Problem 59
Question
For exercises 59-66, use the five steps. Assume that the rate of work does not change if done individually or together. A worker can prune one row of grapevines in \(44 \mathrm{~min}\). Another worker can prune one row in \(33 \mathrm{~min}\). Find the time for these workers to do the job together. Round to the nearest whole number.
Step-by-Step Solution
Verified Answer
19 minutes.
1Step 1 - Identify individual rates of work
The first worker can prune one row in 44 minutes, which means their rate of work is \( \frac{1}{44} \) rows per minute. The second worker can prune one row in 33 minutes, meaning their rate of work is \( \frac{1}{33} \) rows per minute.
2Step 2 - Combine the rates to find the total rate of work
To find the combined rate of work when both workers are working together, add their individual rates: \( \frac{1}{44} + \frac{1}{33} \).
3Step 3 - Find a common denominator
The common denominator for 44 and 33 is 132. Rewrite the rates with the common denominator: \( \frac{1}{44} = \frac{3}{132} \) and \( \frac{1}{33} = \frac{4}{132} \).
4Step 4 - Add the fractions
Add the rewritten rates together: \( \frac{3}{132} + \frac{4}{132} = \frac{7}{132} \). So, the combined rate of work is \( \frac{7}{132} \) rows per minute.
5Step 5 - Calculate the time to prune one row together
To find the time it takes for both workers to prune one row together, take the reciprocal of the combined rate \( \frac{7}{132} \) which gives the time in minutes: \( \frac{132}{7} \approx 18.857 \). Rounding to the nearest whole number, it is 19 minutes.
Key Concepts
rate of workfractionscommon denominatorreciprocal
rate of work
The concept of rate of work is crucial to solving problems where multiple workers are performing a task together. The rate of work is essentially how much of the task a worker can complete in a unit of time. For instance, in the given exercise, one worker can prune one row in 44 minutes. This means their rate of work is \(\frac{1}{44}\) rows per minute. Similarly, the second worker prunes one row in 33 minutes, so their rate is \(\frac{1}{33}\) rows per minute.
Understanding individual rates allows us to combine them to find the total rate when both are working together. When combining rates, consider each worker's contribution over the same time frame. This combined rate helps us determine how long it will take for the workers to complete the task together.
Understanding individual rates allows us to combine them to find the total rate when both are working together. When combining rates, consider each worker's contribution over the same time frame. This combined rate helps us determine how long it will take for the workers to complete the task together.
fractions
In this problem, we use fractions to represent the rate of work. A fraction like \(\frac{1}{44}\) illustrates how much of the job one worker can do in a minute. Fractions are handy for these calculations because they allow us to add rates easily. When we add fractions, we sum the numerators and keep a common denominator.
For example, \(\frac{1}{44} + \frac{1}{33}\), when written with a common denominator, becomes \(\frac{3}{132} + \frac{4}{132}\). The addition is straightforward once the fractions have the same base, simplifying understanding of combined efforts in work-rate problems.
For example, \(\frac{1}{44} + \frac{1}{33}\), when written with a common denominator, becomes \(\frac{3}{132} + \frac{4}{132}\). The addition is straightforward once the fractions have the same base, simplifying understanding of combined efforts in work-rate problems.
common denominator
To add fractions effectively, they need a common denominator, which is a shared multiple of the denominators of the fractions involved. In this exercise, the denominators are 44 and 33. The smallest common multiple of 44 and 33 is 132. Converting each fraction to this common denominator allows for easier addition.
Using the concept of least common multiple, we transform \(\frac{1}{44}\) to \(\frac{3}{132}\) and \(\frac{1}{33}\) to \(\frac{4}{132}\). Now, adding \(\frac{3}{132}\) and \(\frac{4}{132}\) is simple, becoming \(\frac{7}{132}\). This simplification makes combining rates and solving work problems more manageable and less error-prone.
Using the concept of least common multiple, we transform \(\frac{1}{44}\) to \(\frac{3}{132}\) and \(\frac{1}{33}\) to \(\frac{4}{132}\). Now, adding \(\frac{3}{132}\) and \(\frac{4}{132}\) is simple, becoming \(\frac{7}{132}\). This simplification makes combining rates and solving work problems more manageable and less error-prone.
reciprocal
The reciprocal of a fraction is simply flipping its numerator and denominator. For instance, the reciprocal of \(\frac{7}{132}\) is \(\frac{132}{7}\). Using reciprocals in work-rate problems helps convert the combined rate of work back into the required time to complete a task.
In our exercise, once we have the combined rate of \(\frac{7}{132}\) rows per minute, finding its reciprocal \(\frac{132}{7}\) gives the time it takes the workers to prune one row together. The calculation \(\frac{132}{7} \approx 18.857\); when rounded, lets us know it takes about 19 minutes to complete the task together. Reciprocals thus bridge the gap between rate and time, providing a clear pathway from one to the other.
In our exercise, once we have the combined rate of \(\frac{7}{132}\) rows per minute, finding its reciprocal \(\frac{132}{7}\) gives the time it takes the workers to prune one row together. The calculation \(\frac{132}{7} \approx 18.857\); when rounded, lets us know it takes about 19 minutes to complete the task together. Reciprocals thus bridge the gap between rate and time, providing a clear pathway from one to the other.
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