Machine K and Machine N, working simultaneously and independently at their respective constant rates, process 2/3 of the shipment of...

1. Translate the problem requirements: Two machines K and N working together complete \(\frac{2}{3}\) of a shipment in \(1.6\) hours, then machine N alone completes the remaining \(\frac{1}{3}\) in \(2\) hours. We need to find how long machine K alone would take for the entire shipment.
2. Determine machine N's individual rate: Use the fact that N completes \(\frac{1}{3}\) of the shipment in \(2\) hours to find N's rate per hour.
3. Find the combined rate of both machines: Use the fact that both machines together complete \(\frac{2}{3}\) of the shipment in \(1.6\) hours to determine their combined rate.
4. Calculate machine K's individual rate: Subtract machine N's rate from the combined rate to isolate machine K's rate.
5. Determine time for entire shipment: Use machine K's rate to calculate how long it would take to complete the full shipment working alone.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in plain English:

We have two machines, K and N, working on processing a chemical shipment. Think of this like two people working together to complete a job.

• First, both machines work together and complete \(\frac{2}{3}\) of the entire job in \(1.6\) hours
• Then machine K stops, and machine N works alone to finish the remaining \(\frac{1}{3}\) of the job in \(2\) hours
• We need to find: How long would it take machine K to do the entire job by itself?

The key insight is that we can use the information about how much work each machine does to figure out their individual work rates.

Process Skill: TRANSLATE - Converting the problem's language into clear mathematical relationships

2. Determine machine N's individual rate

Let's start with what we know for certain about machine N working alone.

Machine N completes \(\frac{1}{3}\) of the shipment in \(2\) hours when working by itself.

In everyday terms: if machine N can do \(\frac{1}{3}\) of the job in \(2\) hours, then in \(1\) hour it does half of that amount.

So machine N's rate = \(\frac{1}{3} \div 2 = \frac{1}{6}\) of the job per hour

This means machine N completes \(\frac{1}{6}\) of the entire shipment every hour when working alone.

3. Find the combined rate of both machines

Now let's figure out how fast both machines work when they're working together.

Both machines together complete \(\frac{2}{3}\) of the shipment in \(1.6\) hours.

Combined rate = \(\frac{2}{3} \div 1.6\)

Let's convert \(1.6\) to a fraction to make our calculations cleaner: \(1.6 = \frac{16}{10} = \frac{8}{5}\)

Combined rate = \(\frac{2}{3} \div \frac{8}{5} = \frac{2}{3} \times \frac{5}{8} = \frac{10}{24} = \frac{5}{12}\) of the job per hour

So when working together, the machines complete \(\frac{5}{12}\) of the entire shipment every hour.

4. Calculate machine K's individual rate

Here's the key insight: when two machines work together, their combined rate equals the sum of their individual rates.

Combined rate = Machine K's rate + Machine N's rate

We know:
• Combined rate = \(\frac{5}{12}\) of the job per hour
• Machine N's rate = \(\frac{1}{6}\) of the job per hour

So: Machine K's rate = Combined rate - Machine N's rate
Machine K's rate = \(\frac{5}{12} - \frac{1}{6}\)

To subtract these fractions, we need a common denominator. Since \(12\) is a multiple of \(6\):
\(\frac{1}{6} = \frac{2}{12}\)

Machine K's rate = \(\frac{5}{12} - \frac{2}{12} = \frac{3}{12} = \frac{1}{4}\) of the job per hour

This means machine K completes \(\frac{1}{4}\) of the entire shipment every hour when working alone.

5. Determine time for entire shipment

Now we can answer the main question: How long would it take machine K to complete the entire shipment working alone?

If machine K completes \(\frac{1}{4}\) of the job per hour, then to complete the entire job (\(1\) whole job):

Time = \(1 \div \frac{1}{4} = 1 \times 4 = 4\) hours

Let's verify this makes sense: If machine K takes \(4\) hours for the whole job, then in \(1.6\) hours it would complete \(\frac{1.6}{4} = 0.4 = \frac{2}{5}\) of the job. Machine N would complete \(1.6 \times \frac{1}{6} = \frac{1.6}{6} = \frac{4}{15}\) of the job. Together: \(\frac{2}{5} + \frac{4}{15} = \frac{6}{15} + \frac{4}{15} = \frac{10}{15} = \frac{2}{3}\) ✓

Final Answer

Machine K would take \(4.0\) hours to process the entire shipment working alone.

The answer is B. \(4.0\)

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the work sequence

Students often confuse which machine does what work in which time period. They might think both machines work together for the entire job, or misunderstand that Machine N works alone for the final portion. This leads to setting up incorrect equations from the start.

2. Confusion about what fraction represents the remaining work

Students may incorrectly assume that after \(\frac{2}{3}\) of work is completed together, Machine N completes \(\frac{2}{3}\) of the remaining work (instead of the remaining \(\frac{1}{3}\)) in \(2\) hours. This fundamental misreading changes the entire problem setup.

3. Setting up rate equations incorrectly

Students often struggle with the concept that combined rate equals sum of individual rates. They might try to multiply the rates or use other incorrect relationships, leading to wrong mathematical models.

Errors while executing the approach

1. Fraction arithmetic errors

When calculating Machine N's rate (\(\frac{1}{3} \div 2 = \frac{1}{6}\)) or the combined rate (\(\frac{2}{3} \div \frac{8}{5} = \frac{5}{12}\)), students frequently make mistakes with fraction division, especially when converting decimals like \(1.6\) to fractions.

2. Common denominator mistakes

When subtracting fractions to find Machine K's rate (\(\frac{5}{12} - \frac{1}{6}\)), students often struggle with finding common denominators or make errors in the subtraction process, leading to incorrect individual rates.

3. Unit conversion confusion

Students may forget to convert \(1.6\) hours properly to a fraction (\(\frac{8}{5}\)) or make errors in this conversion, which propagates through all subsequent calculations.

Errors while selecting the answer

1. Verification step oversight

Students might arrive at the correct answer of \(4\) hours but fail to verify their solution by checking if the work rates add up correctly for the given time periods, missing the opportunity to catch potential errors.

2. Reciprocal confusion

After finding Machine K's rate as \(\frac{1}{4}\) job per hour, students sometimes mistakenly select \(\frac{1}{4} = 0.25\) as the answer instead of taking the reciprocal (\(4\) hours) to find the time for the complete job.