A fraction becomes 6/5 when 5 is added to its numerator and becomes 1/2 when 4 is added to its denominator. What is the value of the original fraction?

Introduction / Context:
This question tests algebraic thinking using fractions. We are given how the value of a fraction changes when we add specific numbers to its numerator and denominator. From these conditions, we must find the original fraction. Such problems are common in aptitude tests because they combine fractions, equations, and basic reasoning.

Given Data / Assumptions:

- Let the original fraction be x / y, where x and y are positive integers and y is not zero.

- When 5 is added to the numerator, the fraction becomes 6/5.

- When 4 is added to the denominator, the fraction becomes 1/2.

- We must find the original fraction x / y.

Concept / Approach:
We translate the verbal conditions into algebraic equations. Then we solve the simultaneous equations to find x and y. Once we know x and y, we compute the fraction and compare it with the given options. This uses the basic idea that if two fractions are equal, their cross products are equal.

Step-by-Step Solution:
Step 1: Let the fraction be x / y. Step 2: From the first condition, (x + 5) / y = 6 / 5. Cross multiply: 5(x + 5) = 6y. This gives 5x + 25 = 6y. (Equation 1) Step 3: From the second condition, x / (y + 4) = 1 / 2. Cross multiply: 2x = y + 4. So y = 2x - 4. (Equation 2) Step 4: Substitute y from Equation 2 into Equation 1. 5x + 25 = 6(2x - 4). 5x + 25 = 12x - 24. Rearrange: 25 + 24 = 12x - 5x. 49 = 7x, so x = 7. Step 5: From y = 2x - 4, substitute x = 7. y = 2 * 7 - 4 = 14 - 4 = 10. Step 6: Therefore, the original fraction is 7 / 10.

Verification / Alternative check:
Check with the conditions: (7 + 5) / 10 = 12 / 10 = 6 / 5, which matches the first condition. 7 / (10 + 4) = 7 / 14 = 1 / 2, which matches the second condition. Hence the fraction 7 / 10 fully satisfies both conditions.

Why Other Options Are Wrong:
- 8/9: Adding 5 to numerator or 4 to denominator does not give 6/5 or 1/2.
- 7/8: Fails both transformation conditions when tested.
- 6/11: Also does not produce 6/5 and 1/2 under the given operations.

Common Pitfalls:
A frequent mistake is to treat the changes to numerator and denominator incorrectly, for example adding 5 and 4 to both numerator and denominator together. Another common error is forgetting to cross multiply correctly when equating fractions. Also, some students guess from options without checking both conditions, which leads to incorrect answers.

Final Answer:
The original fraction that satisfies both conditions is 7/10.