Question 1203727
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On average, it needs Php 250,000 per year and an additional Php 600,000 every 4 years for major repairs 
to maintain a certain three-storey office building. How much must be paid now to take care 
of these payments on a bank that offers 8% per year?
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<pre>
We follow to this scheme/scenario.

(0)  At the end of the year "0", person A deposits the amount of Php X to the bank.   
     The value of X is unknown and should be determine in the solution.



(1)  On Jan 1 of the year 1, the person withdraws Php 250,000 for the regular annual repair.
     Now, there are (X-250000) Php at the account.

     On Dec 1 of the year 1, these (X-250000) Php become (X-250000)*1.08 Php after compounding.



(2)  On Jan 1 of the year 2, the person withdraws Php 250,000 for the regular annual repair.
     Now, there are (X-250000)*1.08-250000 Php at the account.

     On Dec 1 of the year 2, these (X-250000)*1.08-250000  Php become ((X-250000)*1.08-250000)*1.08 Php after compounding.



(3)  On Jan 1 of the year 3, the person withdraws Php 250,000 for the regular annual repair.
     Now, there are ((X-250000)*1.08-250000)*1.08-250000  Php at the account.

     On Dec 1 of the year 2, these ((X-250000)*1.08-250000)*1.08-250000 Php become (((X-250000)*1.08-250000)*1.08-250000)*1.08 Php after compounding.



(4)  On Jan 1 of the year 4, the person withdraw Php 250,000 for the regular annual repair, and he (or she) should still have
     enough remaining money at the account to cover Php 600,000 of the 4-years repair.


So, this equation should be held

     (((X-250000)*1.08-250000)*1.08-250000)*1.08 - 250000 = 600000.


From this equation, we should find the unknown value of X.


To do it, simplify this equation, step by step.

     (((X-250000)*1.08-250000)*1.08-250000)*1.08 = 600000 + 250000

     (((X-250000)*1.08-250000)*1.08-250000)*1.08 = 850000


Divide both sides by 1.08

     ((X-250000)*1.08-250000)*1.08-250000 = 850000/1.08 

     ((X-250000)*1.08-250000)*1.08-250000 = 787037.04      (I am rounding to the closest cent)


     ((X-250000)*1.08-250000)*1.08 = 250000 + 787037.04   

     ((X-250000)*1.08-250000)*1.08 = 1037037.04   


Divide both sides by 1.08

     (X-250000)*1.08-250000 = 1037037.04/1.08   

     (X-250000)*1.08-250000 = 960219.48                    (I am rounding to the closest cent)


     (X-250000)*1.08 = 250000 + 960219.48   

     (X-250000)*1.08 = 1210219.48


Divide both sides by 1.08

      X-250000 = 1210219.48/1.08

      X-250000 = 1120573.59                                (I am rounding to the closest cent)

         X     = 250000 + 1120573.59

         X     = 1370573.59


<U>ANSWER</U>.  The original / (initial) deposit should be  Php 1,370,573.59.


Compare it with the required amount of Php 600,000 + Php 4*250,000 = Php 1,600,000 if the compounded account is not used.
</pre>

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Solved, with complete and clear explanation.


It can be solved using more complicated technique, but in this post 
my goal was to explain the idea clearly, not the advanced technique.