SOLUTION: Aaron has inherited $250,000 from his aunt. He invests all of this amount at a rate of 5.25% compounded quarterly. How many years will it take for this amount to grow to $300,000

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Aaron has inherited $250,000 from his aunt. He invests all of this amount at a rate of 5.25% compounded quarterly. How many years will it take for this amount to grow to $300,000      Log On


   

Question 282966: Aaron has inherited $250,000 from his aunt. He invests all of this amount at a rate of 5.25% compounded quarterly. How many years will it take for this amount to grow to $300,000?
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 5.25% and the investment is compounded quarterly, your "r" is 0.0525 and your n is 4:
A+=+P%281+%2B+0.0525%2F4%29%5Et
The expression in the parentheses simplifies as follows:
A+=+P%281+%2B+0.013125%29%5Et
A+=+P%281.013125%29%5Et

You are asked to find how long it will take for an investment of 250000 to grow to 300000. So A = 300000 and P = 250000:
300000+=+250000%281.013125%29%5Et

Now we solve for t. We'll start by isolating the base and its exponent. Divide both sides by 250000:
300000%2F250000+=+%281.013125%29%5Et
30%2F25+=+%281.013125%29%5Et
6%2F5+=+%281.013125%29%5Et
1.2+=+%281.013125%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%281.2%29%29+=+log%28%28%281.013125%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%281.2%29%29+=+t%2Alog%28%281.013125%29%29
Now we can divide both sides by log%28%281.013125%29%29:
%28log%28%281.2%29%29%2Flog%28%281.013125%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. Find the log of 1.2
  2. Find the log of 1.013125
  3. Divide the result of step 1 by the result of step 2

The answer you get will be the approximate number of compounding periods (which are quarters of a year in this problem) it will take to reach $300000. Since the problem asks for an answer in years, take your answer and divide it by 4.