Fraction reconstruction — A fraction becomes 1/2 when the numerator is increased by 2 and denominator by 1; it becomes 3/5 when the numerator is increased by 1 and denominator decreased by 2. Find the original fraction.

Introduction / Context:
This algebra problem encodes two conditions on the same unknown fraction. Translating each sentence to an equation and solving the resulting linear system is the most reliable and general approach. The numbers are small, making factoring and checking straightforward.

Given Data / Assumptions:

• Original fraction is n/d with positive integers n and d.
• (n + 2) / (d + 1) = 1/2.
• (n + 1) / (d - 2) = 3/5.

Concept / Approach:
Clear denominators to turn each condition into a linear equation in n and d. Solve the two linear equations simultaneously. After obtaining (n, d), simplify if needed and quickly verify both conditions to eliminate mistakes.

Step-by-Step Solution:
From (n + 2) / (d + 1) = 1/2 → 2(n + 2) = d + 1 → 2n + 4 = d + 1 → d = 2n + 3.From (n + 1) / (d - 2) = 3/5 → 5(n + 1) = 3(d - 2) → 5n + 5 = 3d - 6 → 3d = 5n + 11 → d = (5n + 11)/3.Equate d expressions: 2n + 3 = (5n + 11)/3 → 6n + 9 = 5n + 11 → n = 2.Then d = 2n + 3 = 7, so the fraction is 2/7.

Verification / Alternative check:
(2 + 2)/(7 + 1) = 4/8 = 1/2; and (2 + 1)/(7 - 2) = 3/5. Both conditions hold exactly.

Why Other Options Are Wrong:

• 3/5 / 1/7 / 2/5 / 5/12: Substituting any of these into the two conditions fails at least one requirement; only 2/7 satisfies both simultaneously.

Common Pitfalls:
Sign or algebra slips when clearing denominators; forgetting to verify both constraints; reducing the fraction prematurely and losing track of n and d.

Final Answer:
2/7