Hi Amy,
The problem basically offers you two options, and asks, at the end of 5 years, for which one do you end up with more money. The first option leaves you with just $900 at the end of 5 years, not giving you any chance to earn interest on that amount. The second option is harder to calculate, but may be more profitable. Let's step through what you have with the second option, year by year. Note that to calculate the amount you have at the end of each year, you have to first add 6 percent to what you had at the end of the previous year, which you do by multiplying the amount from the previous year by 1.06. Then you add the extra $150 you get at the end of the year. (For an algebraic treatment of the same topic, scroll down to the bottom.)
End of the first year: 150
End of the second year: 150*1.06 + 150 = 309
End of the third year: 309*1.06 + 150 = 477.54
End of the fourth year: 477.54*1.06 + 150 = 656.19
End of the fifth year: 656.19*1.06 + 150 = 845.56
So with the second option, you'd have $845.56. Since that's less than $900, I'd go with the first option of just getting the $900 at the end of the 5 years (unless I needed money sooner).
Let's repeat that calculation for the case of a 14% return on funds. It's all the same, except this time 1.06 is replaced with 1.14:
End of the first year: 150
End of the second year: 150*1.14 + 150 = 321
End of the third year: 321*1.14 + 150 = 515.94
End of the fourth year: 515.94*1.14 + 150 = 738.17
End of the fifth year: 738.17*1.14 + 150 = 991.52
In this case, you'd end up with $991.52 for the second option, so I'd definitely go with that, unless I felt likely to squander the money before letting the interest accrue!
So in summary, if the interest rate is 6%, then the first option is better, but if it's 14%, the second option is better. The answers are different because the interest rates are different.
PS: in real life, not all investment options/sources of interest are ethical. Some investments are literally destroying the planet for the short-term profits of a few, such as TD Bank's investments in extracting toxic fuels from the Alberta tar sands. Those tar sands mines in Alberta used to be pristine boreal forests which were totally clear-cut, in violation of treaties with indigenous peoples there, to wit. So long story short, if you're going to invest in something, be careful what you're investing in!
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Algebraically: Let r = 1.06 or 1.14, and let's go through the second option year by year:
End of the first year: 150
End of the second year: 150*r + 150 = 150(r+1)
End of the third year: 150(r+1)*r + 150 = 150(r^2+r+1)
End of the fourth year: 150(r^2+r+1)*r + 150 = 150(r^3+r^2+r+1)
End of the fifth year: 150(r^3+r^2+r+1)*r + 150 = 150(r^4+r^3+r^2+r+1)
Thus at the end of 5 years, you'd end up with 150(r^4+r^3+r^2+r+1), which, since there's a geometric series in r there, equals 150(r^5-1)/(r-1).
Plugging in r=1.06 gives $845.56, and r=1.14 gives $991.52. So which option is better depends on the value of r.
PS: can you figure out the value of r that makes both options equivalent? It's hard to do without a calculator: it's r=1.09128, which corresponds to an interest rate of ~9.1%.
