A township expects its population of 5,000 to grow annually at the rate of 5%. The township currently spends $300 per inhabitant, but, as the result of inflation and wage increments, expects the per capita expenditure to grow annually by 7%. How much will the township’s budget be after 10, 15, and 20 years? I know the answers are: $4,805,550 for 10yrs, $8,607,060 for 15yrs, and $15,400,665 for 20yrs but I have no clue as to how to arrive at those conclusions.

In any given year, the total spending will be given by P times C, where P is the total population, and C is the per-capita spending for such year.

But the total population, and the per-capita spending in any given year n, will be, respectively,

P_n \ = \ 5,000(1.05)^n
C_n \ = \ 300(1.07)^n

For example, after 4 years they expect their population and their per-person spending to be
5,000(1.05)^4 \ = \ 6,078
300(1.07)^4 \ = \ 393.24

You can take it from here, but note that the pre-supplied answers involve some significant rounding (hence the phrase, "close enough for government work!").

ArcSine, Thank you for the clarification I truly do appreciate it. After posting this question I walked away from my homework for a bit to allow my brain some time to digest everything I was working on. When I came back to it I ended up figuring it out, but wasn't exactly sure if I truly had or not because my answers where a bit different from the answers in the textbook. Your post definitely helps to verify that my train of thought on how to solve the problem was flowing in the right channel and that the differences were merely because of rounding! Thank you so much for the help and the clarification!

Happy to help, CG. The thing about rounding-off issues is that (especially in exponential-growth situations, such as here) a little rounding can grow into significant differences pretty quick.

Best of luck with your studies.