Recover the original fraction from two scenarios If (x + 1)/(y + 2) = 2/3 and (x + 5)/(y + 1) = 5/4, find the original fraction x/y.

Introduction / Context:Two altered versions of a fraction are provided. Interpreting these as equations allows solving for x and y systematically.

Given Data / Assumptions:

• (x + 1)/(y + 2) = 2/3.
• (x + 5)/(y + 1) = 5/4.
• x/y is the unknown original fraction.

Concept / Approach:Cross-multiply to obtain linear equations, then eliminate one variable. Keep steps exact and consistent with the stated adjustments.

Step-by-Step Solution:From (x + 1)/(y + 2) = 2/3 ⇒ 3x + 3 = 2y + 4 ⇒ 3x − 2y = 1.From (x + 5)/(y + 1) = 5/4 ⇒ 4x + 20 = 5y + 5 ⇒ 4x − 5y = −15.Multiply first by 5: 15x − 10y = 5.Multiply second by 2: 8x − 10y = −30.Subtract equations: 7x = 35 ⇒ x = 5.Use 3x − 2y = 1 ⇒ 15 − 2y = 1 ⇒ 2y = 14 ⇒ y = 7.Original fraction = 5/7.

Verification / Alternative check:Check: (5 + 1)/(7 + 2) = 6/9 = 2/3 and (5 + 5)/(7 + 1) = 10/8 = 5/4. Both match.

Why Other Options Are Wrong:3/7 and 5/8 do not satisfy both conditions; 6/7 is too large; 7/5 is improper and incorrect.

Common Pitfalls:Misreading the increments to numerator/denominator; arithmetic mistakes in elimination.

Final Answer:5/7