Class sections A and B have different student counts. If 10 students shift from section B to section A, the strength of A becomes 3 times the strength of B. However, if instead 10 students shift from section A to section B, then both sections become equal in strength. What is the initial ratio of students in section A to section B?

Introduction / Context:
This classic ratio puzzle tests simultaneous linear equations and careful reading of shifting scenarios between two groups. We must form equations for two different transfers and solve for the original strengths of sections A and B.

Given Data / Assumptions:

• Initially, let A = a and B = b.
• Scenario 1: 10 move B → A, so A becomes 3 times B: a + 10 = 3(b − 10).
• Scenario 2: 10 move A → B, so A and B become equal: a − 10 = b + 10.
• All numbers are whole students, and transfers are feasible.

Concept / Approach:
Translate each scenario into an equation and solve the two-variable system. Finally, express the ratio A : B in simplest terms.

Step-by-Step Solution:
From equality scenario: a − 10 = b + 10 ⇒ a − b = 20 ⇒ a = b + 20. From triple scenario: a + 10 = 3(b − 10) ⇒ a + 10 = 3b − 30 ⇒ a = 3b − 40. Equate a: b + 20 = 3b − 40 ⇒ 2b = 60 ⇒ b = 30. Then a = b + 20 = 50. Hence A : B = 50 : 30 = 5 : 3.

Verification / Alternative check:
Move 10 from B to A → A = 60, B = 20 ⇒ A is 3× B (60 vs 20). Move 10 from A to B → A = 40, B = 40 ⇒ equal. Both conditions hold.

Why Other Options Are Wrong:
2 : 1, 3 : 1, 9 : 4, and 7 : 5 fail at least one scenario when tested, so they cannot be the initial ratio.

Common Pitfalls:
Mixing the direction of transfer or forgetting to subtract/add the 10 in the correct place. Always update both groups consistently.

Final Answer:
5 : 3