Running at the same constant rate, 6 identical machines can produce a total of 270 bottles per minute. At this rate, how many bottles could 10 such machines produce in 4 minutes?

Introduction / Context:
This question tests the concept of proportionality and the chain rule in the context of production rates. Machines working at a constant rate produce a certain number of bottles per minute, and we are asked to find how many bottles will be produced when the number of machines and the time both change. This type of problem appears frequently in aptitude examinations to check reasoning about direct proportion.

Given Data / Assumptions:

• 6 identical machines together produce 270 bottles per minute.
• All machines have the same constant rate of production.
• We need the number of bottles produced by 10 machines running for 4 minutes at the same per machine rate.
• There is no downtime or change in efficiency during the production period.

Concept / Approach:
If 6 machines produce 270 bottles per minute, we can first find the production rate of one machine. Since the relationship is directly proportional, we then multiply by the new number of machines (10) to find the total bottles per minute for the second scenario. Finally, we multiply by the total time in minutes, which is 4, to get the total number of bottles produced in that duration.

Step-by-Step Solution:
Production by 6 machines in 1 minute = 270 bottles Production by 1 machine in 1 minute = 270 / 6 = 45 bottles Production by 10 machines in 1 minute = 10 * 45 = 450 bottles Time given = 4 minutes Total production in 4 minutes = 450 * 4 450 * 4 = 1800 bottles Therefore, 10 machines will produce 1800 bottles in 4 minutes

Verification / Alternative check:
We can also use a direct proportion approach without explicitly finding the single machine rate. The number of bottles is proportional to both the number of machines and the time. So, New bottles = 270 * (10 / 6) * 4 10 / 6 simplifies to 5 / 3 New bottles = 270 * (5 / 3) * 4 270 * (5 / 3) = 270 * 5 / 3 = 90 * 5 = 450 Then 450 * 4 = 1800 bottles This confirms the earlier result.

Why Other Options Are Wrong:
648 bottles is too low and would correspond to ignoring the full 4 minutes or misusing the ratios. 2700 bottles and 10800 bottles are far too high compared to the correct proportional scaling of the original scenario. 1350 bottles is somewhat close but still lower than the correct result and arises from incomplete multiplication of the machine or time factor. Only 1800 bottles match the proportional reasoning based on the given data.

Common Pitfalls:
A common pitfall is to scale only the number of machines and forget to scale the time, or vice versa. Some students mistakenly multiply 270 directly by 4 without adjusting for the increase in machine count. Another error is confusing direct proportion with inverse proportion. Remember that more machines and more time both increase total production in a directly proportional way, which is why we multiply by both factors.

Final Answer:
At the given constant rate, 10 machines running for 4 minutes will produce 1800 bottles.