If the numerator of a fraction is increased by $\frac{1}{4}$ and the denominator is decreased by $\frac{1}{3}$, the new fraction obtained is $\frac{33}{64}$. What was the original fraction?

### Concept & Logic This problem tests the understanding of linear equations with two variables. When absolute values (like $\frac{1}{4}$ and $\frac{1}{3}$) are added to or subtracted from the numerator and denominator of a fraction respectively, it generates a single linear equation with two independent variables. A unique original fraction cannot be determined without a second condition or equation linking the numerator and denominator. ### Step-by-Step Solution * **Given:** * An original fraction exists. Let's call its numerator $x$ and denominator $y$. * The numerator is increased by the absolute value $\frac{1}{4}$. * The denominator is decreased by the absolute value $\frac{1}{3}$. * The resulting fraction is $\frac{33}{64}$. * **Calculation:** 1. Let the original fraction be $\frac{x}{y}$. 2. Set up the equation based on the given conditions: $$ \frac{x + \frac{1}{4}}{y - \frac{1}{3}} = \frac{33}{64} $$ 3. Simplify the complex fraction by finding common denominators in the numerator and denominator expressions: $$ \frac{\frac{4x + 1}{4}}{\frac{3y - 1}{3}} = \frac{33}{64} $$ 4. Multiply by the reciprocal: $$ \frac{3(4x + 1)}{4(3y - 1)} = \frac{33}{64} $$ 5. Cross-multiply to linearize the equation: $$ 64 \times 3(4x + 1) = 33 \times 4(3y - 1) $$ $$ 192(4x + 1) = 132(3y - 1) $$ 6. Divide both sides by 12 to simplify: $$ 16(4x + 1) = 11(3y - 1) $$ $$ 64x + 16 = 33y - 11 $$ $$ 64x - 33y = -27 $$ 7. We are left with one equation and two variables ($x$ and $y$). Without a second equation, it is impossible to find a unique value for the fraction $\frac{x}{y}$. ### Exam Strategy & Shortcut **Logical Deduction:** Whenever a question states that an absolute number (or fixed fraction) is added to or subtracted from the unknown numerator and denominator, immediately look for a relationship given between the original numerator and denominator. If no such relationship (like a ratio or a sum) is provided, the answer is always "Cannot be determined". You do not need to write down any equations. ### Common Pitfall Students often misread "increased by $\frac{1}{4}$" as "increased by $\frac{1}{4}$ of itself" (which would mean multiplying by $\frac{5}{4}$). If it were a proportional increase, the variables could be isolated as a ratio $\frac{x}{y}$, and an answer could be found. Always interpret phrasing literally unless specified otherwise. ### Final Answer **Therefore, the correct answer is Cannot be determined.**