Question 1207430
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A property worth $35,000 is purchased for 10% down and semi-annual payments of $2100 for 12 years. 
What is the nominal annual rate of interest if interest is compounded quarterly?
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        The solution by tutor @Theo has an error  (or a typo). 


        His answer  8.035524952%  is incorrect.


        I came to bring a correct solution,  right numbers and proper answer.



<pre>
Down payment is 10% of $35,000, i.e. 0.1*35000 = 3500 dollars.

Hence, the loan is the rest amount of  $35,000 - $3,500 = $31,500.


    +------------------------------------------------------------+
    |   Notice that, as it is given in the problem, semi-annual  |
    |   payments are desynchronized with quarterly compounding.  |
    +------------------------------------------------------------+


Nevertheless, we can synchronize payments and compounding by considering an EQUIVALENT scheme 
with semi-annual compounding with the effective growth coefficient  'r'  semi-annually.

This coefficient 'r' is not known now, and we should find it from the problem.


We then have a loan of $31,500 with semi-annual payments of $2100 and semi-annual compounding
with the effective semi-annual rate of  r.


Write the standard loan equation for such a loan

    {{{31500}}} = {{{2100*((1-(1+r)^(-24))/r)}}}

    {{{31500/2100}}} = {{{(1-(1+r)^(-24))/r}}}

     15 = {{{(1-(1+r)^(-24))/r}}}


Solve this equation numerically to find 'r'.


I used online calculator https://www.wolframalpha.com/calculators/equation-solver-calculator/


It found the approximate real solution  r = 0.0416015.


Thus, in the equivalent scheme, the semi-annual effective rate is 0.0416015.


    //  Notice that till point my solution coincides with that by @Theo.


Hence, the effective semi-annual growth factor is  1+r = 1.0416015.


It implies that in the basic scheme, the effective quarterly growth factor is the square root of that

    {{{sqrt(1.0416015)}}} = 1.020589.


Hence, the effective quarterly rate is  0.020589.


Then the annual effective rate is four times this, or  4*0.020589 = 0.082388.


Thus the nominal annual compounding interest  is 8.2388%.    <<<---===  <U>ANSWER</U>
</pre>

Solved.