Eighteen binders can bind 900 books in 10 days. Assuming they all work at the same constant rate, how many binders will be required to bind 660 books in 12 days under similar working conditions?

Introduction / Context:
This is a straightforward chain rule or work-rate question. It tests your ability to use the proportionality between number of workers, time, and amount of work. Given one scenario of binders binding books in a certain time, you must compute how many workers are needed in a different scenario.

Given Data / Assumptions:
- 18 binders bind 900 books in 10 days.
- All binders work at the same constant rate.
- We need the number of binders required to bind 660 books in 12 days.
- Work (books bound) is directly proportional to (number of binders) * (number of days).

Concept / Approach:
We can compute the daily output per binder, or directly use proportional reasoning. First we find how many books are bound per binder per day. Then we use that rate to determine how many binders are needed to achieve the target (660 books) in the specified time (12 days).

Step-by-Step Solution:
Step 1: Total binder-days in the original scenario = 18 * 10 = 180 binder-days.Step 2: Total books bound = 900, so books per binder-day = 900 / 180 = 5 books.Step 3: For the new scenario, we need to bind 660 books in 12 days. Let the required number of binders be N.Step 4: Each binder in one day binds 5 books, so in 12 days one binder binds 5 * 12 = 60 books.Step 5: Therefore, N binders in 12 days will bind N * 60 books.Step 6: Set this equal to the required 660 books: N * 60 = 660 → N = 660 / 60 = 11.

Verification / Alternative check:
We can also use direct proportion with a chain rule: required binders = 18 * (900 / 660) * (10 / 12)⁻¹, simplifying to the same value. However, the unitary method using binder-days and books per binder-day is usually more intuitive and less error-prone.

Why Other Options Are Wrong:
Values slightly above or below 11, like 13 or 14, come from arithmetic mistakes such as miscomputing 900 / 180 or 660 / 60. A high value like 22 would imply too many binders relative to the work, and they would finish much earlier than 12 days at the same rate.

Common Pitfalls:
Errors include confusing proportional relationships, for example multiplying when you should divide, or mixing up the 10 and 12 days when scaling. Some students also forget that more time available means fewer binders are needed, not more. Keeping track of units (books, binder-days, days) helps avoid such mistakes.

Final Answer:
11 binders are required to bind 660 books in 12 days.