Working alone at its constant rate, machine K took 3 hours to produce 1/4 of the units produced last Friday....

1. Translate the problem requirements: Machine K alone produces \(\frac{1}{4}\) of Friday's total units in 3 hours. Then K and M together produce the remaining \(\frac{3}{4}\) of Friday's units in 6 hours. We need to find how long M alone would take to produce all of Friday's units.
2. Establish work rates using concrete units: Let Friday's total production be 1 unit, then determine each machine's rate in units per hour based on the given information.
3. Set up the combined work equation: Use the fact that when K and M work together, their rates add up to complete the remaining work in the given time.
4. Solve for M's rate and calculate total time: Find M's individual rate and determine how long it would take M alone to complete all the work.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what happened on Friday in simple terms:

• Machine K worked alone for 3 hours and produced \(\frac{1}{4}\) of all the units made that Friday
• Then Machine M joined in, and both machines worked together for 6 more hours to finish the remaining \(\frac{3}{4}\) of Friday's units
• We need to find: How long would Machine M alone take to produce ALL of Friday's units?

To make this concrete, let's say Friday's total production was 100 units. This means:

• Machine K alone produced 25 units in 3 hours
• Machines K and M together produced the remaining 75 units in 6 hours

Process Skill: TRANSLATE - Converting the problem's language into clear, measurable quantities

2. Establish work rates using concrete units

Now let's figure out how fast each machine works by thinking about their hourly production rates.

For Machine K working alone:
• It produced 25 units in 3 hours
• So Machine K's rate = \(25 \text{ units} \div 3 \text{ hours} = \frac{25}{3} \text{ units per hour}\)

To keep our math simple, let's work with the total as 1 unit instead of 100 units:
• Machine K produced \(\frac{1}{4}\) unit in 3 hours
• So Machine K's rate = \(\frac{1}{4} \div 3 = \frac{1}{12} \text{ units per hour}\)

This means if Machine K worked alone for a full day, it would produce \(\frac{1}{12}\) of the total each hour.

3. Set up the combined work equation

When both machines work together, their speeds add up - just like two people working together get more done than either person alone.

During the 6 hours when both machines worked together:
• They completed \(\frac{3}{4}\) of the total units
• Combined rate = \(\frac{3}{4} \div 6 \text{ hours} = \frac{1}{8} \text{ units per hour}\)

Since rates add when working together:
Machine K's rate + Machine M's rate = Combined rate
\(\frac{1}{12} + \text{Machine M's rate} = \frac{1}{8}\)

This equation tells us that Machine M's contribution plus Machine K's contribution equals their total output together.

4. Solve for M's rate and calculate total time

Now we can find Machine M's individual rate:

Machine M's rate = \(\frac{1}{8} - \frac{1}{12}\)

To subtract these fractions, we need a common denominator. The least common multiple of 8 and 12 is 24:
• \(\frac{1}{8} = \frac{3}{24}\)
• \(\frac{1}{12} = \frac{2}{24}\)

Machine M's rate = \(\frac{3}{24} - \frac{2}{24} = \frac{1}{24} \text{ units per hour}\)

This means Machine M produces \(\frac{1}{24}\) of the total units each hour.

To find how long Machine M would take to produce ALL the units (1 complete unit):
Time = \(\text{Total work} \div \text{Rate} = 1 \div \frac{1}{24} = 24 \text{ hours}\)

Process Skill: MANIPULATE - Working with fractions systematically to find the unknown rate

5. Final Answer

Machine M, working alone at its constant rate, would take 24 hours to produce all of the units produced last Friday.

Verification: Let's check our answer makes sense:
• Machine K's rate: \(\frac{1}{12}\) units/hour → would take 12 hours for all units
• Machine M's rate: \(\frac{1}{24}\) units/hour → would take 24 hours for all units
• Machine M is slower than Machine K, which makes sense since K alone did \(\frac{1}{4}\) in 3 hours

The answer is D. 24

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the work sequence and timing
Students often confuse the timeline: Machine K worked alone for 3 hours first, THEN Machine M joined for 6 more hours. A common error is thinking both machines worked together for the entire 9 hours, or that Machine M worked alone for some period. The phrase "Then machine M started working" clearly indicates a sequential process, not simultaneous work from the beginning.

