how long will it take for $500 to double if it is invested at 5% annual interest, compounded continously

Question

Math problem how many years will it take to double the amount of the 500 how many? ??

Answer 1

Eric C gave an excellent answer. If you're wondering, the P for "amount invested" stands for Principle, a term that is handy and important when paying off debt, when interest is working against you, and the more Principle in your account, the more interest you have to pay. 

 

"e" is one of those mathemagical numbers like Pi that mathematicians have discovered are just what is needed for simplifying calculations of certain values. 
 
It is time consuming, but you may find it instructive to work through a few years of investment interest like this:

500 * .05 = 25  since it's compound interest, we add it to the principle (500+25=525) 

Second year

525*.05 = 26.25 

Add the interest again, third year: 551.25*.05 = 27.5625

and so on (of course, with money, the figures are rounded to the nearest 100th[penny, cent]). you can see how compounding makes the principle grow, which makes the interest rate grow. If the interest rate were "flat" (or you withdrew the interest every year) it would just be figured on the original principal and only be $25 each year, so to make $500 from the original investment that way would take several years longer. 

Answer 2

Hey Tre.

 

Interest that is compounded continuously follows this formula:

 

A = Pe^rt

 

A = amount desired

P = amount invested

r = rate of return

t = time

 

In your case:

 

2*500 = 500*e^0.05t

 

2 = e^0.05t

 

ln(2) = 0.05t

 

t = ln(2)/0.05

 

t = 13.9 years

 

One of the rules of thumb for investing is what's known as "The Rule of 72." This is a good approximation for how long it will take your investment to double. The rule states that:

 

If the interest rate multiplied by the time passed is 72, that's how long it will take for your investment to double.

 

For example:

 

If your interest rate is, say, 3%, then 72/3 = 24 years. It'll take 24 years for your investment to double.

If your interest rate is 6%, then 72/6 = 12 years.

If you're returning 12%, it'll only take 6 years, since 12*6 = 72

 

This comes from the simple fact that e^0.72 is about 2. Since your rate of return is 5%, a good approximation of the years it'll take would be:

 

72 / 5 = 14.4 years.

 

Of course this is just an off-hand approximation, but it's close enough to your answer to know you're in the right ball park. On exam day it'd serve as a quick and easy check for your calculated result.

 

Hope this helps.