Question 1193612
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If a loan of ₱60,000 is to be settled by 3,200 monthly payments for 2 
years, what interest rate compounded monthly is charged on the loan?
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        The solution in the post by the other tutor is  WRONG.


        So,  I came to bring a correct solution.



<pre>
Write a loan equation

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


where 

M is the monthly payment;

L is the loaned amount;

r is the monthly effective rate as a decimal;

n is the number of payments (= the number of months).


In our case this equation takes the form

    3200 = {{{(60000*r)/(1-(1+r)^(-2*12))}}},

or

    {{{3200/60000}}} = {{{r/(1-(1+r)^(-24))}}}.

    0.053333333 = {{{r/(1-(1+r)^(-24))}}}.


In this equation, r is the unknown.


Such equation is unsolvable algebraically, so use the numerical methods.

You may use any of numerous online calculators.

I used DESMOS at  www.desmos.com/calculator


It gave me  r = 0.02077, approximately.


    Here is the link to the DESMOS solution.  

    https://www.desmos.com/calculator/aivq2p1eli

    Click on the intersection point to get its coordinates.



This value  r = 0.02077  is the monthly effective rate - - - so, the annual nominal rate is 12 times this value

    {{{r[annual]}}} = 12*0.02077 = 0.24924,  or about 0.25,

which corresponds to 25%.


<U>ANSWER</U>. In this problem, the annual interest rate is about 25% compounded monthly.

        Surely, from the usual common sense, this rate is extremely high,
        but it tells only that this problem is Math - - - not from real life.
</pre>

Solved.


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<pre>
To check my solution, I substituted this value r= 0.02077

into the loan function  f(r) = {{{r/(1-(1+r)^(-24))}}} = {{{0.02077/(1-1.02077^(-24))}}}.


It returns the value 0.053333396, which is quite close to  0.053333333, so my solution is confirmed.




To check the solution from the other tutor, I substituted his value r= 0.01

into the loan function  f(r) = {{{r/(1-(1+r)^(-24))}}} = {{{0.01/(1-1.01^(-24))}}}.


It returns the value 0.047073, which is not close to  {{{3200/60000}}} = 0.053333333.


So, the other's tutor solution is wrong, which is confirmed.
</pre>