$\frac{1}{(2 \frac{1}{3})} + \frac{1}{(1 \frac{3}{4})}$ is equal to

### Concept & Strategy The key insight is converting mixed fractions to improper fractions before taking their reciprocals. Remember that the reciprocal of a fraction is determined by flipping its numerator and denominator. $$ \text{Reciprocal of } \frac{a}{b} = \frac{b}{a} $$ ### Step-by-Step Solution * Convert the mixed fractions in the denominators into improper fractions: $2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$ $1 \frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}$ * Substitute these improper fractions back into the original expression: $\frac{1}{\frac{7}{3}} + \frac{1}{\frac{7}{4}}$ * Take the reciprocal of each fraction to bring them to the numerator: $\frac{3}{7} + \frac{4}{7}$ * Since the denominators are identically $7$, simply add the numerators together: $\frac{3 + 4}{7} = \frac{7}{7} = 1$ * The calculated answer is $1$, which does not match options (a), (b), or (c). ### Exam Strategy & Shortcut When dealing with expressions structured as "1 over a mixed fraction," immediately convert the mixed fraction to an improper fraction and flip it. Mental math: $2 \frac{1}{3}$ becomes $\frac{7}{3}$, flipped is $\frac{3}{7}$. $1 \frac{3}{4}$ becomes $\frac{7}{4}$, flipped is $\frac{4}{7}$. Adding them gives $\frac{7}{7} = 1$. The entire problem can be solved mentally without writing down a single equation. ### Common Pitfall Students sometimes incorrectly evaluate the reciprocal of a mixed fraction by only flipping the fractional part (for example, assuming $\frac{1}{2 \frac{1}{3}}$ is $2 \frac{3}{1}$). Always convert the entire number to a pure improper fraction before applying the reciprocal. ### Final Answer **Therefore, the correct answer is None of these.**