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Banker's Discount

Important Formulae

Banker’s Gain (B.G.) = (Banker's Discount(B.D)) - (True Discount(T.D)) for the unexpired time.
1. B.D. = Simple Interest(S.I) on bill for unexpired time
2. B.G. = (B.D.) - (T.D.) = S.I. on T.D. = (T.D.)²/P.W(Present worth)
3. T.D. = √P.W.x B.G
4. B.D = Amount X Rate X Time/100
5. T.D = Amount X Rate X Time / 100 + (Rate X Time)
6. Amount = B.D x T.D. / B.D. - T.D
7. T.D = B.G x 100 / Rate X Time

General Tips

Suppose a merchant A buys goods worth, say Rs. 10,000 from another merchant B at a credit of say 5 months. Then, B prepares a bill, called the bill of exchange. A signs this bill and allows B to withdraw the amount from his bank account after exactly 5 months
The date exactly after 5 months is called nominally due date. Three days (known as grace days) are added to it get a date, known as legally due date
Suppose B wants to have the money before the legally due date. Then he can have the money from the banker or a broker, who deducts S.I. on the face vale (i.e., Rs. 10,000 in this case) for the period from the date on which the bill was discounted (i.e., paid by the banker) and the legally due date. This amount is know as Banker's Discount (B.D.)
Thus, B.D. is the S.I. on the face value for the period from the date on which the bill was discounted and the legally due date
Example 1:
The banker’s discount on Rs.1600 at 15% per annum is the same as true discount on Rs.1680 for the same time and at the same rate. The time is:
Solution

S.I on Rs. 1600 =T.D on Rs. 1680
Rs.1600 is the P.W of Rs.1680
Rs.80 is S.I on Rs.1600 at 15%
Time = (100x80 / 600x15) = 1/3 year
= 4 months

Example 2:
The banker's discount on Rs.1800 at 12% per annum is equal to the true discount on Rs.1872 for the same time at the same rate. Find the time?
Solution

S.I on Rs.1800 = T.D on Rs.1872
P.W on Rs.1872 is Rs.1800
Rs.72 is S.I on Rs. 1800 at 12%
Time =(100x72 / 12x1800)
= 1/3 year
= 4 months

Example 3:
The banker's gain of a certain sum due 2 years hence at 10% per annum is Rs.24. The present worth is:
Solution:
T.D = (B.G x 100/Rate x Time)
= Rs.(24x100 / 10x 2)
= Rs.120
P.W = (100 x T.D/Rate x Time)
= Rs.(100x120 / 10x 2)
= Rs.600