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Math Help - Apportionment using Webster's Method

  1. #1
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    Apportionment using Webster's Method

    A small country consists of 7 provinces with the following populations:
    Province
    Population 25,312 19,734 33,407 29,591 13,288 22,751 31,992

    Listed on the table the province G has a population of 31,992

    There are 300 federal judges to be apportioned according to the population of each province. Find the apportionment for province G using Webster's method.

    I got 54 as my answer. First I added up all of the population totals which was 176,075 the I divided by 300 to get 586.916 I rounded 586.916 up to get 586.92. It says for the letter G so I took 31,992 and divided it by 586.92 to get 54.50. I rounded the number to get 54 as my answer.

    Thanks
    Last edited by BreedingtheSpawn; September 10th 2010 at 11:38 AM.
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  2. #2
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    I did not know previously about the Webster's method, but according to what I read in Wikipedia and cut-the-knot, I see several problems with your solution. I agree with your final answer, though.

    I rounded 586.916 up to get 586.92. It says for the letter G so I took 31,992 and divided it by 586.92 to get 54.50. I rounded the number to get 54 as my answer.
    First, why did you round 54.50 down? It is more common to round .50 up. Second, 31,992 / 586.92 = 54.508 > 54.50, so it should definitely be rounded up. In general, when one leaves k decimal places, one should increase the kth digit if the (k+1)th digit in the original number is >= 5. In this case, when rounding to two decimal places, the ratio should be 54.51.

    Third, why did you round 176,075 / 300 = 586.916666... to two decimal places? If you rounded it to 586.9, then 31,992 / 586.9 = 54.51, and again it should be rounded up to 55.

    Most importantly, Webster's method requires finding ratios for all provinces. You found that on average there are d=586.92 people per judge; however, when one divides the population of each province by d, rounds the results and adds them together, the final sum can differ from 300 due to rounding errors. The essence of Webster's method is to change d a little so that the final sum is exactly 300.

    So, what you need to do is to take a number d, starting from 586.916666..., divide the population of each province by d; round the results and add them. If the sum is not 300, change d a little and repeat. This is easier to do in Excel or OpenOffice Calc.
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