Binder requirement from books–men–days proportion: If 18 binders can bind 900 books in 10 days at a steady pace, how many binders are needed to bind 660 books in 12 days, assuming identical efficiency and conditions?

Introduction / Context:
Total output in such problems is proportional to the product of workers and time when individual productivity is constant. We compare two scenarios—one known and one desired—to find the unknown headcount using direct proportion across books, men, and days.

Given Data / Assumptions:

• Scenario 1: 18 binders, 10 days, 900 books
• Scenario 2: ? binders, 12 days, 660 books
• Productivity per binder per day remains constant.

Concept / Approach:
Compute per-binder-per-day output from the first scenario; then solve for the number of binders needed in the second scenario. Alternatively, set up a proportion equating men * days * rate to total books.

Step-by-Step Solution:
Per binder per day = 900 / (18 * 10) = 900 / 180 = 5 booksLet m be binders required: 660 = m * 12 * 5660 = 60m → m = 11

Verification / Alternative check:
Reverse check: 11 binders * 12 days * 5 books/day = 660 books, matching the target exactly.

Why Other Options Are Wrong:
21 and 18 overstate the workforce; 14 is unnecessary; 12 is close but still exceeds what is required at the given productivity.

Common Pitfalls:
Forgetting to multiply by days in the second scenario, or mixing “books per day” with “books per binder per day.”

Final Answer:
11