A School Wants to Distribute 20 Seats on the Student Council Among the Following Three

Discrete Math: Apportionment, Part 1 Examples

A school wants to distribute 20 seats on the student council among the following three classes, with the given populations:

Sophomores: 900 Juniors: 750 Seniors: 650

1. What is the ideal ratio?

Add up the total number of students, and then divide by the number of seats. 900 + 750 + 650 = 2,300 2,300 / 20 = 115

This means that ideally each member of the student council would represent 115 students.

2. What is the quota for each class?

Divide the population of each individual class by the ideal ratio:

Sophomores: 900 / 115 = 7.826 Juniors: 750 / 115 = 6.522 Seniors: 650 / 115 = 5.652

This means that ideally the sophomores would have 7.826 representatives, the juniors would have 6.522, and the seniors would have 5.652. Of course, it’s kind of tough to have a fraction of a representative—we prefer to keep our representatives whole.

3. What is the final apportionment using the Hamilton method?

The Hamilton method begins by truncating the quota for each class:

Soph. 7.862  7 Jr.: 6.522  6 Sr.: 5.652  5

Add the total number of representatives so far: 7 + 6 + 5 = 18 We need 20 representatives, though, so we need to add 2 more. Now look at only the decimal portion of each quota. The class with the highest decimal portion gets the next representative:

Soph. 7.862  7 Jr.: 6.522  6 Sr.: 5.652  5

The sophomores have the highest decimal portion (.862), so they get an additional representative, for a total of 8. The seniors have the next highest decimal portion (.652), so they get the 20th representative.

The final apportionment is: Soph. 8 Jr.: 6 Sr.: 6

4. What is the final apportionment using the Jefferson method?

The Jefferson method starts out the same as the Hamilton method: Calculate the ideal ratio and then the quota for each class. Truncate the quotas. Check to see if you have the correct number of representatives.

Soph. 7.862  7 Jr.: 6.522  6 Sr.: 5.652  5

Once again, we have only 18 seats assigned so far, and must assign 2 more. To determine who gets the next seat(s) divide each class population by 1 more than the current allocation for each class. For example, the sophomore class currently has 7 seats; divide its population by 8. The junior class has 6 seats; divide its population by 7, etc. This will determine what level of representation a class would have if it gained 1 more seat.

Soph.: 900 / 8 = 112.50 Jr.: 750 / 7 = 107.14 Sr.: 650 / 6 = 108.3

These are the adjusted ratios for each class. The class with an adjusted ratio closest to the ideal ratio gets the next seat. The ideal ratio is 115, so the sophomores are closest to that value and get the next seat. The seniors are the next closest and get the last seat. Once again the final apportionment is: Soph.: 8 Jr.: 6 Sr.: 6 A county wants to distribute 20 seats on the county board among the following four cities, with the given populations:

Alapso: 35,000 Bozedra: 41,000 Calihandra: 64,000 Desmoville 43,000

5. What is the ideal ratio?

Add up the populations and divide by the number of seats.

35,000 + 41,000 + 64,000 + 43,000 = 183,000 183,000 / 20 = 9150

6. What is the quota for each city?

A: 35,000 / 9150 = 3.825 B: 41,000 / 9150 = 4.481 C: 64,000 / 9150 = 6.995 D: 43,000 / 9150 = 4.699

7. What is the final apportionment using the Hamilton method?

Determine the initial apportionment by truncating the quotas.

A: 3.825  3 B: 4.481  4 C: 6.995  6 D: 4.699  4

Add the current number of seats: 3 + 4 + 6 + 4 = 17

Award an additional seat to the cities with the highest decimal portion until all 20 seats have been awarded.

Calihandra has the highest decimal portion (.995) so it gets the 18th seat. Alapso has the next highest decimal portion (.825) so it gets the 19th seat. Desmoville has the next highest decimal portion (.699) so it gets the 20th and final seat. The final apportionment is: Alapso: 4 Bozedra: 4 Calihandra: 7 Desmoville: 5 8. What is the final apportionment using the Jefferson method?

Pick up at the point of truncation in the Hamilton method:

A: 3.825  3 B: 4.481  4 C: 6.995  6 D: 4.699  4

Divide each city’s population by one more than its current allocation to determine the adjusted ratio.

A: 35,000 / 4 = 8750 B: 41,000 / 5 = 8200 C: 64,000 / 7 = 9142.9 D: 43,000 / 5 = 8600

Notice that all of the adjusted ratios are lower than the ideal ratio of 9150. The city with the adjusted ratio closest to the ideal ratio gets the next seat. Calihandra is the closest (9142.9), so it gets the 18th seat. Alapso is the next closest (8750), so it gets the 19th seat. Desmoville is the next closest (8600), so it gets the 20th seat. This gives us a final apportionment of:

Alapso: 4 Bozedra: 4 Calihandra: 7 Desmoville: 5