Question 1193768
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A P38,000 loan bears interest at 10% compounded semi-annually and is to be repaid 
in semi-annual payments of P2,000 each. 
a. How many semi-annual payments must be the debtor make?
b. What smaller final payment should he make six months after the last payment 
of P2,000 is made?
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<pre>
Use the standard formula for the semi-annual payment for a loan

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


where L is the loan amount; r = {{{0.1/2}}} is the effective semi-annual compounding interest rate;
n is the number of payments; P is the semi-annual payment.


In this problem  P = $2000;  r = {{{0.1/2}}} = 0.05.


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

    2000 = {{{38000*(0.05/(1-1.05^(-n))))}}}.


In this equation, n is the unknown: we should find n from this equation.


Simplify step by step

    {{{2000/38000}}} = {{{(0.05/(1-1.05^(-n)))}}},

    0.052631579 = {{{(0.05/(1-1.05^(-n)))}}},

    {{{0.052631579/0.05}}} = {{{(1/(1-1.05^(-n)))}}},

    1.05263158 = {{{(1/(1-1.05^(-n)))}}},

    {{{1/1.05263158}}} = {{{1-1.05^(-n)}}},

    0.95 = {{{1-1.05^(-n)}}},

    {{{1.05^(-n)}}} = 1 - 0.95,

    {{{1.05^(-n)}}} = 0.05,

    {{{1/1.05^n}}} = 0.05,

    1.05^n = 1/0.05,

    1.05^n = 20,

    n*log(1.05) = log(20),

    n = {{{log((20))/log((1.05))}}} = 61.4.


So, 61 full semi-annual payments should be made of 2,000 each,
and then the last,62-th payment, should be made of the lesser amount.


<U>ANSWER</U>.  61 full semi-annual payments should be made of 2,000 each,
         and then the last, 62-th payment, should be made of the lesser amount.

         The total number of semi-annual payments is 62.
</pre>

Solved.