SOLUTION: How long does it take $875 to double if it is invested at 8% compounded monthly?

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Question 436823: How long does it take $875 to double if it is invested at 8% compounded monthly?
Found 5 solutions by mananth, ikleyn, josgarithmetic, n3, MathTherapy:
Answer by mananth(16949) About Me  (Show Source):
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Principal P = 875
Amount= 1750
years=n
compounded (t) 12
Rate = 8 0.01
Amount =P*((1+r/t))^n*t

1750 =875*(1+0.01)^ n*t
2 = *( 1 + 0.01 )^ n* 12
ln 2 = 12 n *ln 1.01
0.69 = 12* 0.01 *n
8.69 = n

Answer by ikleyn(53427) About Me  (Show Source):
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.
How long does it take $875 to double if it is invested at 8% compounded monthly?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


        The formulas in the post by @mananth are incorrect;  the answer is incorrect
        and is given in the incorrect form.

        I came to bring a correct solution. 


We start from the standard formula for the future value of this compounded account

    FV = 875%2A%281%2B0.08%2F12%29%5En,

where n is the number of monthly compounding.  The value of 'n' is the unknown and is the subject for finding.


We write this equation for the doubled future value

    1750 = 875%2A%281%2B0.08%2F12%29%5En.      (1)


We simplify equation (1) step by step

    1750%2F875 = %281%2B0.08%2F12%29%5En.

    2 = %281%2B0.08%2F12%29%5En.


Take logarithm of both sides 

    log(2) = n*log(1+0.08/12).


Express and calculate 'n'

    n = log%28%282%29%29%2Flog%28%281%2B0.08%2F12%29%29 = 104.318  (approximately).


The number of compounding is an integer number - so, we must round this decimal 104.318
to the closest GREATER integer 105 in order for the bank be in position to make the last compounding.


ANSWER.  First time the compounded amount will exceed the doubled principal in 105 months, 

         or 8 years and 9 months.

Solved correctly.



Answer by josgarithmetic(39702) About Me  (Show Source):
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875%2A%280.08%2F12%2B1%29%5Em=2%2A875 m is the number of months

875%2A1.0066666%5Em=2%2A875

1.0066666%5Em=2
m%2Alog%28%281.0066666%29%29=log%28%282%29%29
m=log%28%282%29%29%2Flog%28%281.0066666%29%29

0.30103%2F0.0028856=104.32

about 8 years and 8 months

Answer by n3(5) About Me  (Show Source):
You can put this solution on YOUR website!
.

The post by @josgarithmetic with his answer  " about 8 years and 8 months "  is

        (1)   incorrect

    and

        (2)   distorts the meaning of the problem and the meaning of my solution.

This problem is not seeking for an approximate solution.
What josgarithmetic calls  " an approximate solution ",  is  NOT  a solution and is  NOT  an appropriate solution.
The problem seeks for a  PRECISE  solution,  instead,  which is achieved via the proper rounding.

The proper rounding of the resulting decimal number to the closest greater integer number
is an integral and essential part of the solution,  which can not be neglected/ignored/omitted.

Regarding all the rest in the post by @josgarithmetic,  it is a reduced re-writing from the solution by @ikleyn.

Therefore,  my advise and instruction to a reader is fully ignore the post by @josgaritmetic.


*********************************************************************

        @josgarithmetic, rewriting from others is a very low-level practice,
                            so I advise you to stop doing it.

*********************************************************************



Answer by MathTherapy(10587) About Me  (Show Source):
You can put this solution on YOUR website!
How long does it take $875 to double if it is invested at 8% compounded monthly?

Future value of $1 formula: A+=+P%281+%2B+i%2Fm%29%5E%28mt%29%29

Doesn't matter what P is, A will ALWAYS be 2 (DOUBLE). 
So, substituting 2 for A (Accumulated amount/Future Value), 1 for P (Principal, or Initial Investment/Amount), .08 for
i (interest rate, as a percent/decimal), 12 for m (number of annual coumpounding periods), t (time, in years) is UNKNOWN.

With that, A+=+P%281+%2B+i%2Fm%29%5E%28mt%29%29 now becomes: 
           2+=+%281+%2B+.08%2F12%29%5E%2812t%29%29 
           2+=+%281+%2B+.02%2F3%29%5E%2812t%29%29
           2+=+%28%283+%2B+.02%29%2F3%29%5E%2812t%29%29
           2+=+%283.02%2F3%29%5E%2812t%29%29
         12t+=+log%28%283.02%2F3%29%2C+%282%29%29%29 ----- Converting to LOGARITHMIC form
  Time, or t+=+%28log%28%283.02%2F3%29%2C+%282%29%29%2F12%29, or approximately 8.693189 years, or 8 years, 8.3183 months, or 104.3183 months. 
This amount is then ROUNDED to a time of 105 months, or 8 years, and 9 months.

As stated by Tutor @IKLEYN, the 8.3183 years, or 104.3813 months MUST be ROUNDED UP to the next INTEGER, which is 105.

Note that at the 104th-month, or 8-year, 8-month juncture, the amount will NOT have doubled. One has to wait until the 105th
month to see the invested amount DOUBLE.