For a certain fraction, if 1 is added to its denominator the value of the fraction becomes 1/2, and if 1 is added to its numerator the value of the fraction becomes 1. What is the original fraction?

Introduction / Context:
This question tests understanding of algebraic representation of fractions and how changes to the numerator or denominator affect the fraction's value. We are given two different transformations of the same fraction and the resulting values. By modeling these conditions using algebra, we can form two equations and solve for the numerator and denominator of the original fraction. Such fraction puzzles are common in aptitude and school-level mathematics.

Given Data / Assumptions:

• Let the original fraction be x / y.
• When 1 is added to the denominator, the fraction becomes 1/2, so x / (y + 1) = 1/2.
• When 1 is added to the numerator, the fraction becomes 1, so (x + 1) / y = 1.
• We must find the original fraction x / y.

Concept / Approach:
We translate the two conditions into algebraic equations. From x / (y + 1) = 1/2, cross multiplication gives 2x = y + 1. From (x + 1) / y = 1, we obtain x + 1 = y. These two equations in x and y can be solved simultaneously. Once we find x and y, we can express the original fraction and match it with the options given. This approach uses elementary algebra and reasoning with fractions.

Step-by-Step Solution:
Step 1: Let the original fraction be x / y. Step 2: From the first condition, x / (y + 1) = 1/2. Step 3: Cross multiply: 2x = y + 1. Call this Equation (1). Step 4: From the second condition, (x + 1) / y = 1. Step 5: Cross multiply: x + 1 = y. Call this Equation (2). Step 6: Substitute y from Equation (2) into Equation (1). Step 7: Equation (2) gives y = x + 1. Step 8: Substitute into Equation (1): 2x = (x + 1) + 1. Step 9: Simplify the right side: (x + 1) + 1 = x + 2, so 2x = x + 2. Step 10: Subtract x from both sides: x = 2. Step 11: Substitute x = 2 into y = x + 1 to get y = 2 + 1 = 3. Step 12: The original fraction is x / y = 2 / 3.

Verification / Alternative check:
Check the first condition using 2/3. Adding 1 to the denominator gives 2 / (3 + 1) = 2 / 4 = 1/2, which matches. Check the second condition. Adding 1 to the numerator gives (2 + 1) / 3 = 3 / 3 = 1, which also matches. Thus, 2/3 satisfies both conditions and is the correct original fraction.

Why Other Options Are Wrong:
Option (a) 1/3: Adding 1 to numerator or denominator does not produce 1/2 and 1 as required.
Option (c) 4/3 and option (d) 5/3: These are improper fractions and do not satisfy both transformation conditions when checked.

Common Pitfalls:
Common mistakes include mixing up numerator and denominator when applying the changes or writing the conditions as x + 1 / y + 1 instead of adjusting only one part at a time. Another error is failing to cross multiply correctly, leading to wrong equations. Careful translation of the word statement into algebraic form and systematic solving of the two equations avoids these problems.

Final Answer:
The original fraction is 2/3.