Solve for the original fraction from two conditions If 3 is added to the denominator, the fraction becomes 1/3; if 4 is added to the numerator, it becomes 3/4. Find the original fraction.

Introduction / Context:Classic fraction-reconstruction problems translate words into equations in x/y, then solve a simple linear system. Here, two alterations to numerator or denominator yield target fractions 1/3 and 3/4.

Given Data / Assumptions:

• Original fraction = x/y, with positive integers x, y.
• x/(y + 3) = 1/3 when 3 is added to denominator.
• (x + 4)/y = 3/4 when 4 is added to numerator.

Concept / Approach:Convert to linear equations and solve simultaneously. Keep arithmetic exact to avoid rounding errors.

Step-by-Step Solution:From x/(y + 3) = 1/3 ⇒ 3x = y + 3 ⇒ y = 3x − 3.From (x + 4)/y = 3/4 ⇒ 4x + 16 = 3y.Substitute y: 4x + 16 = 3(3x − 3) = 9x − 9.Rearrange: 16 + 9 = 9x − 4x ⇒ 25 = 5x ⇒ x = 5.Then y = 3*5 − 3 = 12. Fraction = 5/12.

Verification / Alternative check:Check 5/(12 + 3) = 5/15 = 1/3 and (5 + 4)/12 = 9/12 = 3/4. Both conditions hold.

Why Other Options Are Wrong:4/9, 3/20, 7/24, 2/9 do not satisfy both transformation rules when tested.

Common Pitfalls:Mixing which change applies to numerator vs. denominator; arithmetic slips when substituting.

Final Answer:5/12