If x - 4y = 0 and x + 2y = 24, then what is the value of the fraction (2x + 3y)/(2x - 3y)?

Introduction / Context:
This question combines solving simultaneous linear equations with evaluating a rational expression involving the same variables. Problems of this type are very common in aptitude tests because they check both algebraic manipulation skills and the ability to substitute correctly into a derived expression without making mistakes in fractions.

Given Data / Assumptions:

• x - 4y = 0
• x + 2y = 24
• Required value: (2x + 3y)/(2x - 3y)
• x and y are real numbers that satisfy both equations simultaneously

Concept / Approach:
The approach is straightforward: first solve the two linear equations in x and y using substitution or elimination. Once x and y are known, substitute them into the numerator and denominator of the given fraction. Finally, simplify the resulting numerical fraction to its lowest terms to match one of the options provided.

Step-by-Step Solution:
Step 1: From x - 4y = 0, express x in terms of y as x = 4y. Step 2: Substitute x = 4y into the second equation x + 2y = 24. Step 3: This gives 4y + 2y = 24, so 6y = 24 and therefore y = 4. Step 4: Substitute y = 4 back into x = 4y to find x = 4 × 4 = 16. Step 5: Compute the numerator 2x + 3y = 2 × 16 + 3 × 4 = 32 + 12 = 44. Step 6: Compute the denominator 2x - 3y = 2 × 16 - 3 × 4 = 32 - 12 = 20. Step 7: Form the fraction (2x + 3y)/(2x - 3y) = 44/20. Step 8: Simplify 44/20 by dividing numerator and denominator by 4 to get 11/5.

Verification / Alternative check:
You can verify by checking that x = 16 and y = 4 satisfy both original equations. Then quickly recompute the ratio 44/20 and again simplify it to 11/5. Since both equations are linear and consistent, the solution pair (16, 4) is unique, and therefore the computed ratio is also unique and correct.

Why Other Options Are Wrong:
9/7 and 13/7 are incorrect because they come from arithmetic mistakes in forming or simplifying the fraction. The value 9/5 could appear if someone incorrectly uses 2x + y instead of 2x + 3y or miscalculates 2x - 3y. The value 5/3 is just an arbitrary fraction that does not arise from the correct substitution and simplification.

Common Pitfalls:
A common error is to forget that both equations must hold simultaneously and to treat each equation independently. Another issue is simplifying the fraction 44/20 incorrectly, such as dividing only the numerator or denominator by the common factor. Always check that the simplified fraction is fully reduced and corresponds exactly to one of the options.

Final Answer:
Therefore, (2x + 3y)/(2x - 3y) = 11/5.