Determine the original fraction from two transformed forms (x + 2)/(y + 1) = 5/8 and (x + 3)/(y + 1) = 3/4. Find the original fraction x/y.

Introduction / Context:
This problem provides two versions of the same fraction after specific changes to numerator and denominator. By translating to equations in x and y, we can solve the resulting linear system uniquely.

Given Data / Assumptions:

• (x + 2)/(y + 1) = 5/8.
• (x + 3)/(y + 1) = 3/4.
• Original fraction is x/y in lowest terms.

Concept / Approach:
Cross-multiply each equation to create two linear relations. Solve simultaneously to obtain x and y.

Step-by-Step Solution:
From (x + 2)/(y + 1) = 5/8 ⇒ 8x + 16 = 5y + 5 ⇒ 8x − 5y = −11.From (x + 3)/(y + 1) = 3/4 ⇒ 4x + 12 = 3y + 3 ⇒ 4x − 3y = −9.Multiply the second by 2: 8x − 6y = −18.Subtract: (8x − 5y) − (8x − 6y) = −11 − (−18) ⇒ y = 7.Use 4x − 3y = −9 ⇒ 4x − 21 = −9 ⇒ 4x = 12 ⇒ x = 3.Original fraction = 3/7.

Verification / Alternative check:
Check: (3 + 2)/(7 + 1) = 5/8 and (3 + 3)/(7 + 1) = 6/8 = 3/4. Both conditions match.

Why Other Options Are Wrong:
2/7 and 4/7 fail at least one of the conditions; “Data inadequate” is incorrect since the system has a unique solution.

Common Pitfalls:
Sign errors when moving terms; forgetting to increase the denominator by the same amount in both equations as stated.

Final Answer:
3/7