The difference between $\frac{3}{4}$th of $\frac{4}{5}$th of a number and $\frac{1}{6}$th of $\frac{2}{5}$th of the same number is 648. What is the number?

### Concept & Formula This problem involves finding an unknown number by translating fractional phrases into algebraic operations. The word "of" indicates multiplication. $$\text{First Fractional Product} - \text{Second Fractional Product} = 648$$ ### Step-by-Step Solution * **Define the variable:** Let the unknown number be $x$. * **Translate the operations:** First part: $\frac{3}{4} \times \frac{4}{5} \times x = \frac{3}{5}x$ Second part: $\frac{1}{6} \times \frac{2}{5} \times x = \frac{1}{15}x$ * **Set up the linear equation:** $$\frac{3}{5}x - \frac{1}{15}x = 648$$ * **Solve for x:** Find the common denominator (15): $$\frac{9}{15}x - \frac{1}{15}x = 648$$ $$\frac{8}{15}x = 648$$ $$x = \frac{648 \times 15}{8}$$ Divide 648 by 8: $648 \div 8 = 81$. $$x = 81 \times 15 = 1215$$ ### Exam Strategy & Shortcut Simplify fractions immediately before setting up the equation. Notice that $\frac{8}{15}x = 648$. To find the final answer quickly without complete scratch calculation, analyze the unit digits: $81 \times 15$ must end in a 5 ($1 \times 5 = 5$). Looking at the options, 1215 matches option (b). ### Common Pitfall Students sometimes read the prompt incorrectly and subtract the numbers in the wrong order or forget to multiply the individual fractions before calculating their differences. Always group fractions associated with a single term using parenthesis if necessary. ### Final Answer **Therefore, the correct answer is 1215.**