True Discount:
Suppose a man has to pay Rs. 142 after 3 years and the rate of interest is 14% per annum. Clearly, Rs. 100 at 14% will amount to R.142 in 3 years.
So, the payment of Rs. now will clear off the debt of Rs. 142 due 3 years hence.
Sum due = Rs. 142 due 3 years hence;
Present Worth (P.W.) = Rs. 100;
True Discount (T.D.) = (Sum due) - (P.W.)= Rs. (142 - 100) = Rs. 42
We define: T.D. = Interest on P.W.; Amount = (P.W.) + (T.D.)
Interest is based on P.W. and true discount is based on the amount.
Formulae:
Let rate = R% per annum and Time = T years. Then,
P.W. = 100 x Amount/(100 + (R x T)) = 100 x T.D./R x T
T.D. = (P.W.) x R x T/100 = Amount x R x T/(100 + (R x T))
Sum = (S.I.) x (T.D.)/(S.I.) - (T.D.)
(S.I.) - (T.D.) = S.I. on T.D.
When the sum is put at compound interest, then P.W. = Amount/[(1 + (R/100)] T
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