A student was asked to solve the fraction $$\frac{\frac{7}{3} + 1\frac{1}{2} \text{ of } \frac{5}{3}}{2 + 1\frac{2}{3}}$$ and his answer was $\frac{1}{4}$. By how much was his answer wrong?

### Concept & Strategy First, compute the correct value of the given complex fraction using the standard **BODMAS** rules. Then, find the absolute difference between the correct value and the student's incorrect answer ($\frac{1}{4}$) to determine by how much it was wrong. ### Step-by-Step Solution * **Step 1: Simplify the Numerator** $$\text{Numerator} = \frac{7}{3} + 1\frac{1}{2} \text{ of } \frac{5}{3}$$ Convert the mixed fraction: $1\frac{1}{2} = \frac{3}{2}$ $$\text{Numerator} = \frac{7}{3} + \left(\frac{3}{2} \times \frac{5}{3}\right) = \frac{7}{3} + \frac{5}{2}$$ Find the common denominator (6): $$\text{Numerator} = \frac{14 + 15}{6} = \frac{29}{6}$$ * **Step 2: Simplify the Denominator** $$\text{Denominator} = 2 + 1\frac{2}{3} = 2 + \frac{5}{3} = \frac{6 + 5}{3} = \frac{11}{3}$$ * **Step 3: Calculate the Correct Fraction Value** $$\text{Correct Value} = \frac{\frac{29}{6}}{\frac{11}{3}} = \frac{29}{6} \times \frac{3}{11} = \frac{29}{2 \times 11} = \frac{29}{22}$$ * **Step 4: Find the Difference** $$\text{Difference} = \text{Correct Value} - \text{Student's Answer}$$ $$\text{Difference} = \frac{29}{22} - \frac{1}{4}$$ The LCM of 22 and 4 is 44: $$\text{Difference} = \frac{29 \times 2 - 1 \times 11}{44} = \frac{58 - 11}{44} = \frac{47}{44} = 1\frac{3}{44}$$ Looking at the choices, none of (a), (b), or (c) match $\frac{47}{44}$. Thus, the correct choice is (d). ### Exam Strategy & Shortcut Always solve numerator and denominator separately in layered fractions to minimize computational errors. Since the correct value is greater than 1 ($\frac{29}{22} > 1$), subtracting $\frac{1}{4}$ will yield a value around $1.32 - 0.25 = 1.07$, which instantly eliminates options (b) and (c). ### Common Pitfall Students often calculate the difference in reverse order or misinterpret "by how much was his answer wrong" as a percentage error rather than an absolute difference. ### Final Answer **Therefore, the correct answer is None of these.**