SOLUTION: John invests $27,500 at 4.75% interest compounded semi-annually. How many years will it take for him to earn $2000 in INTEREST?

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Question 282967: John invests $27,500 at 4.75% interest compounded semi-annually. How many years will it take for him to earn $2000 in INTEREST?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The formula for compound interest is:
A+=+P%281+%2B+r%2Fn%29%5Et
where
P = Principal (the amount of the initial investment)
r = annual rate of interest (as a decimal or fraction)
n = the number of compounding periods per year
t = the total number of compounding periods of the investment
A = Amount (the value of the investment after t compounding periods)

Since your rate is 4.75% and the investment is compounded semi-annually, your "r" is 0.0475 and your n is 2:
A+=+P%281+%2B+0.0475%2F2%29%5Et
The expression in the parentheses simplifies as follows:
A+=+P%281+%2B+0.02375%29%5Et
A+=+P%281.02375%29%5Et

You are asked to find how long it will take for an investment of 27500 to earn 2000 in interest. So P = 27500 and A, since the interest is 2000, will be 27500 + 2000 = 29500:
29500+=+27500%281.02375%29%5Et
Now we solve for t. We'll start by isolating the base and its exponent. Divide both sides by 27500:
29500%2F27500+=+%281.02375%29%5Et
295%2F275+=+%281.02375%29%5Et
59%2F55+=+%281.02375%29%5Et
Solving for a variable in an exponent usually involves logarithms. So we'll find the logarithm of each side. (Any base of logarithm can be used. But if you want a decimal approximation of the answer it is best to use a base your calculator "knows" (like base 10 or base e (ln))). We'll use base 10:
log%28%2859%2F55%29%29+=+log%28%28%281.02375%29%5Et%29%29
Now we can use a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, to move the exponent out in front. (This property, with its ability to change an exponent into a coefficient, is the very reason we use logarithms on equations where the variable is in an exponent.)
log%28%2859%2F55%29%29+=+t%2Alog%28%281.02375%29%29
Now we can divide both sides by log%28%281.02375%29%29:
%28log%28%2859%2F55%29%29%2Flog%28%281.02375%29%29%29+=+t
This is an exact expression of the answer. You probably want a decimal approximation so use your calculator on this. If your calculator has keys for parentheses then you can pretty much type in what you see with the parentheses. If not, then
  1. Divide 59 by 55
  2. Find the log of the answer from step 1.
  3. Find the log of 1.02375
  4. Divide the result of step 2 by the result of step 3

The answer you get will be the approximate number of compounding periods (which are half-years in this problem) it will take to for the interest to reach $2000. Since the problem asks for an answer in years, you will have to divide the above answer by 2.