Alright, let's go through this step-by-step to determine the appropriate values for each cell in the table.
1. Rate of mowing for Roni:
Roni can mow the entire field in 30 minutes. Therefore, his rate of mowing is \(\frac{1}{30}\) fields per minute.
2. Rate of mowing for Allie:
Allie can mow the entire field in 75 minutes. Therefore, her rate of mowing is \(\frac{1}{75}\) fields per minute.
3. Combined rate of mowing:
When Roni and Allie work together, their combined rate of mowing is the sum of their individual rates:
[tex]\[\frac{1}{30} + \frac{1}{75} = 0.03333333333333333 + 0.013333333333333334 = 0.04666666666666667 \][/tex]
fields per minute.
4. Time taken to mow the field together:
To find the time it takes for both to mow the entire field together, we take the reciprocal of their combined rate:
[tex]\[\frac{1}{0.04666666666666667} = 21.428571428571427 \text{ minutes}\][/tex]
5. Part of the field Roni would mow when working together:
Since Roni’s rate is \(\frac{1}{30}\) fields per minute, the part of the field he would mow can be calculated by:
[tex]\[\frac{\frac{1}{30}}{0.04666666666666667} = 0.7142857142857142\][/tex]
With these calculations, we can fill out the table as follows:
\begin{tabular}{|c|c|c|c|}
\hline & \begin{tabular}{c}
Rate \\
(part/min)
\end{tabular} & \begin{tabular}{c}
Time \\
(min)
\end{tabular} & \begin{tabular}{c}
Part of Field \\
Mowed
\end{tabular} \\
\hline Roni & 0.03333333333333333 & 30 & 0.7142857142857142 \\
\hline Allie & 0.013333333333333334 & 75 & 0.2857142857142857 \\
\hline
\end{tabular}
To answer the multiple-choice question:
If Roni and Allie work together to mow the field, the part of the field Roni would mow would be about 0.71 (or more precisely, \(0.7142857142857142\)).
Thus, the correct answer is:
about 0.71 of the field
