \cup S| = 100$.
- $|G|$ represents the total number of students participating in Golf. We are given $|G| = 60$.
- $|S|$ represents the total number of students participating in Skating. We are given $|S| = 56$.
- $|G \cap S|$ represents the number of students participating in both Golf and Skating.
Finding Students in Both Golf and Skating
The principle of inclusion-exclusion for two sets states:
$$|G \cup S| = |G| + |S| - |G \cap S|$$
We can plug in the values we know:
$$100 = 60 + 56 - |G \cap S|$$
Now, we solve for $|G \cap S|$:
$$100 = 116 - |G \cap S|$$
$$|G \cap S| = 116 - 100$$
$$|G \cap S| = 16$$
So, 16 students participate in both Golf and Skating.
Calculating Students Only in Skating
Students who participate only in Skating are those who are in the Skating set but not in the Golf set. In set notation, this is $|S \setminus G|$, which can be calculated as the total number of students in Skating minus the number of students in both Golf and Skating:
$$\text{Only Skating} = |S| - |G \cap S|$$
Using the values we have:
$$\text{Only Skating} = 56 - 16$$
$$\text{Only Skating} = 40$$
Therefore, 40 students participate only in Skating.
Verification
Let's check our numbers:
- Students only in Golf = $|G| - |G \cap S| = 60 - 16 = 44$
- Students only in Skating = 40 (our answer)
- Students in both = 16
- Total students in Golf or Skating or both = (Only Golf) + (Only Skating) + (Both) = 44 + 40 + 16 = 100. This matches the given information.
- Students participating in neither activity = Total students - Students in Golf or Skating or both = 120 - 100 = 20.
- Total students = (Only Golf) + (Only Skating) + (Both) + (Neither) = 44 + 40 + 16 + 20 = 120. This matches the total number of students in the class.
The calculation for students participating only in Skating is consistent with all the given data.
Summary of Participation Numbers
| Category | Number of Students |
|---|---|
| Total Students | 120 |
| Participate in Golf or Skating or Both ($|G \cup S|$) | 100 |
| Participate in Golf ($|G|$) | 60 |
| Participate in Skating ($|S|$) | 56 |
| Participate in Both Golf and Skating ($|G \cap S|$) | 16 |
| Participate Only in Golf ($|G| - |G \cap S|$) | 44 |
| Participate Only in Skating ($|S| - |G \cap S|$) | 40 |
| Participate in Neither | 20 |
Conclusion
Based on the calculations using set theory, the number of students who participate only in Skating is 40.
Revision Table: Key Concepts Review
Reviewing the key terms and formulas used in this problem:
| Concept | Explanation | Formula Used |
|---|---|---|
| Set Union ($A \cup B$) | Elements in set A OR set B OR both. | $|A \cup B|$ |
| Set Intersection ($A \cap B$) | Elements in set A AND set B (in both). | $|A \cap B|$ |
| Principle of Inclusion-Exclusion (for 2 sets) | Relates the sizes of the union, intersection, and individual sets. | $|A \cup B| = |A| + |B| - |A \cap B|$ |
| Elements Only in Set B | Elements in set B but not in set A. | $|B| - |A \cap B|$ |
Additional Information: Solving Set Problems
Problems involving students participating in activities can often be solved effectively using either set theory formulas or by drawing a Venn diagram. Both methods rely on the same underlying logic.
- Venn Diagram Method: Draw two overlapping circles (one for Golf, one for Skating). The overlapping region is for students in both. Start by filling in the number in the intersection. Then, calculate the "only" parts by subtracting the intersection from the total for each activity. Finally, add the "only" parts and the "both" part to get the total in the union. Subtract this from the total number of students to find those in neither. This visual method can help clarify the relationships between the groups.
- Formula Method: Directly apply the Principle of Inclusion-Exclusion to find the intersection, and then use the intersection value to find the "only" parts. This method is quicker once you are comfortable with the formulas.
Both approaches lead to the same correct answer for problems like this one, helping you find the number of students only in Skating or other specific groups.
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