Problem 45
Question
In Example \(1,\) one crew could drywall a house in 4 days, and another crew could drywall the same house in 5 days. We were asked to find how long it would take them to drywall the house working together. Explain why each of the following approaches is incorrect. The time it would take to drywall the house: is the sum of the lengths of time it takes each crew to drywall the house: 4 days \(+5\) days \(=9\) days. drywall the house: 4 days \(+5\) days \(=9\) days. is the difference in lengths of time it takes each crew to drywall the house: 5 days \(-4\) days \(=1\) day. is the average of the lengths of time it takes each crew to drywall the house: \(\frac{4 \text { days }+5 \text { days }}{2}=\frac{9}{2}\) days \(=4 \frac{1}{2}\) days.
Step-by-Step Solution
Verified Answer
Summing, subtracting, or averaging their working times isn't logical; they should be combined using their work rates.
1Step 1: Understanding Combined Work
When two or more parties work together to complete a task, we cannot simply add or subtract the individual times it takes them separately to complete the same task. Instead, we consider the rate of work for each and then determine the combined rate to find the time it takes them to work together.
2Step 2: Explaining Incorrect Approach: Addition
The incorrect method suggested is summing the times it takes each crew to do the job separately, assuming they would take 9 days together. This is incorrect because adding their individual times implies they somehow slow each other down, which isn't logical as working together should reduce the time.
3Step 3: Explaining Incorrect Approach: Subtraction
The second incorrect approach is taking the difference of the times, implying that one crew can complete the drywalling almost instantaneously after the other starts. This doesn't make sense as it doesn't reflect any logical form of teamwork or combined effort in reducing the time needed.
4Step 4: Explaining Incorrect Approach: Averaging
The last incorrect method is taking an average of their individual working days, assuming this average is their combined working day. Averaging neglects the fact that combined rates add up, not average out, which would result in less time than both individually take, not some middle ground.
Key Concepts
Understanding Rate of Work in Combined TasksRecognizing Incorrect Approaches in Combined WorkAccurately Calculating Time in Collaborative Tasks
Understanding Rate of Work in Combined Tasks
To solve combined work problems, it's important to understand the concept of "rate of work." Think of rate of work as the speed at which a task is done. It measures the part of a task completed per unit of time. When different crews or individuals work together, their rates combine. This means you need to know not how long each takes alone, but how much of the task each can complete in a given time frame, such as one day.
For instance, if one crew can drywall a house in 4 days, their rate of work is \(\frac{1}{4}\) of the house per day. If the second crew takes 5 days, their rate is \(\frac{1}{5}\) of the house per day. Working together, these rates add up. This means they collectively complete \(\frac{1}{4} + \frac{1}{5} = \frac{5+4}{20} = \frac{9}{20}\) of the house per day.
For instance, if one crew can drywall a house in 4 days, their rate of work is \(\frac{1}{4}\) of the house per day. If the second crew takes 5 days, their rate is \(\frac{1}{5}\) of the house per day. Working together, these rates add up. This means they collectively complete \(\frac{1}{4} + \frac{1}{5} = \frac{5+4}{20} = \frac{9}{20}\) of the house per day.
- Identify each crew's rate: Crew 1 is \(\frac{1}{4}\), Crew 2 is \(\frac{1}{5}\).
- Add the rates together: \(\frac{1}{4} + \frac{1}{5}\).
- Find the combined rate: \(\frac{9}{20}\) of a house per day.
Recognizing Incorrect Approaches in Combined Work
When solving problems involving collaborative work, certain incorrect ideas can come up. Let's explore why they are wrong.
Addition of Individual Times: Some might think adding the times from each crew gives the time needed together (4 days \(+ 5\) days = 9 days). This method mistakenly implies the crews slow each other down. Actually, working together should be faster, not slower.
Subtraction of Times: Another incorrect approach is subtracting the times (5 days \(- 4\) days = 1 day). This method incorrectly assumes that the faster crew will magically speed up the entire process. Realistically, both contribute to finishing the drywalling together, not as a race.
Average of Times: Calculating the average \(\frac{4 \,\text{days} + 5 \,\text{days}}{2} = 4.5\) days might seem logical since averages provide a kind of "middle" value. However, it fails in combined work problems because combining efforts usually results in less time than even the fastest individual takes. Always rely on the sum of rates rather than simplistic arithmetic operations for these problems.
Addition of Individual Times: Some might think adding the times from each crew gives the time needed together (4 days \(+ 5\) days = 9 days). This method mistakenly implies the crews slow each other down. Actually, working together should be faster, not slower.
Subtraction of Times: Another incorrect approach is subtracting the times (5 days \(- 4\) days = 1 day). This method incorrectly assumes that the faster crew will magically speed up the entire process. Realistically, both contribute to finishing the drywalling together, not as a race.
Average of Times: Calculating the average \(\frac{4 \,\text{days} + 5 \,\text{days}}{2} = 4.5\) days might seem logical since averages provide a kind of "middle" value. However, it fails in combined work problems because combining efforts usually results in less time than even the fastest individual takes. Always rely on the sum of rates rather than simplistic arithmetic operations for these problems.
Accurately Calculating Time in Collaborative Tasks
With the importance of work rates established, let's delve into the correct method to figure out actual time needed for the crews working together.
Begin with the combined rate, which we calculated as \(\frac{9}{20}\) of the house per day. This means each day, they jointly finish \(\frac{9}{20}\) of the drywalling task.
To find out how many days are required for the whole task, you use the formula:\[\text{Time} = \frac{\text{Total Task}}{\text{Combined Rate of Work}}\]Since the total task is the entire house, represented by 1 unit, here's how you calculate:\[\text{Time} = \frac{1}{\frac{9}{20}} = \frac{20}{9} \approx 2.22 \,\text{days}\]This calculation tells us the time both crews will take together. It neatly ties into the idea that combining efforts reduces time. While doing this, remember to focus on total units produced or finished per day. This approach allows for an accurate calculation of the day's number required, facilitating better planning and organization of collaborative work.
Begin with the combined rate, which we calculated as \(\frac{9}{20}\) of the house per day. This means each day, they jointly finish \(\frac{9}{20}\) of the drywalling task.
To find out how many days are required for the whole task, you use the formula:\[\text{Time} = \frac{\text{Total Task}}{\text{Combined Rate of Work}}\]Since the total task is the entire house, represented by 1 unit, here's how you calculate:\[\text{Time} = \frac{1}{\frac{9}{20}} = \frac{20}{9} \approx 2.22 \,\text{days}\]This calculation tells us the time both crews will take together. It neatly ties into the idea that combining efforts reduces time. While doing this, remember to focus on total units produced or finished per day. This approach allows for an accurate calculation of the day's number required, facilitating better planning and organization of collaborative work.
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