Five machines at a certain factory operate at the same constant rate. If four of these machines, operating simultaneously, take...

1. Translate the problem requirements: We need to find the difference in time between using 4 machines versus 5 machines for the same production order. All machines work at identical constant rates.
2. Establish the work rate relationship: Use the given information about 4 machines taking 30 hours to determine how much work one machine can do.
3. Calculate time for 5 machines: Apply the inverse relationship between number of machines and time needed to find how long 5 machines take.
4. Find the time difference: Subtract the time for 5 machines from the time for 4 machines to get the answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're being asked to find. We have five identical machines, but we're comparing two scenarios:

• Scenario 1: 4 machines working together take 30 hours to complete a production order
• Scenario 2: All 5 machines working together to complete the same production order

We need to find how many fewer hours it takes when we use all 5 machines instead of just 4 machines. This means we'll calculate the time for 5 machines, then subtract that from 30 hours.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical requirements

2. Establish the work rate relationship

Let's think about this intuitively. If 4 machines working together can finish the job in 30 hours, we can figure out how much work gets done per hour.

Think of it this way: the total amount of work needed is like filling a big container. When 4 machines work for 30 hours, they completely fill this container.

So the total work = \(4 \text{ machines} \times 30 \text{ hours} = 120 \text{ machine-hours}\) of work

This tells us that no matter how we arrange the machines, we always need exactly 120 machine-hours to complete this production order.

3. Calculate time for 5 machines

Now we know the total work required is 120 machine-hours. If we use all 5 machines working simultaneously, we can find the time needed:

\(\text{Time needed} = \text{Total work} \div \text{Number of machines}\)
\(\text{Time needed} = 120 \text{ machine-hours} \div 5 \text{ machines} = 24 \text{ hours}\)

This makes intuitive sense: more machines working together means the job gets done faster. With 5 machines instead of 4, we have 25% more working power, so the job should take less time.

4. Find the time difference

Now we can answer the original question. We need to find how many fewer hours it takes with 5 machines compared to 4 machines:

\(\text{Time difference} = \text{Time with 4 machines} - \text{Time with 5 machines}\)
\(\text{Time difference} = 30 \text{ hours} - 24 \text{ hours} = 6 \text{ hours}\)

Final Answer

Using all 5 machines takes 6 fewer hours than using just 4 machines.

The answer is (C) 6.

Common Faltering Points

Errors while devising the approach

• Misinterpreting "how many fewer hours": Students may misread the question and think they need to find just the time for 5 machines, rather than the difference between the time for 4 machines and 5 machines. This leads them to calculate only one scenario instead of comparing both.
• Confusion about the work rate concept: Students might struggle with the idea that total work remains constant regardless of how many machines are used. They may incorrectly think that using more machines means more total work needs to be done, rather than understanding that the same amount of work gets distributed among more machines.
• Setting up incorrect proportional relationships: Instead of using the machine-hours concept, students might try to set up direct proportions incorrectly, such as thinking "4 machines : 30 hours = 5 machines : x hours" without properly understanding the inverse relationship between number of machines and time required.

Errors while executing the approach

• Arithmetic errors in basic calculations: Students may make simple computational mistakes when calculating \(120 \div 5 = 24\), or when subtracting \(30 - 24 = 6\), especially under time pressure during the exam.
• Incorrect calculation of total work: Students might miscalculate the total machine-hours as something other than \(4 \times 30 = 120\), possibly by adding instead of multiplying, or by making other basic arithmetic errors in this crucial step.

Errors while selecting the answer

• Selecting the time for 5 machines instead of the difference: After correctly calculating that 5 machines take 24 hours, students might select answer choice (E) 24, forgetting that the question asks for "how many fewer hours" rather than the absolute time for 5 machines.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a smart number for total production work

Let's assign the total production order a value of 60 units of work. This number is chosen because it's divisible by both 4 and 5 (the number of machines), making our calculations clean.

Step 2: Calculate the rate of work per machine

If 4 machines take 30 hours to complete 60 units of work:
• Total machine-hours = \(4 \text{ machines} \times 30 \text{ hours} = 120 \text{ machine-hours}\)
• Rate per machine = \(60 \text{ units} \div 120 \text{ machine-hours} = 0.5 \text{ units per machine per hour}\)

Step 3: Calculate time needed for 5 machines

With 5 machines working simultaneously:
• Combined rate = \(5 \text{ machines} \times 0.5 \text{ units per machine per hour} = 2.5 \text{ units per hour}\)
• Time needed = \(60 \text{ units} \div 2.5 \text{ units per hour} = 24 \text{ hours}\)

Step 4: Find the difference

Time difference = \(30 \text{ hours} - 24 \text{ hours} = 6 \text{ hours}\)

Why this smart number works: By choosing 60 as our total work units, we created clean divisions that make the arithmetic straightforward while maintaining the same proportional relationships as the original problem.