The sum of the numerator and denominator of a fraction is 11. If 1 is added to the numerator and 2 is subtracted from the denominator, it becomes $\frac{2}{3}$. The fraction is

### Concept & Strategy This problem translates simple arithmetic operations on the components of a fraction into algebraic equations. However, because we are looking for a specific starting fraction, checking the provided multiple-choice options against the conditions is the absolute fastest and most reliable method. Let the fraction be $\frac{n}{d}$. ### Step-by-Step Solution * **Given:** 1. $n + d = 11$ 2. $\frac{n + 1}{d - 2} = \frac{2}{3}$ * **Calculation (Option Verification):** Let's check the options to see which satisfies both conditions. * **Check Option A ($\frac{5}{6}$):** * Sum: $5 + 6 = 11$ (Condition 1 satisfied). * New fraction: $\frac{5 + 1}{6 - 2} = \frac{6}{4} = \frac{3}{2}$. This is not $\frac{2}{3}$. * **Check Option C ($\frac{3}{8}$):** (Note: Checking C before B because $\frac{3}{8}$ looks more standard as a proper fraction, though any order works). * Sum: $3 + 8 = 11$ (Condition 1 satisfied). * New fraction: $\frac{3 + 1}{8 - 2} = \frac{4}{6} = \frac{2}{3}$. * This perfectly satisfies both conditions. *(For completion, if we solved algebraically: From $n+d=11$, we get $d = 11-n$. Substitute into the second equation: $\frac{n+1}{11-n-2} = \frac{2}{3} \implies \frac{n+1}{9-n} = \frac{2}{3} \implies 3(n+1) = 2(9-n) \implies 3n+3 = 18-2n \implies 5n = 15 \implies n=3$. If $n=3$, then $d=11-3=8$. The fraction is $\frac{3}{8}$.)* ### Exam Strategy & Shortcut Use the "Sum" condition to eliminate options instantly. Option B ($\frac{6}{5}$) has sum $11$. Option D ($\frac{8}{3}$) has sum $11$. Wait, all options sum to $11$! So the first condition doesn't help eliminate. Proceed directly to the second condition mentally: "Add $1$ to top, subtract $2$ from bottom." Apply to A: $\rightarrow \frac{6}{4}$ (No) Apply to C: $\rightarrow \frac{4}{6}$ (Yes, simplifies to $\frac{2}{3}$). Done in 10 seconds. ### Common Pitfall When doing algebra, students might write the second equation incorrectly, such as $\frac{n+1}{d} - 2 = \frac{2}{3}$, misinterpreting "subtracted from the denominator" as subtracted from the whole fraction. Always ensure operations on numerators/denominators are properly grouped in the fraction. ### Final Answer **Therefore, the correct answer is $\frac{3}{8}$.**