Two examination rooms A and B have unknown student counts. If 10 students move from A to B, the rooms then have the same number. If instead 20 students move from B to A, room A then has twice as many students as room B. How many students are in room A?

Given data

• Let A = x, B = y.
• x − 10 = y + 10 (after sending 10 from A to B, both equal).
• x + 20 = 2(y − 20) (after sending 20 from B to A, A is double B).

Concept / Approach

• Form two linear equations in x and y and solve simultaneously.

Step-by-step calculation
From x − 10 = y + 10 ⇒ x − y = 20 … (1)From x + 20 = 2(y − 20) ⇒ x − 2y = −60 … (2)Subtract (2) from (1): (x − y) − (x − 2y) = 20 − (−60)⇒ y = 80Then x = y + 20 = 100

Verification
Move 10: (100 − 10, 80 + 10) = (90, 90) ✔Move 20: (100 + 20, 80 − 20) = (120, 60), and 120 = 2 × 60 ✔

Common pitfalls

• Adding or averaging student counts instead of setting up equations.
• Sign mistakes when moving students between rooms.

Final Answer
100