May you help me please

Question

A manufacturing company has 100 forklifts to apportion among three factories. The forklifts are to be apportioned based on the number of employees at each factory as shown in the table below.

Factory	A	B	C	Total
Employees	3,822	7,818	28,360	40,000

Apportion the forklifts using Hamilton’s method. 

Suppose 1 year later the factories have the following number of employees. If the 100 forklifts are reapportioned to the factories, does the population paradox occur using Hamilton’s method? Which answer below best represents the problem?

Factory	A	B	C	Total
Employees	3,861	7,896	28,360	40,117

a. )40,000 employees; A has 10 forklifts, B has 19 forklifts, C has 71 forklifts. 

40,117 employees; A has 9 forklifts, B has 20 forklifts, C has 71 forklifts. 

Yes, the population paradox occurs because A lost a forklift to factory B even though A’s population increased at a faster rate than the population of B.

b.) 40,000 employees; A has 10 forklifts, B has 19 forklifts, C has 71 forklifts. 

40,117 employees; A has 10 forklifts, B has 19 forklifts, C has 71 forklifts. 

No, the population paradox did not occur because no factory lost any forklift.

 c.) 40,000 employees; A has 10 forklifts, B has 19 forklifts, C has 71 forklifts. 

40,117 employees; A has 10 forklifts, B has 20 forklifts, C has 70 forklifts. 

No, the population paradox did not occur because only C lost one and their population never changed.

d.) 40,000 employees; A has 10 forklifts, B has 19 forklifts, C has 71 forklifts. 

40,117 employees; A has 10 forklifts, B has 19 forklifts, C has 71 forklifts. 

Yes, the population paradox occurs because B lost a forklift to factory A even though B’s population increased at a faster rate than the population of A.

Answer 1

Cool, let's do a few rounds of Hamilton's Method!

From the first table, we see there are 40,000 employees to start, and we'd like to apportion 100 forklifts between them. This means ideally we have 1 forklift per every 40,000/100 = 400 employees. If the distribution of the employees all were in multiples of 400, we'd be able to easily apportion! With numbers that aren't so nicely divisible by 400, though, we'll look to Hamilton for how to make up the difference.

I basically think of the method like this: each factory has a specific percentage of the total number of employees, so ideally they should get that percentage of the forklifts, too. Let's find those percentages, then, and take that percentage of the number of forklifts:

Factory A has 3822/40000 of the employees; 3822/40000 * 100 forklifts = 9.555. We definitely know Factory A will get at least 9 forklifts, and we'll keep track of the decimals to see how they compare to the other factories.

Factory B has 7818/40000 of the employees; 7818/40000 * 100 = 19.54500. We're definitely giving at least 19 forklifts to Factory B, but with a smaller number after the decimal point than Factory A, they're later in the running for any remaining ones.

Factory C has 28360/40000 of the employees; 28360/40000 * 100 = 70.9. Factory C gets at least 70 forklifts, and since .9 is bigger than either of the other two factories' decimals, they're first in line for an additional forklift should this first-go leave any unapportioned.

So far, we've accounted for 9+19+70 = 98 of the 100 forklifts; we're able to apportion an additional forklift to the top two factories: Factory C with its decimal of .9, then Factory A with its decimal of .555. There unfortunately isn't a third forklift to give Factory B, with its low decimal of .545.

We're left with 9+1 = 10 forklifts for Factory A, 19 for Factory B, and 70+1 = 71 for Factory C. Notice 10+19+71 indeed is 100 total, so all the forklifts are now apportioned.

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A year goes by, and the employee totals have shifted. We can do the same process again:

Factory A's apportionment is now based on 3861/40117*100 ≈ 9.6243

Factory B's comes from 7896/40117*100 ≈ 19.6824

and Factory C's from 28360/40117*100 ≈ 70.6932

Based on the numbers before the decimals, we again start by apportioning 9 to A, 19 to B, and 70 to C.

This time, though, of the 100-9-19-70 = 2 remaining forklifts, notice that Factory B's decimal is now higher than Factory A's; the priority order will this time be C with .69, then B with .68, and A is last with .62.

So the final totals are A with 9, B with 19+1 = 20, and C with 70+1 = 71.

At this point, multiple choice tells us the answer must be (a). We can further explore its implication around the Population Paradox:

Factory A's growth multiplier was 3861/3822 ≈ 1.01020408, or a little over 1%

Factory B's growth, by comparison, was 7896/7818 ≈ 1.00997698, a little under 1%

You'd expect that with a faster-growing population, Factory A should at least keep the number of forklifts it was apportioned before, if not receive even more. But instead, we took a forklift away from Factory A to give to Factory B this round.

Alexander Hamilton originally came up with this method in the hopes it was a fair way of distributing a fixed number of government representatives across each state's population. How do these results leave you feeling about his sense of fairness? About non-mathematically-literate politicians determining how government resources should be distributed, more generally?

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