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Homework answers / question archive / A woman leaves an inheritance of $ 90,000 for her three children today
A woman leaves an inheritance of $ 90,000 for her three children today. The estate of $90,000 is to be placed in a fund yielding 10% converted semiannually from which each child is to receive an equal amount from the estate upon reaching age 21. After many years the woman dies when the children aged 13, 15, and 19. How much does each child get when reaching age
Let P be the equal payment each child gets,
the Estate is incorporated upon the death of the woman, at the time of which the children are aged 13,15 and 19.
So the eldest child (aged 19 ) will received the share in estate in 2 years
the second eldest child (aged 15 ) will received the share in estate in 6 years
the second eldest child (aged 13 ) will received the share in estate in 8 years
90000 invested at 10% pa compounded semi annually
so the effective annual interest rate will be
= ( 1+ r) ^t - 1 ; t is the number of times compounded.
since semi annual t =2
effective annual rate = ( 1+ 10% )^2 - 1
= 0.1025 or 10.25%
the future value of 90000 at end of 2 years when the first payment is due will be,
FV = PV * (1+r )^n r =10.25%
FV = 90000 * ( 1+ 10.25%)^2
= 109,395.56
after Payment of P, the balance at end of year 2 = 109,395.56 - P
this balance earns 10% semi annual interest until the end of year 6, when the next payment is due to the 2nd eldest child.
FV of above balance before payment
= ( 109,395.56 - P ) * ( 1+10.25%)^4
n = 4 years, ( 6 -2 years)
= ( 109,395.56 - P ) * 1.477455444
= 161627.07 - 1.477455444P
now, a payment of P is made to 2nd child, so the balance after payment will be
= 161627.07 - 1.477455444P - P
= 161627.07 - 2.477455444P
this balance earns 10% semi annual interest until the end of year 8, when the next payment is due to the youngest child
FV of above balance before payment
= ( 161627.07 - 2.477455444P ) * ( 1+10.25%)^2
n = 2 years, ( 8 - 6 years)
= ( 161627.07 - 2.477455444P ) * 1.21550625
= 196458.71 - 3.0113625763 P
now, a payment of P is made to 3rd child, so the balance after payment will be
= 196458.71 - 3.0113625763 P - P
= 196458.71 - 4.0113625763 P
the balance after this final payment must be 0, hence equating this to 0 we solve for P
4.0113625763 P = 196458.71
P = 196458.71 / 4.0113625763
= $48,975.56
