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The Scotia Metropolitan Area Rapid Transit Service (SMARTS) operates six bus routes (A, B, C, D, E, and F) with a total of 125 buses. The number of buses apportioned to each route is based on the number of passengers riding that route. The table below shows the daily average ridership on each route.

[tex]
\[
\begin{array}{|l|c|c|c|c|c|c|}
\hline
\text{Route} & A & B & C & D & E & F \\
\hline
\text{Ridership} & 45,300 & 31,070 & 20,490 & 14,160 & 10,260 & 8,720 \\
\hline
\end{array}
\]
[/tex]

Find the apportionment under Hamilton's method of the SMARTS buses to the six bus routes.

- A: [tex]$\square$[/tex]
- B: [tex]$\square$[/tex]
- C: [tex]$\square$[/tex]
- D: [tex]$\square$[/tex]
- E: [tex]$\square$[/tex]
- F: [tex]$\square$[/tex]

Answer :

To solve this problem using Hamilton's method, we need to allocate the 125 buses to the six bus routes based on the ridership numbers provided. Here's a step-by-step guide to how it's done:

1. Calculate Total Ridership: First, add up the daily average ridership for all routes.
- Ridership for Route A: 45,300
- Ridership for Route B: 31,070
- Ridership for Route C: 20,490
- Ridership for Route D: 14,160
- Ridership for Route E: 10,260
- Ridership for Route F: 8,720
- Total Ridership = 45,300 + 31,070 + 20,490 + 14,160 + 10,260 + 8,720 = 130,000

2. Calculate Standard Quota: For each route, calculate the standard quota (ideal number of buses) using the formula:
[tex]\[
\text{Standard Quota} = \left(\frac{\text{Ridership for the route}}{\text{Total Ridership}}\right) \times \text{Total Buses}
\][/tex]

- Route A: [tex]\((45,300 / 130,000) \times 125 = 43.5577\)[/tex]
- Route B: [tex]\((31,070 / 130,000) \times 125 = 29.875\)[/tex]
- Route C: [tex]\((20,490 / 130,000) \times 125 = 19.7019\)[/tex]
- Route D: [tex]\((14,160 / 130,000) \times 125 = 13.6154\)[/tex]
- Route E: [tex]\((10,260 / 130,000) \times 125 = 9.8654\)[/tex]
- Route F: [tex]\((8,720 / 130,000) \times 125 = 8.3846\)[/tex]

3. Initial Apportionment: Allocate each route the whole number part of its standard quota.
- Route A: 43
- Route B: 29
- Route C: 19
- Route D: 13
- Route E: 9
- Route F: 8

4. Calculate Remaining Buses: Add up the initial apportionment to see how many buses are left.
- Total initially allocated: 43 + 29 + 19 + 13 + 9 + 8 = 121
- Buses remaining: 125 - 121 = 4

5. Distribute Remaining Buses: Allocate the remaining buses to the routes with the largest fractional parts of their standard quotas.

- Route A: 0.5577
- Route B: 0.875 (highest, gets 1 more bus)
- Route C: 0.7019 (gets 1 more bus)
- Route D: 0.6154 (gets 1 more bus)
- Route E: 0.8654 (second highest, gets 1 more bus)
- Route F: 0.3846

6. Final Apportionment: Add the extra buses based on the largest fractional parts to the initial apportionment.
- Route A: 43
- Route B: 30
- Route C: 20
- Route D: 14
- Route E: 10
- Route F: 8

This final distribution of buses under Hamilton's method ensures an equitable allocation based on ridership.

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