Mathematical Puzzle for Students: Intersecting Sets and Instrumental Skills
In a group of 142 pupils, 90 can play the guitar only and a certain number can play the piano only. If there are 12 who play neither instrument, we can determine how many pupils can play both instruments. This puzzle not only tests your understanding of set theory but also enhances your problem-solving skills. For more math-related questions and strategies, our Quora Space on Mathematics provides support and discussions, and celebrating mathematical skills and exploring the world of numbers with a community of math enthusiasts is an engaging experience.
Understanding the Problem
The problem presented here involves understanding the intersection and union of sets, a fundamental concept in set theory. Let's break down the given numbers to find the number of students who can play both the guitar and the piano.
Given:
Total number of students: 142 Students who can play the guitar only: 90 Students who play the piano only: Let's denote this as x Students who play neither: 12What to Find:
The number of students who can play both instruments.
Calculation:
To find the number of students who play both instruments, you need to subtract the total number of guitar players and those who play neither from the overall group. The remaining students must be those who play both the guitar and the piano. Let's define the equation:
Total number of students Number of guitar players Number of piano players - Number of students who play both instruments Number of students who play neither
Therefore, we have:
142 90 x - y 12
where y is the number of students who play both instruments. Simplifying this, we get:
142 - 90 - 12 x - y
40 x - y
x 40 y
Since y is the number of students who play both instruments, we substitute x 40 y back into the equation:
142 90 (40 y) - y 12
142 90 40 y - y 12
142 142
From the above calculation, we can see that:
y 40 - x
Given that x 40 y, let's solve for y by isolating it:
y 40 - x
Since we know the total number of students who play the guitar only (90) and neither (12), and x y 40, the number of students who play both instruments is 40.
Therefore, y 40.
Conclusion
There are 40 students who can play both the guitar and the piano.
If you need more help with math-related questions or problem-solving strategies, our Quora Space on Mathematics provides support and discussions. Join the community of math enthusiasts, sharpen your mathematical skills, and explore the world of numbers.
Wrapping Up
The mathematical puzzle presented here involves understanding set theory and problem-solving. Whether you're a student, a teacher, or someone who enjoys puzzles, this type of problem encourages logical thinking and enhances your mathematical skills. For more such questions and discussions, our space on Mathematics is a great place to find support and engagement.