math problem

Question

How long (in years and months) will it take for an investment to double at 12% compounded monthly

Answer 1

Hi Joe. Think of compounding as your money multiplying.  This is an amazing and wonderful thing, so awesome in fact that Albert Einstein has been quoted as saying that compound interest is the Eighth Wonder of the World. 

 

So you need to first decide what is 12% of your investment.  Consider what you would do if it were 50% interest.  (If it was, please call me and let me in on that!)  You would simply multiply your investment amount by 0.50 to get the interest amount earned.  Well, in this case, it is 12% so let's multiply by 0.12 instead.  

 

Now that you have the interest amount, add that to your original investment amount (often called the "principle").  Problem solved, right?  Not exactly.  We compounded the interest just one time, as if it were compounded one time per year, or annually. You need to spread out that 12% over 12 months.  So you would do 0.12 divided out across 12 months, or 0.12 ⁄ 12.  That's 0.10.  So multiply your principle amount by 0.10 and add whatever that interest would be back to your principle. Problem is, you'll have to repeat that 12 times!  Yikes.  That would be a lot of button pushing.

 

Thankfully, there a faster way.  Let's say your principle was $100 and the interest was 10%.  We would have done this according to the steps I outlined above. 

100 times 0.10 plus 100, or

10 plus 100, which equals 

110

 

If we replace the $100 with P, it looks like this 

P times (interest rate) plus P

we can then factor out the P and it changes to

P(interest rate plus 1)

 

You will find the formula in any text or online as 

Future Value or Amount = Principle (1 + r/n)^nt

where most of that should look familiar compared to what I outlined above. 

The nt in the exponent is the total number of times you are compounding.  

If you were compounding monthly for two years, that would be n=12 times t=2 or 24 total compounds. 

 

Good luck, and remember Strength in Numbers, my friend. 

Answer 2

The model equation is A = P(1 + r/n)^nt

 

Let A = 2P since the question is asking how long it would take the initial investment, P, to double.

      r = 0.12 since r represents the interest rate.

      n = 12 since n represents the number of times the interest is compounded per year.

      t = time (in years), we are solving for t

 

Thus,   2P = P(1 +0.12/12)^12t,  dividing both sides by gives

             2 = (1.01)^12t, since we are solving for t (an exponent) we must use logarithms.

          log 2 = log (1.01)^12t, now using the power rule of logarithms gives

          log 2 = (12t) (log 1.01) , dividing both sides by log (1.01) gives

        (log 2) / (log 1.01) = 12t and now by dividing both sides by 12 or multiplying by (1/12) gives

          (1/12)[(log 2) / (log 1.01)] = t

               t = 5.8 years (rounded correct to the nearest tenth)   

 

 

Answer 3

2a_o = a₀(1 + 0.12/12)^12t

2 = (1 + 0.01)^12t

2 = 1.01^12t

ln 2 = 12t ln 1.01

0.693 ≈ 12t (0.00995)

69.648 ≈ 12t

5.80 ≈ t

Answer 4

A = P(1+r/n)^nt

 

where

A = the final amount, which for us will be 2P

P = the initial amount

R = the interest, 12% which is written as .12

n = how many periods, since it's monthly this is 12

t = what we're solving for

 

2P = P(1+.12/12)^12t

divide out the P's

2 = (1+.12/12)^12t

2 = (1+.01)^12t

2 = 1.01^12t

to solve for t, use logs of each side of the equation and rules of logs:

log2 = log 1.01^12t

log2 = 12t (log 1.01)

log2/(12)(log 1.01) = t

compute it!