A fraction becomes 2/3 when 1 is added to both its numerator and denominator, and it becomes 1/2 when 1 is subtracted from both the numerator and the denominator. What is the original fraction?

Introduction:
This is a classic fraction puzzle where the same unknown fraction is modified in two different ways: adding 1 to both numerator and denominator gives one known value, and subtracting 1 from both gives another. The question tests algebraic setup using two equations in two variables (numerator and denominator) and solving them accurately without confusion between the different transformations.

Given Data / Assumptions:

• Let the original fraction be n/d, with n and d positive integers and d ≠ 0.
• (n + 1)/(d + 1) = 2/3 (after adding 1).
• (n - 1)/(d - 1) = 1/2 (after subtracting 1).
• We must find n and d, hence the fraction n/d.

Concept / Approach:
Translate each condition into an equation and solve the resulting system. Both equations are rational, but cross-multiplication will reduce them to simple linear relationships between n and d. Solving the pair of equations simultaneously will give a unique solution for n and d, provided the fraction is well-defined.

Step-by-Step Solution:
From (n + 1)/(d + 1) = 2/3, cross-multiply: 3(n + 1) = 2(d + 1).3n + 3 = 2d + 2 => 3n + 1 = 2d … (1).From (n - 1)/(d - 1) = 1/2, cross-multiply: 2(n - 1) = d - 1.2n - 2 = d - 1 => d = 2n - 1 … (2).Substitute (2) into (1): 3n + 1 = 2(2n - 1) = 4n - 2.Rearrange: 4n - 2 - 3n - 1 = 0 => n - 3 = 0 => n = 3.Then d = 2n - 1 = 2*3 - 1 = 5.Original fraction = 3/5.

Verification / Alternative check:
Check adding 1: (3 + 1)/(5 + 1) = 4/6 = 2/3, correct. Check subtracting 1: (3 - 1)/(5 - 1) = 2/4 = 1/2, also correct. Thus 3/5 satisfies both transformed conditions perfectly.

Why Other Options Are Wrong:
1/3, 2/5, 4/7, 5/7: none of these fractions give 2/3 when you add 1 to numerator and denominator and 1/2 when you subtract 1 from both. At least one of the two conditions fails in each case.

Common Pitfalls:
Forgetting to cross-multiply correctly when forming the equations.Using the same modified fraction expression in both equations instead of the original n/d.Solving (1) and (2) with arithmetic mistakes while isolating n and d.

Final Answer:
3/5