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Machine K and Machine N, working simultaneously and independently at their respective constant rates, process \(\frac{2}{3}\) of the shipment of a certain chemical product in \(1.6\) hours. Then machine K stopped working, and Machine N, working alone at it's constant rate, processed the rest of the shipment in \(2\) hours. How many hours would it have taken Machine K, working alone at it's constant rate, to process the entire shipment?
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
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.
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.
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.
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: \(\frac{2}{5} + \frac{4}{15} = \frac{6}{15} + \frac{4}{15} = \frac{10}{15} = \frac{2}{3}\) ✓
Machine K would take \(4.0\) hours to process the entire shipment working alone.
The answer is B. \(4.0\)
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.
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.
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.
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.
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.
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.
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.
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.