21 binders can bind 1400 books in 15 days. How many binders will be required to bind 800 books in 20 days?

### Concept & Chain Rule Formula This problem requires the fundamental Chain Rule (or unitary method) relating Men (binders), Days, and Work (books). $$ \frac{M_1 \times D_1}{W_1} = \frac{M_2 \times D_2}{W_2} $$ ### Step-by-Step Solution 1. **Identify the variables:** - Initial scenario: $M_1 = 21$ binders, $W_1 = 1400$ books, $D_1 = 15$ days. - Target scenario: $W_2 = 800$ books, $D_2 = 20$ days. Let the required binders be $M_2$. 2. **Apply the formula:** - $\frac{21 \times 15}{1400} = \frac{M_2 \times 20}{800}$ 3. **Simplify the equation:** - Simplify the denominators by dividing both by 200: - $\frac{21 \times 15}{7} = \frac{M_2 \times 20}{4}$ - $3 \times 15 = M_2 \times 5$ - $45 = 5 \times M_2$ 4. **Solve for $M_2$:** - $M_2 = \frac{45}{5} = 9$ ### Exam Strategy & Shortcut You can set up a direct proportion chain: Binders needed = Initial Binders $\times$ (Ratio of Books) $\times$ (Inverse Ratio of Days). $M_2 = 21 \times (\frac{800}{1400}) \times (\frac{15}{20}) = 21 \times \frac{4}{7} \times \frac{3}{4} = 3 \times 3 = 9$. ### Common Pitfall A frequent error is misplacing the "Work" parameter in the numerator instead of the denominator. Always remember that work done is inversely proportional to the time and effort required. ### Final Answer Therefore, the correct answer is **9**.