Eighteen binders can bind 900 books in 10 days. How many binders will be required to bind 660 books in 12 days, assuming each binder works at the same rate?

Introduction / Context:
This work and time problem asks how many workers are required to complete a smaller amount of work over a longer period when the rate of each worker is constant. The situation involves binders and books instead of generic workers and tasks, but the mathematics is the same. Understanding proportional relationships makes such questions quite manageable.

Given Data / Assumptions:

• Eighteen binders can bind 900 books in 10 days.
• Each binder works at a constant daily rate.
• We need to find the number of binders required to bind 660 books in 12 days.
• The nature of the books and binding work does not change between the two scenarios.

Concept / Approach:
First, we determine the rate at which one binder works, expressed as books per binder per day. Using this rate, we can find how many binder days are needed for 660 books and then divide by the available 12 days to get the required number of binders. This solution uses the unitary method and basic proportional reasoning.

Step-by-Step Solution:
Step 1: Compute the total binder days in the first situation: 18 binders * 10 days = 180 binder days to bind 900 books.Step 2: Find the binding rate of one binder per day. One binder day therefore corresponds to 900 / 180 = 5 books per binder per day.Step 3: For 660 books, determine total binder days needed. Since each binder day handles 5 books, total binder days required are 660 / 5 = 132 binder days.Step 4: Let N be the number of binders required to finish 660 books in 12 days. Then N * 12 = 132 binder days.Step 5: Solve for N: N = 132 / 12 = 11 binders.
Verification / Alternative check:
An alternative method is to use direct proportion. The amount of work in the second scenario as a fraction of the first is 660 / 900 = 11 / 15. The time available changes from 10 days to 12 days. Since work is proportional to men * days, we have men2 = men1 * (work2 / work1) * (time1 / time2) = 18 * (11 / 15) * (10 / 12). Simplifying, we get 18 * 11 / 15 * 10 / 12 = 11, which matches the unitary method result.

Why Other Options Are Wrong:
Twenty two binders would double the workforce, providing 22 * 12 = 264 binder days, which is much more than the 132 binder days needed.Fourteen binders would yield 14 * 12 = 168 binder days, more than necessary and therefore not minimal.Thirteen binders would supply 13 * 12 = 156 binder days, again exceeding the required 132 binder days.
Common Pitfalls:
Students sometimes mistakenly treat the relationship as direct without accounting for both the change in number of books and the change in days. Others may attempt to plug values into an incorrect chain rule formula without seeing the simple logic behind binder days. Focusing on the basic idea that total work equals rate times time helps structure the reasoning and avoid calculation errors.

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