Hamilton A. answered • 01/19/16
Tutor
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Degrees in Math, 10+ Years Tutoring
You're asking a number of fairly basic business math questions here, Brenda. Hopefully, you're starting to see the pattern: interest causes money to be worth more in the future.
In particular, if I have x dollars right now, and there's an i% interest rate per period, I have:
after 1 period: x * (1 + i)
after 2 periods: (after 1 period) * (1 + i) = x * (1 + i) * (1 + i) = x * (1 + i)2
after 3 periods: (after 2 periods) * (1 + i) = x * (1 + i)3
In general, x now will be worth x * (1 + i)n after n periods of interest compounding at an effective interest rate i.
NOTE: In the above equations, we express i as a decimal. So an interest rate of 3% becomes 0.03. An interest rate of 8% becomes 0.08. Etc.
But this question is turning things around: we know how much we'll have in a few years, and we want to know how much that amount of money is worth now.
Pop quiz: Do you expect the answer to be less than $4370.91, equal to $4370.91, or greater than $4370.91?
Hopefully you said "less than $4370.91"; if money becomes worth more as we travel forward in time, then it should be worth less as we travel backwards in time.
With that intuition in place, let's go ahead and solve the problem now. Let x = present value of the sum. By the definition of how interest works, we have:
x * (1 + i)3 = $4370.91
This equation is crucial to understand. It says that my present value, after three periods of compounding interest, will be worth the desired amount. Which is exactly what the question is asking.
And we know that there's a 3% interest rate for each period, i.e. i = 0.03. So we have:
x * (1.03)3 = $4370.91
Solving for x involves dividing both sides by (1.03)3 = 1.092727, and reveals that x = $4000.
You'll note that this is less than $4370.91, which matches our intuition, and it's fairly easy to check our answer:
$4000 * (1.03)3 does indeed equal $4370.91, as desired.
