Question 282951
The formula for compound interest is:
{{{A = P(1 + r/n)^t}}}
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)<br>
Since your rate is 7% and the investment is compounded monthly, your "r" is 0.07 and your n is 12:
{{{A = P(1 + 0.07/12)^t}}}
The expression in the parentheses simplifies as follows:
{{{A = P(12/12 + 0.07/12)^t}}}
{{{A = P(12.07/12)^t}}}<br>
You are asked to find how long it will take for an investment of 2500 to grow to 4000. So A = 4000 and P = 2500:
{{{4000 = 2500(12.07/12)^t}}}
Now we solve for t. We'll start by isolating the base and its exponent. Divide both sides by 2500:
{{{4000/2500 = (12.07/12)^t}}}
{{{40/25 = (12.07/12)^t}}}
{{{8/5 = (12.07/12)^t}}}
{{{1.6 = (12.07/12)^t}}}
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((1.6)) = log(((12.07/12)^t))}}}
Now we can use a property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}, 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((1.6)) = t*log((12.07/12))}}}
Now we can divide both sides by {{{log((12.07/12))}}}:
{{{(log((1.6))/log((12.07/12))) = 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 <i>with the parentheses</i>. If not, then<ol><li>Divide 12.07 by 12</li><li>Find the log of the answer from step 1.</li><li>Find the log of 1.6</li><li>Divide the result of step 3 by the result of step 2</li></ol>
The answer you get will be the approximate number of compounding periods (which are months in this problem) it will take to reach $4000.