Question 1207428
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For how many years will Prasad make payments on the $28,000 he borrowed to start his machine shop 
if he makes payments of $3400 at the end of every three months and interest is 8.08% compounded semi-annually?
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        The scheme of payments/compounding is non-standard and it is not totally clear how to it really works.


        So,  in my solution I will make some assumptions,  but I am not sure if they are adequate.


        In any case,  I will try.



<pre>
It is equivalent to semi-annual payments on the loan with semi-annual payments of 2*3400 = 6800 dollars,
while the loan is compounded semi-annually.

 
So, use the formula for semi-annual payments for a loan

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


where P is the loan amount; r = {{{0.0808/2}}} is the effective interest rate semi-annually;
n is the number of payments (same as the number of semi-annual periods); 
M is the semi-annual payment of $6800.


In this problem  M = 6800;  P = $28000;  r = {{{0.0808/2}}}.


Substitute these values into the formula and get for semi-annual payments

   6800 = {{{28000*(((0.0808/2))/(1-(1+0.0808/2)^(-n)))}}}.


We should find "n" from this equation.

Divide both sides by 28000.  You will get

    {{{6800/28000}}} = {{{((0.0808/2))/(1-(1+0.0808/2)^(-n))}}},

or

    0.242857143 = {{{((0.0808/2))/(1-(1+0.0808/2)^(-n))}}}.


It implies, step by step

    {{{0.242857143*(2/0.0808)}}} = {{{1/(1-(1+0.0808/2)^(-n))}}},

    6.011315417 = {{{1/(1-(1+0.0808/2)^(-n))}}},

    {{{1/6.011315417}}} = {{{1-(1+0.0808/2)^(-n)}}},

    0.166352941 = {{{1-(1+0.0808/2)^(-n)}}},
    
    {{{(1+0.0808/2)^(-n)}}} = 1 - 0.166352941,

    {{{(1+0.0808/2)^(-n)}}} = 0.833647059,

    {{{(1+0.0808/2)^n}}} = {{{1/0.833647059}}},

    {{{(1+0.0808/2)^n}}} = 1.199548405.


Take logarithm base 10

    n*log(1+0.0808/2) = ln(1.199548405)

and find "n"

    n = {{{log((1.199548405))/log((1+0.0808/2))}}} = 4.59.


From these calculations, I make the conclusion that input data in the problem are incorrect,
since they lead to the non-integer number of semi-annual payments.
</pre>