Two-fifth of one-fourth of three-seventh of a number is 15. What is half of that number?

### Concept & Formula Translate the word problem into an algebraic equation. "Of" represents multiplication. Let the unknown number be $x$. The statement can be written as: $$\frac{2}{5} \times \frac{1}{4} \times \frac{3}{7} \times x = 15$$ Once $x$ is found, calculate $\frac{x}{2}$. ### Step-by-Step Solution * **Set up the equation:** $$\left(\frac{2}{5} \times \frac{1}{4} \times \frac{3}{7}\right) \times x = 15$$ * **Simplify the fraction coefficient:** $$\frac{2 \times 1 \times 3}{5 \times 4 \times 7} \times x = 15 \implies \frac{6}{140} \times x = 15 \implies \frac{3}{70} \times x = 15$$ * **Solve for $x$:** $$x = 15 \times \frac{70}{3} = 5 \times 70 = 350$$ * **Find half of the number:** $$\text{Required Value} = \frac{350}{2} = 175$$ * **Match with options:** Since 175 is not listed in options (a), (b), (c), or (d), the answer is "None of these". ### Exam Strategy & Shortcut Instead of finding $x$ completely, solve directly for $\frac{x}{2}$: $$\frac{3}{70} \times x = 15 \implies \frac{3}{35} \times \left(\frac{x}{2}\right) = 15 \implies \frac{x}{2} = 15 \times \frac{35}{3} = 5 \times 35 = 175$$ Quickly noticing that 175 is missing lets you confidently pick option (e) within seconds. ### Common Pitfall Forgetting to divide by 2 at the end. Students frequently calculate the full number (350), notice it's not there, or mistakenly pick a close option if they made an arithmetic slip. Always re-read what the final question asks for. ### Final Answer **Therefore, the correct answer is None of these.**