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. Approach a banker or broker, which pays 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).

Where, BD = Bankers Discount
T = Time in Years
F = Face Value of the Bill,

Example 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?
Solution:

Face value of the bill = Rs. 6000.
Date on which the bill was drawn = july 14 ar 5 months.
Nominally duw date = December 14. Legally duw date = December 17.
Date on which the bill was discounted = October 5.
Unexpected time: 26(Oct) + 30(Nov) + Dec(17) = 73 days = [latex]\frac{1}{5}[/latex]years.
Therefore, B.D = S.I. on Rs. 6000 for [latex]\frac{1}{5}[/latex] year = Rs.(6000 x 10 x [latex]\frac{1}{5}[/latex] x [latex]\frac{1}{100}[/latex]) = Rs. 120.

Example 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 the same rate?
Solution:

B.G. = S.I. on T.D. = Rs. (120 x 15 x [latex]\frac{1}{2}[/latex] x [latex]\frac{1}{100}[/latex]) = Rs 9.
Therefore, (B.D.) - (T.D.) = Rs. 9
B.D. = Rs. (120 + 9) = Rs. 129.

Where, B.G. = Banker’s Gain
B.D. = Bankers Discount
T.D. = True Discount
P.W. = True Present Worth

Example 1: The present worth of a certain sum due sometime hence is Rs. 1600 and the true discount is Rs. 160. The banker's gain is?
Solution:

B.G. = [latex]\frac{(T.D.)^{2}}{P.W.}[/latex] = Rs. ([latex]\frac{160 \times 60}{1600}[/latex]) = Rs. 16.

Example 2: The Present worth of a certain bill due sometime hence is Rs. 800 and the true discount is Rs. 36. The banker's discount is:
Solution:

B.G. = [latex]\frac{(T.D.)^{2}}{P.W.}[/latex] =Rs. ([latex]\frac{36 \times 36}{800}[/latex]) = Rs. 1.62.
Therefore, B.D. = (T.D. + B.G.) = Rs. (36 + 1.62) = Rs. 37.62.

Example: The present worth of a sum due sometime hence is Rs. 576 and the banker's gain is Rs. 16. The true discount is:
Solution:

T.D. = [latex]\sqrt{P.W. \times B.G.}[/latex] = [latex]\sqrt{576 \times 16}[/latex] = 96.

Example: The banker's discount on a bill due 4 months hence at 15% is Rs. 420. The true discount is:
Solution:

T.D. = [latex]\frac{B.D. \times 100}{100 + (R \times T)}[/latex] = Rs. [latex][\frac{420 \times 100}{100 + (15 \times \frac{1}{3})}][/latex] = Rs. ([latex]\frac{420 \times 100}{105}[/latex]) = Rs. 400.

Example 1: The Banker's gain on a bill due 1 year hence at 12% per annum is Rs. 6. The true discount is:
Solution:

T.D. = [latex]\frac{B.G. \times 100}{R \times T}[/latex] = Rs. [latex](\frac{6 \times 100}{12 \times 1})[/latex] = Rs. 50.

Example 2: The banker's gain on a sum due 3 years hence at 12% per annum is Rs. 270. The banker's discount is:
Solution:

T.D. = [latex](\frac{B.G. \times 100}{R \times T})[/latex] = Rs. [latex](\frac{270 \times 100}{12 \times 3})[/latex] = Rs. 750.
Therefore, B.D. = Rs. (750 + 270) = Rs. 1020.

1. Banker's Gain(B.G.) = (Simple Interest)S.I. on bill for unexpected time.
2. Banker's Gain(B.G.) = Banker's discount - True discount = Simple interest on true discount = [latex]\frac{(true discount)^2}{Present worth}[/latex]
3. True discount = [latex]\sqrt{P.W. * B.G.}[/latex]
4. B.D. = [latex]\frac{Amount * Rate * Time}{100}[/latex]
5. T.D = [latex]\frac{Amount * Rate * Time}{100 + (Rate * Time)}[/latex]
6. Amount = [latex]\frac{B.D. * T.D.}{B.D. - T.D.}[/latex]
7. T.D. = [latex]\frac{B.G. * 100}{Rate * Time}[/latex]

1. The present worth of a bill due something hence is Rs. 1200 and the true discount on the bill is Rs. 120. Find the banker's discount and the banker's gain?
Solution:

Given that,
True discount = 120
Present worth = 1200
Now,
True discount = [latex]\sqrt{P.W. * B.G.}[/latex]
⇒ B.G. = [latex]\frac{(T.d)^2}{P.W.}[/latex]
⇒ B.G. = [latex]\frac{(120 * 120}{1200}[/latex]
⇒ B.G. = Rs. 12
Therefore, B.D. = (T.D. + B.G.) = Rs. (120 + 12) =Rs. Rs. 132

2. The banker's discount on Rs. 1600 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:

Given that,
S.I. on Rs. 1600 = T.D. on Rs. 1872
Therefore, P.W. of Rs. 1872 is Rs. 1600
Therefore, Rs.72 is S.I. on Rs. 1600 at 12%
here, B.G. = Rs. 72
Rate = 12%
T.D. = Rs. 1600
Now, consider
T.D. = [latex]\frac{B.G. * 100}{Rate * Time}[/latex]
⇒ Time = [latex]\frac{B.G. * 100}{Rate * T.D.}[/latex]
⇒ Time = [latex]\frac{72 * 100}{12 * 1600}[/latex]
⇒ Time = [latex]\frac{3}{8}[/latex] years

3. What rate percent does a man get for his money when in discounting a bill due 10 months hence, deduce 20% of the amount of the bill?
Solution:

Given that,
Rate = 10%
Time = 10 months
Let amount of the bill = Rs. 100.
Money deducted = Rs. 20
Money received by the holder of the bill = Rs. (100 - 20) = Rs. 80.
Therefore, S.I. on Rs. 80 for 10 months = Rs. 20
Therefore, Rate = [latex]\frac{B.G. * 100}{Time * T.D.}[/latex]
⇒ Rate = [latex]\frac{20 * 100}{\frac{10}{12} * 80}[/latex]
⇒ Rate = 30%

4. If the true discount on a certain sum due 6 months hence at 20% is Rs. 140, 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. on Rs. (140 x 20 x [latex]\frac{1}{2}[/latex] x [latex]\frac{1}{100}[/latex]) = Rs. 14
Therefore, B.G. - T.D. = Rs. 14
B.G. = Rs. (140 + 14) =Rs. 154.

5. The banker's discount on Rs. 1850 due to certain time hence is Rs. 185. Find the true discount and the banker's gain?
Solution:

Given that
B.G. = Rs. 185
Sum = Rs. 1850
Now, Sum = [latex]\frac{B.G. * T.D.}{B.G. - T.D.}[/latex] = [latex]\frac{B.D. * T.D.}{B.G.}[/latex]
Therefore, [latex]\frac{T.D.}{B.G.}[/latex] = [latex]\frac{sum}{B.D.}[/latex] = [latex]\frac{1850}{185}[/latex] = 10
Thus, if B.G. is Re. 1, T.D. = Rs. 10.
If B.D. is Rs. 185, T.D. = Rs.([latex]\frac{10}{11}[/latex] x 185) = Rs. 169
B.G. = Rs. (185 - 169) = Rs. 16.