A fraction becomes 4 when 1 is added to both numerator and denominator, and it becomes 7 when 1 is subtracted from both numerator and denominator. What is the numerator of the original fraction?

Introduction / Context:
Transformations that add or subtract the same amount to both the numerator and denominator create linear equations in the original numerator and denominator. Solving these simultaneously reveals the original fraction’s parts without guesswork.

Given Data / Assumptions:

• Let the fraction be n / d with positive integers n and d.
• (n + 1) / (d + 1) = 4.
• (n − 1) / (d − 1) = 7.

Concept / Approach:
Convert each condition to an equation in n and d, then solve the pair. Because both outcomes are integers (4 and 7), n and d will be small whole numbers that satisfy both equations. Substitution or elimination are straightforward here.

Step-by-Step Solution:

Verification / Alternative check:
Original fraction 15/3 = 5. Check: (15 + 1)/(3 + 1) = 16/4 = 4 and (15 − 1)/(3 − 1) = 14/2 = 7, both matching the statements.

Why Other Options Are Wrong:
2, 3, 7, and 9 do not satisfy both linear conditions simultaneously when paired with an integer denominator.

Common Pitfalls:
Adding 1 to only the numerator or denominator, or assuming the result must remain a proper fraction rather than allowing integer results.

Final Answer:
15