Answered

Working alone, it would take Shandra 40 minutes to sand a table. To sand the same table, also working alone, it would take Kelly 1 hour. If they work together, the situation can be modeled by the following equation, where $t$ is the time it would take to sand the table working together:



$\frac{1}{40}+\frac{1}{60}=\frac{1}{t}$



Shandra claims it will take them about 50 minutes working together, because 50 is the mean of 40 and 60. Which best describes Shandra's claim?


AShandra's claim is correct.
BShandra's estimate is too high because she should have found the differences instead of the mean.
CShandra's estimate is too high because it will take them about 24 minutes.
DShandra's estimate is too low because it will take them about 100 minutes.

Verified Answer :

  • Identify the work rates of Shandra (\(\frac{1}{40}\) task/min) and Kelly (\(\frac{1}{60}\) task/min).
  • Combine the rates and solve for the total time \(t\): \(\frac{1}{40} + \frac{1}{60} = \frac{3}{120} + \frac{2}{120} = \frac{5}{120} = \frac{1}{24}\).
  • Determine that the actual time is \(t = 24\) minutes, proving Shandra's estimate of 50 minutes is incorrect.
  • Conclude that the best description is: \(\boxed{\text{Shandra's estimate is too high because it will take them about 24 minutes.}}\)

Explanation

  1. Problem Analysis and Formulation

Shandra takes 40 minutes to complete a task working alone, meaning her work rate is \(\frac{1}{40}\) of the task per minute. Kelly takes 1 hour, which is 60 minutes, to complete the same task alone, so her work rate is \(\frac{1}{60}\) of the task per minute. When working together, their combined rate is the sum of their individual rates, \(\frac{1}{40} + \frac{1}{60}\). If they take \(t\) minutes to finish the task together, their combined rate must also equal \(\frac{1}{t}\), leading to the model equation \(\frac{1}{40} + \frac{1}{60} = \frac{1}{t}\). Shandra claims the time taken is the mean of their individual times, 50 minutes, which we must evaluate.

  1. Calculating the Combined Work Time

To find the correct time \(t\), we first simplify the expression \(\frac{1}{40} + \frac{1}{60}\). We find a common denominator for the fractions, which is 120. Converting the fractions:
\[\frac{1}{40} = \frac{3}{120} \quad \text{and} \quad \frac{1}{60} = \frac{2}{120}\]
Adding these gives:
\[\frac{3}{120} + \frac{2}{120} = \frac{5}{120}\]
So, the equation becomes \(\frac{1}{t} = \frac{5}{120}\). We can simplify the fraction on the right by dividing both numerator and denominator by 5:
\[\frac{5}{120} = \frac{1}{24}\]
Therefore, \(\frac{1}{t} = \frac{1}{24}\), which implies \(t = 24\) minutes.

  1. Evaluation of the Claim and Final Conclusion

Shandra estimated that working together would take 50 minutes, which is the arithmetic mean of 40 and 60 minutes. However, our calculation shows that the combined time required is 24 minutes. Since 50 minutes is significantly larger than the actual 24 minutes needed, Shandra's estimate is too high. Comparing this to the given choices, the statement that "Shandra's estimate is too high because it will take them about 24 minutes" correctly identifies the error and provides the accurate result.

Examples

Understanding work rates is crucial for project management and resource allocation. For example, if two employees work on a software development project at different speeds, knowing their combined efficiency helps managers estimate realistic deadlines. Just as with the table sanding task, simply averaging completion times often leads to incorrect predictions; instead, summing their individual work capacities allows for accurate scheduling, ensuring teams meet goals effectively.