Question 256355: i'm doing business algebra online in college, and we're now talking about continuous compound interest. The formula i'm struggling with is the Pert formula. I'm stuck on how to solve a problem when i know what the P, r and t is, but what is the e for? I've never figured out how to come up with the same answer as my study guide example.
For example: Use the continuous compound interest formula to find the indicated value. A=$7,700; r=8.78%; t=10 years; P=? P=_______(round to two decimal places as needed).
I look forward to getting a reply back soon. Hopefully I will get this figured out soon. Thank you.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! with continuous compounding formula, you get:
f = future value
p = present amount
r = annual interest rate
t = number of years
the e is the symbol for the scientific constant of 2.718281828...
This is an irrational number (never ending non repeating fractional part).
It's called Napier's constant.
the formula for the base of e to an exponent is  where x is the exponent.
the inverse formula of is equal to  which means the natural log of x.
the basis definition of is:
 if and only if  .
this is nothing more than your normal logarithm functions except you have a base of e.
It's no different than the statement:
 if and only if 
for example:
let x = 3
{y = 10^3}}} = 10*10*10 = 1000
the statement becomes:
 if and only if  .
you can confirm that by using your calculator and taking the log of 3.
you will get 1000.
the base e is nothing more than another base to work with.
In your problem, here's how you would work it.
formula is .
you are given:
p = $7,700
r = 8.78%
t = 10 years
first thing you need to do is take all the dollar signs and commas out of p to get:
p = 7700
next thing you need to do is divide 8.78% by 100% to get .0878.
in the formula, you need to work with the rate, not the percent.
since r is an innual interest rate already, you do not need to adjust it any further.
since t is already specified in years, you do not need to adjust it any further.
plug these values into your formula to get:
becomes
simplify to get:
use your calculator to get  = 2.406082726
alternatively, you can substitute the constant of 2.718281828 to get:
solve for f to get:
f = 18526.83699
that's equivalent to $18,526.84.
for some good examples of continuous compounding, check out the web.
do a search on "continuous compounding".
one such website is http://moneychimp.com/articles/finworks/continuous_compounding.htm
It has a calculator that lets you calculate different numbers and see the effect of continuous compounding compared to discrete compounding such as yearly, monthly, daily, and hourly.
It also has a box where you can put in the number of time periods you want.
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