### Chapter: True Discount

Concepts Of Banker's Discount:

Assume that a merchant A purchases goods worth, say Rs.1000 from another merchant B at a credit of say 4 months.

Then B prepares a bill called bill of exchange (also called Hundi). On receipts of goods, A gives an agreement by signing on the bill allowing B to withdraw the money from A’s bank exactly after 4 months of the date of the bill.

The date exactly after 4 months is known as nominally due date. Three more days (called grace days) are added to this date to get a date known as legally due date.

The amount given on the bill is called the Face Value (F) which is Rs.1000 in this case.

Assume that B needs this money before the legally due date. He can approach a banker or broker who pays him the money against the bill, but somewhat less than the face value. The banker deducts the simple interest on the face value for the unexpired time. This deduction is known as Bankers Discount (BD). In another words, Bank Discount (BD) is the simple interest on the face value for the period from the date on which the bill was discounted and the legally due date.

The present value is the amount which, if placed at a particular rate for a specified period will amount to that sum of money at the end of the specified period. The interest on the present value is called the True Discount (TD). If the banker deducts the true discount on the face value for the unexpired time, he will not gain anything.

Banker’s Gain (BG) is the difference between banker’s discount and the true discount for the unexpired time.

Let's understand True Discount by taking an example so that it will be easy to understand:

Suppose a man has to pay Rs.156 after 4 years and the rate of interest is 14%per annum.
Clearly ,Rs.100 at 14% will amount to Rs156 in 4 years
So,the payment of Rs.100 now wil cear off the debt of Rs156 due 4 years hence
We say that Sum due = Rs156 due 4 years hence Present Worth =Rs100
True Discount=Rs.156-Rs100=Rs56=Sumdue-PW
We define
TrueDiscount= Interest on Present Worth(PW) Amount = PresentWorth+TrueDiscount

Very Important Formulae on True Discount:

Let rate=R%per annum, Time= T years

1.Present worth (PW) = (100*Amount)/(100+(R*T))
= (100* TrueDiscount)/(R*T)

2.TrueDiscount (TD) = (P.W*R*T)/100
= (Amount*R*T)/(100+(R*T))

3.Sum =(SimpeInterest*TrueDiscount)/(SimpleInterest-TrueDiscount)

4.Simple Interest-True Discount=Simple Interest on True Discount

5.When the sum is put at Compound Interest,then

PresentWorth=Amount/(1+(R/100))^T

For solved problems on above formulas please visit below sections: