Two exam halls P and Q — After moving 10 students from P to Q, the numbers become equal. After moving 20 students from Q to P, P becomes double Q. Find the original numbers in P and Q.

Introduction / Context:
This is a system-of-equations problem describing two different transfers of students between halls P and Q. Translating the narrative statements precisely into algebra will let you solve for the original counts without guesswork.

Given Data / Assumptions:

• Let initial counts be p (P) and q (Q).
• After 10 shift from P to Q: p - 10 = q + 10.
• After 20 shift from Q to P: p + 20 = 2 * (q - 20).
• All numbers are nonnegative integers.

Concept / Approach:
From the first transfer, express p in terms of q. Substitute into the second transfer equation and solve. This straightforward linear system yields unique integer solutions that match the scenario conditions.

Step-by-Step Solution:
From p - 10 = q + 10 → p - q = 20 → p = q + 20.Use the second condition: p + 20 = 2(q - 20).Substitute p: (q + 20) + 20 = 2(q - 20) → q + 40 = 2q - 40 → q = 80.Then p = q + 20 = 100.

Verification / Alternative check:
Check first transfer: 100 - 10 = 90 and 80 + 10 = 90 → equal. Check second transfer: 100 + 20 = 120; 80 - 20 = 60; indeed 120 = 2 * 60.

Why Other Options Are Wrong:

• 60, 40 / 70, 50 / 80, 60 / 90, 70: None satisfy both transfer conditions simultaneously.

Common Pitfalls:
Reversing the direction of transfer; misinterpreting “P becomes double Q” after the second move; algebraic slips when substituting p = q + 20.

Final Answer:
100, 80