2. Setting up incorrect work relationships
Many students struggle with the fractional relationships. They may incorrectly assume that since Machine K produced \(\frac{1}{4}\) of the units in 3 hours, it would take 12 hours to produce all units, and then incorrectly use this as a basis for comparison. The key insight they miss is that we need to find individual rates first, then use the combined work equation.

3. Confusing what the question is asking for
The question asks how long Machine M would take to produce ALL of Friday's units working alone. Students sometimes get confused and try to find how long Machine M took to produce just the remaining \(\frac{3}{4}\) of the units, or how long the combined work took, rather than Machine M's time for the complete job.

Errors while executing the approach

1. Fraction arithmetic errors when finding common denominators
When calculating Machine M's rate as \(\frac{1}{8} - \frac{1}{12}\), students frequently make errors finding the least common multiple of 8 and 12, or incorrectly convert to equivalent fractions. They might use 12 or 16 as the common denominator instead of 24, leading to wrong calculations like \(\frac{3}{12} - \frac{1}{12} = \frac{2}{12} = \frac{1}{6}\).

2. Incorrect rate calculations from work and time
Students often struggle with the rate formula (Work = Rate × Time, so Rate = Work ÷ Time). For Machine K, they might incorrectly calculate the rate as \(3 \div \frac{1}{4} = 12\) instead of \(\frac{1}{4} \div 3 = \frac{1}{12}\). This fundamental confusion between numerator and denominator leads to completely wrong rates.

3. Errors in setting up the combined work equation
When both machines work together producing \(\frac{3}{4}\) units in 6 hours, students may incorrectly calculate the combined rate. They might use \(6 \div \frac{3}{4}\) instead of \(\frac{3}{4} \div 6\), or forget that combined rate equals the sum of individual rates, leading to incorrect equations.

Errors while selecting the answer

1. Selecting Machine K's time instead of Machine M's time
After calculating that Machine K would take 12 hours alone and Machine M would take 24 hours alone, students sometimes select answer choice B (12) because they confuse which machine the question is asking about. The solution process involves finding both machines' capabilities, making this mix-up common.

2. Providing the reciprocal or incorrectly inverted final calculation
When students find Machine M's rate as \(\frac{1}{24}\) units per hour, they sometimes forget the final step of converting rate to total time. They might select a small number thinking the rate is the answer, or incorrectly calculate \(24 \div 1\) instead of \(1 \div \frac{1}{24}\), especially if they're rushed or unclear about what the rate represents.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a smart number for total production

Let's say the total units produced on Friday = 12 units (chosen because it's easily divisible by 4, making the fractions simple to work with)

Step 2: Calculate Machine K's rate

Machine K produces \(\frac{1}{4}\) of total units in 3 hours
Machine K produces: \(\frac{1}{4} \times 12 = 3\) units in 3 hours
Machine K's rate = \(3 \text{ units} \div 3 \text{ hours} = 1\) unit per hour

Step 3: Analyze the combined work phase

Remaining work after K's solo phase = \(\frac{3}{4} \times 12 = 9\) units
Machines K and M together complete these 9 units in 6 hours
Combined rate of K and M = \(9 \text{ units} \div 6 \text{ hours} = 1.5\) units per hour

Step 4: Find Machine M's rate

Since combined rate = K's rate + M's rate
\(1.5 = 1 + \text{M's rate}\)
M's rate = 0.5 units per hour

Step 5: Calculate time for M to produce all units

Time for M alone to produce all 12 units = \(12 \text{ units} \div 0.5 \text{ units per hour} = 24\) hours

Verification: Our smart number choice of 12 units worked perfectly because it made all calculations clean and avoided complex fractions throughout the solution process.