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Six machines at a certain factory operate at the same constant rate. If four of these machines, operating simultaneously, take 27 hours to fill a certain production order, how many fewer hours does it take all six machines, operating simultaneously, to fill the same production order?
Let's start by understanding what we're being asked to find. We have six identical machines, but only four of them are used initially. These four machines take 27 hours to complete a production order. We want to know: if we use all six machines instead, how many fewer hours will it take?
The key insight here is that all machines work at exactly the same rate - they're identical. This means we can think about this problem in terms of "machine-hours" of work needed.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding
Let's think about this step by step using everyday reasoning. If 4 machines working together take 27 hours, then the total amount of work can be measured in "machine-hours."
Total work = \(4 \text{ machines} \times 27 \text{ hours} = 108 \text{ machine-hours}\)
This means the production order requires exactly 108 machine-hours of work to complete, regardless of how we distribute those hours among the machines.
Think of it like this: if you have 4 people painting a fence and it takes them 27 hours, the "person-hours" of work needed is \(4 \times 27 = 108\). Whether you use 4 people or 6 people, you still need to do the same total amount of painting work.
Now that we know the total work required is 108 machine-hours, we can figure out how long it takes 6 machines to do this same work.
If 6 machines are working together, and we need 108 machine-hours total:
Time needed = Total work ÷ Number of machines
Time needed = \(108 \text{ machine-hours} \div 6 \text{ machines} = 18 \text{ hours}\)
This makes intuitive sense: more machines working on the same job means the job gets done faster.
The question asks how many fewer hours it takes with 6 machines compared to 4 machines.
Time with 4 machines = 27 hours
Time with 6 machines = 18 hours
Difference = \(27 - 18 = 9\) hours
So using all 6 machines takes 9 fewer hours than using just 4 machines.
The answer is 9 hours, which corresponds to choice A.
To verify: 4 machines take 27 hours (108 machine-hours total), 6 machines take 18 hours (also 108 machine-hours total), and the difference is \(27 - 18 = 9\) hours.
Students often miss that the question asks "how many fewer hours" rather than just "how many hours." This leads them to calculate only the time for 6 machines (18 hours) without finding the difference. They might select answer choice D (18) instead of the correct answer A (9).
2. Confusing work rate relationshipsSome students struggle with the inverse relationship between number of machines and time. They might incorrectly think that if 4 machines take 27 hours, then 6 machines would take more time (like \(27 \times \frac{6}{4} = 40.5\) hours), failing to recognize that more machines should complete the work faster.
3. Not recognizing the constant total work conceptStudents may not grasp that the total amount of work (measured in machine-hours) remains constant regardless of how many machines are used. Without this key insight, they cannot set up the proper equation to solve the problem systematically.
Even with the correct approach, students make computational mistakes such as: incorrectly calculating \(4 \times 27 = 108\), getting \(108 \div 6 = 18\) wrong, or making errors in the final subtraction \(27 - 18 = 9\).
2. Setting up incorrect proportionsStudents might set up the wrong ratio when comparing machines to time. For example, they could write \(\frac{4}{27} = \frac{6}{x}\) and solve to get \(x = 40.5\), not realizing this gives an incorrect relationship since they're treating machines and time as directly proportional rather than inversely proportional.
Students correctly calculate that 6 machines take 18 hours, but then select answer choice D (18) instead of recognizing they need to find the difference between 27 and 18 hours to get the final answer of 9 hours (choice A).
Step 1: Choose a smart number for total work
Let's say the total production order requires 108 units of work. This number is chosen because it's divisible by both 4 (number of machines in first scenario) and 6 (number of machines in second scenario), making our calculations clean.
Step 2: Find the work rate per machine
If 4 machines take 27 hours to complete 108 units of work:
• Total machine-hours = \(4 \text{ machines} \times 27 \text{ hours} = 108 \text{ machine-hours}\)
• Work rate per machine = \(108 \text{ units} \div 108 \text{ machine-hours} = 1 \text{ unit per machine-hour}\)
Step 3: Calculate time for 6 machines
With 6 machines working at 1 unit per machine-hour:
• Combined rate = 6 units per hour
• Time needed = \(108 \text{ units} \div 6 \text{ units per hour} = 18 \text{ hours}\)
Step 4: Find the difference
Difference in time = 27 hours (for 4 machines) - 18 hours (for 6 machines) = 9 hours
The smart number approach gives us the same answer: 9 hours fewer.