Question 1207429
.
Keys Company has a target of establishing a fund of $50,000. If $10,000 is deposited 
at the end of every six months, and the fund earns interest at 4% compounded quarterly, 
how long will it take to reach the target?
~~~~~~~~~~~~~~~~~~~~~


<pre>
It is a non-traditional accumulative saving plan with $10,000 deposited semi-annually and compounded 
quarterly at 4% per annum. 

It means that the quarterly effective rate is  0.04/4 = 0.01 and the equivalent
semi-annual effective rate is {{{(1+0.01)^2}}} = {{{1.01^2}}} = 1.0201.


    +-------------------------------------------------------------------+
    |  So, this non-traditional accumulative saving plan is equivalent  |
    |   to the ordinary annuity with semi-annual deposits of $10,000    |
    |   and the semi-annual effective rate of compounding r = 1.0201.   |
    +-------------------------------------------------------------------+


Now use the general formula for a classic Ordinary Annuity saving plan


    FV = {{{P*(((1+r)^n-1)/r)}}},    (1)


where  FV is the future value of the annuity;  P is the semi-annual deposit; r is the semi-annual 
effective percentage yield presented as a decimal; n is the number of deposits.


Under the given conditions, P = 10000;  r = 0.0201.  So, according to (1), the formula for
the future value is


    FV = {{{10000*(((1+0.0201)^n-1)/0.0201)}}}.


So, we should find n, the number of deposits (or the number of semi-annual periods)  from this equation

    50000 = {{{10000*((1.0201^n-1)/0.0201)}}}.


Simplify it by dividing both sides by 10000

    {{{50000/10000}}} = {{{((1.0201^n-1)/0.0201)}}},

or

    5 = {{{((1.0201^n-1)/0.0201)}}}.


Simplify it further, step by step

    5*0.0201 = {{{1.0201^n-1}}},

    0.1005 = {{{1.0201^n-1}}},

    0.1005 + 1 = {{{1.0201^n}}},

    1.1005 = {{{1.0201^n}}}.


Take logarithm base 10 of both sides

    log(1.1005) = n*log(1.0201)

and find n

    n = {{{log((1.1005))/log((1.0201))}}} = 4.81  (approximately).


Finally, round the decimal value of 4.81 to the closest GREATER integer value of 5 
in order for the bank be in position to complete the last semi-annual compounding.


At this point, the solution is complete.


The <U>ANSWER</U> is: 5 semi-annual periods are needed.
</pre>

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On Ordinary Annuity saving plans, &nbsp;see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Ordinary-Annuity-saving-plans-and-geometric-progressions.lesson>Ordinary Annuity saving plans and geometric progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problem-on-Ordinary-Annuity-saving-plans.lesson>Solved problems on Ordinary Annuity saving plans</A>

in this site.


The lessons contain &nbsp;EVERYTHING &nbsp;you need to know about this subject, &nbsp;in clear and compact form.


When you learn from these lessons, &nbsp;you will be able to do similar calculations in semi-automatic mode.