Question 282967
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 4.75% and the investment is compounded semi-annually, your "r" is 0.0475 and your n is 2:
{{{A = P(1 + 0.0475/2)^t}}}
The expression in the parentheses simplifies as follows:
{{{A = P(1 + 0.02375)^t}}}
{{{A = P(1.02375)^t}}}<br>
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(1.02375)^t}}}
Now we solve for t. We'll start by isolating the base and its exponent. Divide both sides by 27500:
{{{29500/27500 = (1.02375)^t}}}
{{{295/275 = (1.02375)^t}}}
{{{59/55 = (1.02375)^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((59/55)) = log(((1.02375)^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((59/55)) = t*log((1.02375))}}}
Now we can divide both sides by {{{log((1.02375))}}}:
{{{(log((59/55))/log((1.02375))) = 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 59 by 55</li><li>Find the log of the answer from step 1.</li><li>Find the log of 1.02375</li><li>Divide the result of step 2 by the result of step 3</li></ol>
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.