33. BANKER'S DISCOUNT
IMPORTANT CONCEPTS
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 to 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 value (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
known 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.
IMPORTANT FORMULAE
1.
B.D. = S.I. on bill for unexpired time.
2. B.G. = (B.D.)-(T.D.) =
S.I. on T.D. =(T.D)2 /P.W .
3. T.D. = Ö(P.W.xB.G.)
4. B.D. =[(Amount *Rate *Time)/100]
5.Amount=[(B.D.
x T.D.)/(B.D.-T.D.)]
6. T.D.= [(Amount x Rate x
Time)/(100+(Rate*Time))]
7.
T.D.=[( B.G. x 100)/(Rate x Time)]
SOLVED EXAMPLES
Ex. 1.
A bill for Rs. 6000 is drawn on July 14 at 5 months. It is discounted on 5th
October at 10%. Find the banker's discount, true discount, banker's gain and
the money that the holder of the bill receives.
Sol.
Face value of the bill = Rs. 6000.
Date
on which the bill was drawn = July 14 at 5 months. Nominally due date = December 14.
Legally
due date = December 17.
Date
on which the bill was discounted = October 5.
Unexpired
time : Oct. Nov. Dec.
26
+ 30 +
17 = 73 days =1/ 5Years
B.D. = S.I. on Rs. 6000 for 1/5 year
= Rs.
(6000 x 10 x1/5 x1/100)= Rs. 120.
T.D.
= Rs.[(6000 x 10 x1/5)/(100+(10*1/5))]
=Rs.(12000/102)=Rs. 117.64.
B.G.
= (B.D.) - (T.D.) = Rs. (120 - 117.64) = Rs. 2.36.
Money
received by the holder of the bill = Rs. (6000 - 120)
=
Rs. 5880.
Ex. 2. If the true discount on a
certain sum due 6 months hence at 15% is Rs. 120, what is the banker's discount
on the same sum for the same time and at the same rate?
Sol.
B.G. = S.I. on T.D.
= Rs.(120 x 15 x 1/2 x 1/100)
= Rs. 9.
(B.D.)
- (T.D.) = Rs. 9.
B.D.
= Rs. (120 + 9) = Rs. 129.
Ex. 3.
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.
Sol.
S.I.
on Rs. 1800 = T.D. on Rs. 1872.
P.W.
of Rs. 1872 is Rs. 1800.
Rs.
72 is S.I. on Rs. 1800 at 12%.
Time
=[(100 x 72)/ (12x1800)]year
1/3year
= 4 months.
Ex. 4.
The banker's discount and the true discount on a sum of money due 8 months
hence are Rs. 120 and Rs. 110 respectively. Find the sum and the rate percent.
Sol.
Sum
=[( B.D.*T.D.)/(B.D.-T.D.)]
= Rs.[(120x110)/(120-110)]
= Rs. 1320.
Since
B.D. is S.I. on sum due, so S.I. on Rs. 1320 for 8 months is Rs. 120.
Rate
=[(100 x120)/( 1320 x 2/3)%
= 13
7/11%.
Ex. 5.
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 banker's
gain.
Sol. T.D. =Ö(P.W.*B.G)
B.G.
=(T.D.)2/ P.W.
= Rs.[(110x110)/ 1100]
= Rs. 11.
B.D.=
(T.D. + B.G.) = Rs. (110 + 11) = Rs. 121.
Ex. 6.
The banker's discount on Rs. 1650 due a certain time hence is Rs. 165. Find the
true discount and the banker's gain.
Sol.
Sum = [(B.D.xT.D.)/ (B.D.-T.D.)]
= [(B.D.xT.D.)/B.G.]
T.D./B.G. = Sum/ B.D.
=1650/165
=10/1
Thus, if B.G. is Re 1, T.D. = Rs. 10.
If
B.D.is Rs. ll, T.D.=Rs. 10.
If
B.D. is Rs. 165, T.D. = Rs. [(10/11)xl65]
=Rs.150
And, B.G. = Rs. (165 - 150) = Rs, 15.
Ex. 7. What rate percent does a man get
for his money when in discounting a bill due 10 months hence, he deducts
10% of the amount of the bill?
Solution: Let amount of the bill =
Rs.100
Money deducted =Rs.10
Money received by the holder of the
bill = Rs.100-10 = Rs.90
SI on Rs.90 for 10 months = Rs.10
Rate =[(100*10)/(90*10/12)%=13 1/3%
