Find the original fraction using two changes Given (x + 2)/(y + 3) = 7/9 and (x − 1)/(y − 1) = 4/5, determine x/y.

Introduction / Context:
Two independent transformations of a fraction are given. Solving the resulting pair of linear equations in x and y yields the original fraction.

Given Data / Assumptions:

• (x + 2)/(y + 3) = 7/9.
• (x − 1)/(y − 1) = 4/5.
• x, y are positive integers representing the original fraction x/y.

Concept / Approach:
Cross-multiply both equations to create linear relations and solve simultaneously using elimination or substitution.

Step-by-Step Solution:
From (x + 2)/(y + 3) = 7/9 ⇒ 9x + 18 = 7y + 21 ⇒ 9x − 7y = 3.From (x − 1)/(y − 1) = 4/5 ⇒ 5x − 5 = 4y − 4 ⇒ 5x − 4y = 1.Eliminate y: multiply the second equation by 7 ⇒ 35x − 28y = 7.Multiply the first by 4 ⇒ 36x − 28y = 12.Subtract: (36x − 28y) − (35x − 28y) = 12 − 7 ⇒ x = 5.Substitute: 5x − 4y = 1 ⇒ 25 − 4y = 1 ⇒ y = 6.Original fraction = 5/6.

Verification / Alternative check:
Check both: (5 + 2)/(6 + 3) = 7/9 and (5 − 1)/(6 − 1) = 4/5. Works exactly.

Why Other Options Are Wrong:
13/16, 9/11, and 17/21 do not satisfy both equations; 7/9 is a target fraction after a change, not the original.

Common Pitfalls:
Arithmetic slips when aligning coefficients; mixing which equation corresponds to which adjustment.

Final Answer:
5/6