A machine can put c caps on bottles in m minutes at a constant working rate. Assuming the same rate is maintained, how many hours will the machine take to put caps on b bottles in total?

Introduction / Context:
This question tests basic proportional reasoning involving rates of work and unit conversion from minutes to hours. The machine's rate of putting caps on bottles is given, and we must calculate how long it will take to cap a different number of bottles, expressing the result in hours.

Given Data / Assumptions:

• The machine puts c caps on bottles in m minutes.
• Its working rate is constant.
• We need the time required to put caps on b bottles.
• The final answer should be in hours, not minutes.

Concept / Approach:
First, we determine the machine's rate of work in bottles per minute. Then we use that rate to compute the total time in minutes for b bottles. Finally, we convert this time from minutes to hours by dividing by 60. Working symbolically allows a general algebraic expression in terms of b, c and m.

Step-by-Step Solution:
Step 1: Find the rate of work in bottles per minute. Since c caps are put in m minutes, the rate = c / m caps per minute. Step 2: Time in minutes to put caps on b bottles. Required caps = b, rate = c / m, so time in minutes = b / (c / m) = b * m / c. Step 3: Convert minutes to hours. There are 60 minutes in an hour, so time in hours = (b * m / c) / 60. Simplify: time = bm / (60c) hours. Thus the required time in hours is bm / 60c.

Verification / Alternative check:
As a simple check, suppose c = 30 caps in m = 10 minutes. Then the rate is 3 caps per minute. For b = 90 bottles, time is 90 / 3 = 30 minutes, which is 30 / 60 = 0.5 hours. Using the formula bm / 60c gives 90 * 10 / (60 * 30) = 900 / 1800 = 0.5 hours, confirming the expression is correct.

Why Other Options Are Wrong:
Expression 60bm/c multiplies by 60 instead of dividing, giving a value in minutes squared per unit, which is dimensionally incorrect. The forms bc/60m and 60b/cm invert the role of m and c or misplace the 60 factor, leading to wrong units and incorrect times. The option bm/c gives the time in minutes, not hours, and therefore misses the required conversion factor of 1/60.

Common Pitfalls:
Learners often forget to convert units at the final step or confuse which quantity should be in the numerator or denominator when working with rates. Always remember: time = work / rate, and converting minutes to hours requires dividing by 60, not multiplying.

Final Answer:
The machine will take bm / 60c hours to put caps on b bottles.