Problem

The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift. Apportion 14 salespeople using Jefferson's method given the information below. \begin{tabular}{|c|c|c|c|c|} \hline Shift & Morning & Midday & Afternoon & Evening \\ \hline Average number of customers & 135 & 325 & 385 & 495 \\ \hline Salespeople to assign & & & & \\ \hline \end{tabular} What modified divisor did you use?

Solution

Step 1 :Calculate the total number of customers: \(135 + 325 + 385 + 495 = 1340\)

Step 2 :Calculate the standard divisor: \(\frac{1340}{14} = 95.71428571428571\)

Step 3 :Calculate the standard quotas: \(\frac{135}{95.71428571428571} = 1.410958904109589\), \(\frac{325}{95.71428571428571} = 3.397260273972603\), \(\frac{385}{95.71428571428571} = 4.020547945205479\), \(\frac{495}{95.71428571428571} = 5.171232876712329\)

Step 4 :Round down the standard quotas to get the initial allocation: Morning salespeople = 1, Midday salespeople = 3, Afternoon salespeople = 4, Evening salespeople = 5

Step 5 :Calculate the total number of salespeople assigned so far: \(1 + 3 + 4 + 5 = 13\)

Step 6 :Since the total number of salespeople assigned is less than the total number of salespeople available, we need to adjust the divisor to allocate the remaining salespeople. This is done by lowering the divisor until the total number of salespeople assigned equals the total number of salespeople available. This new divisor is the modified divisor.

Step 7 :After some trial and error, we find that a modified divisor of approximately 94.44 results in the following allocation: Morning salespeople = 1, Midday salespeople = 3, Afternoon salespeople = 4, Evening salespeople = 6

Step 8 :This results in a total of 14 salespeople assigned, which is the total number of salespeople available. So, the modified divisor used is approximately \(\boxed{94.44}\)

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