If x / y = 4 / 9, find the simplified value of the expression (7x^2 − 19xy + 11y^2) divided by y^2.

Introduction / Context:

This algebraic simplification question tests how well you can use a given ratio to reduce a more complicated expression. Instead of trying to find x and y separately, you can work with the ratio x / y and substitute directly into the expression. This approach is common in aptitude tests because it saves time and avoids unnecessary calculations.

Given Data / Assumptions:

• The ratio x / y equals 4 / 9.
• You must evaluate (7x^2 − 19xy + 11y^2) / y^2.
• x and y are nonzero real numbers so that the ratio and division by y^2 are valid.

Concept / Approach:

The key idea is to express everything in terms of the single ratio t = x / y. Once you divide all the terms in the numerator by y^2, x^2 / y^2 becomes t^2 and xy / y^2 becomes t. This transforms the original expression into a simple quadratic expression in t, which is much easier to simplify and evaluate.

Step-by-Step Solution:

Verification / Alternative check:

As a check, you can pick actual values satisfying x / y = 4 / 9, for example x = 4 and y = 9. Substituting into (7x^2 − 19xy + 11y^2) / y^2 gives (7*16 − 19*36 + 11*81) / 81 = (112 − 684 + 891) / 81 = 319 / 81, confirming the result.

Why Other Options Are Wrong:

The fractions 59/81, 100/27, 913/81, and 59/27 come from incomplete simplification, incorrect substitution of t, or mistakes with common denominators. Only 319/81 matches both the algebraic derivation and the numeric check with sample values.

Common Pitfalls:

Students sometimes plug in arbitrary values of x and y that do not respect the ratio, or they try to find explicit x and y by solving two unknowns from a single ratio. Another frequent mistake is failing to convert all terms to a common denominator when adding fractions, leading to arithmetic errors.

Final Answer:

The simplified value of the expression is 319/81.