When the numerator of a fraction increases by 4, the fraction increases by $\frac{2}{3}$. The denominator of the fraction is

### Concept & Formula This problem tests the algebraic relationship between modifying a fraction's numerator and the corresponding change in the fraction's overall value. When you add a value $k$ to the numerator of a fraction $\frac{x}{y}$, the new fraction can be split into the original fraction plus a new term: $\frac{x + k}{y} = \frac{x}{y} + \frac{k}{y}$. The increase in the fraction's value is exactly represented by this new term, $\frac{k}{y}$. ### Step-by-Step Solution * **Given:** * Increase in numerator = 4 * Overall increase in the fraction's value = $\frac{2}{3}$ * **Calculation:** 1. Let the original fraction be $\frac{x}{y}$. 2. The new numerator is $x + 4$. 3. The new fraction is $\frac{x + 4}{y}$. 4. According to the problem, the new fraction equals the original fraction plus $\frac{2}{3}$: $$ \frac{x + 4}{y} = \frac{x}{y} + \frac{2}{3} $$ 5. Separate the terms on the left side: $$ \frac{x}{y} + \frac{4}{y} = \frac{x}{y} + \frac{2}{3} $$ 6. Subtract $\frac{x}{y}$ from both sides to isolate $y$: $$ \frac{4}{y} = \frac{2}{3} $$ 7. Cross-multiply to solve for $y$: $$ 2y = 12 $$ $$ y = 6 $$ 8. The denominator $y$ is 6. ### Exam Strategy & Shortcut **Direct Concept Application:** Recognize immediately that adding to a numerator splits the fraction. An addition of 4 to the numerator creates an extra value of $\frac{4}{\text{Denominator}}$. The problem states this extra value is $\frac{2}{3}$. Mentally set up: $\frac{4}{D} = \frac{2}{3}$. Scale the numerator of $\frac{2}{3}$ to match 4 by multiplying top and bottom by 2. $\frac{2 \times 2}{3 \times 2} = \frac{4}{6}$. By direct comparison, the denominator $D$ must be 6. ### Common Pitfall A common mistake is trying to assign arbitrary values to the original numerator $x$ to test options, which wastes time. The original numerator $x$ is completely irrelevant to finding the denominator in this specific scenario because it cancels out algebraically. ### Final Answer **Therefore, the correct answer is 6.**