IMPORTANT FORMULAE:
1.B.D. = S.I. on bill for unexpired time.
2.B.G. = (B.D.) - (T.D.) = S.I. on T.D. =$ \dfrac{\left(T.D\right)^2}{P.W}$
3.T.D. =$\sqrt{P.W\times B.G}$
4.B.D. = $\left[\dfrac{Amount \times Rate\times Time}{100}\right]$
5.T.D. = $\left[\dfrac{Amount \times Rate\times Time}{100\ + \left( Rate \times Time \right)}\right] $
6. Amount = $\left[\dfrac{B.D\times T.D}{B.T - T.D}\right]$
7.T.D. = $\left[\dfrac{B.D\times 100}{Rate \times time}\right]$
Banker's Discount Important Concepts & Examples:
Banker's Discount:
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.
Banker's Gain (B.G.) = (B.D.) - (T.D.)for the unexpired time.
Note:
When the date of the bill is not given, grace days are not to be added.
Note:
When the date of the bill is not given, grace days are not to be added.
Question 1:
The true discount on a bill Rs.1860 due after 8 months is Rs.60. Find the rate, the banker’s discount and the banker’s gain.
Solution :
Amount = Rs.1860, T.D.= RS.60
P.W. = Rs.(1860-60)
= Rs. 1800
S.I. on Rs. 1800 for 8 months
= Rs.60
Rate= 1800×601800×23 %
= 5% B.G.=(T.D.)2(P.W.)
=Rs.60×601800 = Rs. 2
B.D.= (T.D.) + (B.G.)
= Rs.(60+2)
= Rs.62.
Question 2:
The present worth of a bill due sometime hence is Rs. 1100 and the true discount on the bill is Rs. 110. Find the banker’s discount and the extra gain the banker would make in the transaction.
Solution :
T.D.=$\sqrt{(P.W.)×(B.G.)}$
Or B.G.=$\dfrac{(T.D.)^2}{P.W.}$
= Rs.$\dfrac{110×110}{1100}$
= Rs.11
B.D.= B.G. + T.D.
= Rs.(11+110)
= Rs.121
Question 3:
The banker’s discount and the true discount on a sum of money due 8 months hence are Rs. 52 and Rs. 50, respectively. Find the sum and the rate per cent.
Solution :
Sum = $\dfrac{(B.D.)×(T.D.)}{(B.D.)-(B.D.)}$
= Rs.$\dfrac{52×50}{2}$
= Rs.1300
Since B.D. is S.I. on sum due. So S.I. on Rs. 1300 for 8 months is Rs. 52, Consequently.
Rate=$\dfrac{100×52}{1300×\dfrac{2}{3}}$ %=6%
Question 4:
The banker’s discount on Rs. 1800 at 5% is equal to the true discount on Rs. 1830 for the same time and at the same rate, Find the time.
Solution :
S.I. on Rs. 1800
= T.D.on Rs. 1830
P.W. of Rs. 1830 is Rs. 1800
i.e.., Rs. 30 is S.I. on Rs. 1800 at 5%
Time=$\dfrac{100×30}{1800×5}$ years
= $\dfrac{1}{3} years$
=4 months
Question 5;
If the true discount on a certain sum due 6 months hence at 6% is Rs. 36, what is the banker’s discount on the same sum for the same time and at the same rate ?
Solution :
B.G.= S.I. on T.D.
= Rs.$\dfrac{3600×6×1}{100×2}$
= Rs.1.08
(B.D.)–(T.D.)=Rs. 1.08
Or B.D.=(T.D.)+Rs.1.08
= Rs.(36+1.08)
= Rs.37.08