1.16 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICSArithmetic
Formula:
(i) The present value V of an annuity P to continue for n years is given by
V=Pi{1 1 i− +( )−n} Where i = interest per rupee per annum.
(ii) The Present value V of an annuity P payable half-yearly, then
2P i 2n
V i 1 1 2− (^)
= − +
(^)
(ii) The Present value V of an annuity P payable quarterly, then
4P i 4n
V i 1 1 4
− (^)
= − +
(^)
SOLVED EXAMPLES:
Example 27 :
A man decides to deposit 20,000 at the end of each year in a bank which pays 10% p.a. compound interest. If the instalments are allowed to accumulate, what will be the total accumulation at the end of 9 years? Solution: LetA be the total accumulation at the end of 9 years. Then we have
A=Pi{(1 i 1+)n−}
Here P = 20,000, i = 10010 = 0.1, n = 9 years. ∴ A = 000,20.0 1 {(1 .1 1 2,00,000 1.1 1+ )^9 −}= {( )^9 −} = 2,00,000 (2.3579– 1) = 2,00,000 x 1.3579 = 2,71,590
∴ The required total accumulation = 2,71,590. Example 28 : A truck is purchased on instalment basis, such that10,000 is to be paid on the signing of the contract and
five yearly instalments of 5,000 each payable at the end of 1st, 2nd, 3rd, 4th and 5th years. If interest is charged at 10% per annum what would be the cash down price? Solution: Let V be the present value of the annuity of5,000 for 5 years at 10% p.a. compound interest, then cash
down price of the truck is `(10,000+ V).
Now, V=pi {1 (1 i)− + −n}. Here, P= 5,000, n= 5, i= 10010 =0.1
{ ( ) }
5 5
5
V 5,000 1 1.1 50,000 (1.1) 1 50,000 1.61051 1
0.1 (1.1) 1.61051
∴ = − − =^ −^ =^ −^
(^)
= 50,000×1.610510.61051 = 50,000x 0.379079
Hence the required cash down price of the truck= (18,953.95+ 10,000) =28,953.95