Students in halls A and B: if 10 go from A to B, they become equal. If 20 go from B to A, A becomes double B. Find the original numbers.
A: \(100\) students; B: \(80\) students.
Step 1: Let the original number of students in Hall A be \(a\).
Let the original number of students in Hall B be \(b\).
First condition:
If 10 students go from A to B, then
Students in A = \(a - 10\)
Students in B = \(b + 10\)
According to the question, they become equal:
\(a - 10 = b + 10\)
Rearranging:
\(a - b = 20\) … (1)
Second condition:
If 20 students go from B to A, then
Students in A = \(a + 20\)
Students in B = \(b - 20\)
According to the question, A becomes double of B:
\(a + 20 = 2(b - 20)\)
Expand the right side:
\(a + 20 = 2b - 40\)
Rearranging:
\(a - 2b = -60\) … (2)
Step 2: Solve the equations.
From (1): \(a = b + 20\)
Substitute into (2):
\((b + 20) - 2b = -60\)
\(-b + 20 = -60\)
\(-b = -80\)
\(b = 80\)
Now, put \(b = 80\) into (1):
\(a - 80 = 20\)
\(a = 100\)
Final Answer:
Hall A has 100 students and Hall B has 80 students.