Three machines operating independently, simultaneously, and at the same constant rate can fill a certain production order in 36 hours....

1. Translate the problem requirements: We need to understand that "operating independently, simultaneously, and at the same constant rate" means each machine contributes equally to the work, and we're looking for the time difference between using 3 machines versus 4 machines for the same job.
2. Establish the work-rate relationship: Determine how much work each individual machine can complete and use this to find the total work required.
3. Calculate time with additional machine: Find how long 4 machines would take to complete the same amount of work.
4. Find the time difference: Subtract the 4-machine time from the 3-machine time to get the answer.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this problem is telling us in plain English:

• We have 3 machines that work at the same speed
• When all 3 work together, they complete a job in 36 hours
• We want to know: if we add 1 more machine (making it 4 total), how much time would we save?

The key insight here is that "operating independently, simultaneously, and at the same constant rate" simply means each machine does an equal share of the work. Think of it like having 3 people painting a fence together - each person paints the same amount.

Process Skill: TRANSLATE - Converting the problem language into a clear understanding of what we need to find

2. Establish the work-rate relationship

Let's think about this step by step using everyday reasoning:

If 3 machines working together complete the job in 36 hours, then each machine is doing \(\frac{1}{3}\) of the total work.

Now here's the key insight: if each machine works for 36 hours and does \(\frac{1}{3}\) of the job, then one single machine would need \(3 \times 36 = 108\) hours to complete the entire job by itself.

We can think of the "total amount of work" as 108 machine-hours. This means:

• \(3 \text{ machines} \times 36 \text{ hours} = 108 \text{ machine-hours}\) of work
• This represents the complete job

Technically, we can say: Total Work = 108 machine-hours

3. Calculate time with additional machine

Now let's figure out how long 4 machines would take to do this same job:

We know the total work needed is 108 machine-hours.
If we have 4 machines working together, each contributing equally:

Time needed = Total work ÷ Number of machines
Time needed = \(108 \text{ machine-hours} \div 4 \text{ machines} = 27 \text{ hours}\)

This makes intuitive sense: more machines working together means the job gets done faster!

4. Find the time difference

Finally, we need to find how many fewer hours it would take:

Original time (3 machines) = 36 hours
New time (4 machines) = 27 hours

Time saved = \(36 - 27 = 9\) hours

Final Answer

The production order could be fulfilled in 9 fewer hours with the additional machine.

Verifying against the answer choices: This matches choice B: 9.

The answer is B.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "operating independently, simultaneously"

Students often get confused by this phrase and think the machines work separately rather than together. They might calculate how long ONE machine takes (36 hours) instead of recognizing that THREE machines working together take 36 hours. This leads to completely incorrect setup.

2. Confusing rate and time relationships

Students may struggle with the inverse relationship between number of machines and time. They might incorrectly think that adding one more machine means the time increases proportionally (like \(36 \times \frac{4}{3}\)) rather than decreases proportionally (\(36 \times \frac{3}{4}\)).

3. Setting up incorrect work equations

Some students try to solve this using complex rate equations with variables instead of using the simpler "total work = rate × time" concept. They might define individual machine rates as variables unnecessarily, making the problem more complicated than needed.

Errors while executing the approach

1. Arithmetic errors in basic calculations

Students might make simple calculation mistakes like \(3 \times 36 = 118\) instead of 108, or \(108 \div 4 = 24\) instead of 27. These errors cascade through the entire solution.

2. Incorrectly applying the work formula

Even with the right approach, students might confuse which numbers to multiply or divide. For example, calculating \(36 \div 4 = 9\) directly instead of first finding the total work (108) and then dividing by 4 machines.

3. Using wrong proportional relationships

Students might set up the proportion incorrectly as \(\frac{3}{36} = \frac{4}{x}\) and solve for x, getting x = 48, which leads them to think 4 machines take 48 hours instead of 27 hours.

Errors while selecting the answer

1. Selecting the time for 4 machines instead of the time saved

The most common error here is calculating correctly that 4 machines take 27 hours, but then selecting 27 (choice D) as the final answer instead of recognizing that the question asks for "how many fewer hours" which is \(36 - 27 = 9\).

2. Confusing the direction of time difference

Students might calculate \(27 - 36 = -9\) and then select 9, or get confused about whether they should subtract the smaller from larger or vice versa, especially if they misread what the question is asking for.

Alternate Solutions

Smart Numbers Approach

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 3 (number of original machines) and 4 (number of machines with the addition), making our calculations clean.

Step 2: Calculate individual machine rate
If 3 machines complete 108 units in 36 hours:
Total work rate of 3 machines = \(108 \div 36 = 3\) units per hour
Individual machine rate = \(3 \div 3 = 1\) unit per hour

Step 3: Calculate time with 4 machines
With 4 machines working at 1 unit per hour each:
Combined rate of 4 machines = \(4 \times 1 = 4\) units per hour
Time to complete 108 units = \(108 \div 4 = 27\) hours

Step 4: Find the time difference
Original time with 3 machines: 36 hours
New time with 4 machines: 27 hours
Time saved = \(36 - 27 = 9\) hours

Why this smart number works:
108 was strategically chosen because it's the LCM of the key numbers in our problem (36, 3, and 4), ensuring all our divisions result in whole numbers and making the arithmetic straightforward.