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The Brushstrokes Art Academy is organizing a special project where students and teachers collaborate on paintings. The academy has 50 participants in total, including both students and teachers. Each student will create one painting with every other student and one painting with each teacher.
When two students collaborate, they create a small canvas painting. When a student collaborates with a teacher, they create a large canvas painting.
At the end of the project, there are 157 large canvas paintings.
How many small canvas paintings (student-student collaborations) were created during this project?
Let’s approach this step-by-step:
1) Let x be the number of teachers. Then, (50 – x) is the number of students.
2) We know there are 157 large canvas paintings. Each of these represents a student-teacher collaboration. So:
x(50 – x) = 157
3) Expanding this equation:
50x – x² = 157
4) Rearranging:
x² – 50x + 157 = 0
5) This is a quadratic equation. We can solve it using the quadratic formula:
x = [-(-50) ± √((-50)² – 4(1)(157))] / (2(1))
x = [50 ± √(2500 – 628)] / 2
x = (50 ± √1872) / 2
x = (50 ± 43.27) / 2
6) This gives us two solutions: x ≈ 46.63 or x ≈ 3.37
Since x represents the number of teachers, it must be a whole number.
The only sensible solution is x = 3 teachers.
7) Therefore, there are 3 teachers and 47 students.
8) To find the number of small canvas paintings (student-student collaborations):
Number of small canvases = [47 × (47 – 1)] / 2 = 1081
Therefore, 1081 small canvas paintings were created during this project.
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