$\frac{3}{5}$ of $\frac{4}{7}$ of $\frac{5}{9}$ of $\frac{21}{24}$ of $504 = x$

### Concept & Strategy When a problem strings together multiple fractions using the word "of", treat it as one continuous multiplication sequence. The most efficient approach is to simplify everything as a single combined fraction before attempting to multiply by the final whole number. ### Step-by-Step Solution * Given expression: $\frac{3}{5}$ of $\frac{4}{7}$ of $\frac{5}{9}$ of $\frac{21}{24}$ of $504$ * Convert all the "of" operators directly to multiplication: $\frac{3}{5} \times \frac{4}{7} \times \frac{5}{9} \times \frac{21}{24} \times 504$ * Isolate the continuous fraction part to cross-cancel and simplify: $\frac{3 \times 4 \times 5 \times 21}{5 \times 7 \times 9 \times 24}$ * Begin cancelling terms between the collective numerator and denominator: Cancel the $5$ from both the top and bottom. Simplify $\frac{4}{24}$ to $\frac{1}{6}$. Simplify $\frac{21}{7}$ to $3$. * The remaining consolidated fraction expression is: $\frac{3 \times 3}{9 \times 6}$ * Further simplify this: $\frac{9}{9 \times 6} = \frac{1}{6}$ * Finally, multiply this resulting fraction by the whole number $504$: $\frac{1}{6} \times 504 = \frac{504}{6} = 84$ * The correctly calculated answer is $84$, which is not listed in options (a), (b), (c), or (d). ### Exam Strategy & Shortcut Write out the entire multiplication string as a single tall fraction block. Visually cross out pairs immediately: strike through the $5$s. Strike through $3$ and $4$ on top against the $24$ on the bottom (since $3 \times 4 = 12$, and $12 / 24$ instantly reduces to $1/2$). Then simplify $21 / 7 = 3$. You are left with $(3 \times 1) / (9 \times 2) = 3 / 18 = 1/6$. From there, evaluating $1/6$ of $504$ yields $84$ in seconds. ### Common Pitfall Solving the sequence sequentially from left to right creates messy, unsimplified intermediate fractions (e.g., $12/35$). This heavily wastes time and drastically increases the chance of minor arithmetic errors. Always simplify globally across the entire chain instead of locally. ### Final Answer **Therefore, the correct answer is None of these.**