# Banker's Discount

A bill for Rs. 3000 is drawn on 14th July at 5 months. It is discounted on 5th October at 10%. What is the Banker’s Discount?

F = Rs. 3000, R = 10%

Date on which the bill is drawn = 14th July at 5 months

Nominally Due Date = 14th December

Legally Due Date = 14th December + 3 days = 17th December

Date on which the bill is discounted = 5th October

Unexpired Time
= [6th to 31st of October] + [30 Days in November] + [1st to 17th of December]
= 26 + 30 + 17
= 73 Days
= $$\frac{73}{365}$$ year = $$\frac{1}{5}$$ year

BD = Simple Interest on the face value of the bill for unexpired time
= $$\frac{F \times T \times R}{100}$$
= $$\frac{3000 \times \frac{1}{5} \times 10}{100}$$
= $$30 \times \frac{1}{5} \times 10$$
= Rs. 60

What is the banker’s discount if the true discount on a bill of Rs. 540 is Rs. 90 ?

Present Worth, PW = F - TD = 540 - 90 = Rs. 450

Simple Interest on the Present Worth = True Discount
Hence Simple Interest on 450 = 90 ........(Equation 1)

Simple Interest on the face value = Bankers Discount
⇒ Simple Interest on 540 = Bankers Discount

From Equation 1, Simple Interest on 450 = 90
Hence, Simple Interest on 540 = $$\frac{90}{450} \times 540$$ = $$\frac{540}{5}$$ = Rs. 108
⇒ Bankers Discount = Rs. 108

The present worth of a sum due sometimes hence is Rs. 5760 and the banker’s gain is Rs. 10. What is the true discount?

$$TD = \sqrt{PW \times BG}$$
$$TD = \sqrt{5760 \times 10}$$
$$TD = \sqrt{57600}$$
TD = Rs. 240

The banker’s discount on a certain amount due 2 years hence is 1110 of the true discount. What is the rate percent?

Let TD = Rs. 1

Then BD = $$\frac{11}{10} \times 1$$ = Rs. $$\frac{11}{10} T = 2, R = ? F = [latex]\frac{BD \times TD}{BD - TD}$$
F = $$\frac{\frac{11}{10} \times 1}{\frac{11}{10} - 1}$$
F = $$\frac{\frac{11}{10}}{\frac{1}{10}}$$ = Rs. 11

BD = $$\frac{F \times T \times R}{100}$$

⇒ $$\frac{11}{10} = \frac{11 \times 2 \times R}{100}$$

⇒ 110 = 22R

⇒ R = $$\frac{110}{22}$$ = 5%

The banker’s discount on a bill due 4 months hence at 15% is Rs. 420. What is the true discount?

$$TD = \frac{BD \times 100}{100 + TR}$$

$$TD = \frac{420 \times 100}{100 + \left (\frac{4}{12} \times 15 \right )}$$

$$TD= \frac{420 \times 100}{100 + \left (\frac{1}{3} \times 15 \right )} = \frac{420 \times 100}{100 + 5}$$

$$TD = \frac{420 \times 100}{105}$$

$$TD = \frac{84 \times 100}{21}$$ = 4 × 100 = 400

The banker’s gain on a bill due 1 year hence at 10% per annum is Rs. 20. What is the true discount?

$$TD = \frac{BG \times 100}{TR}$$
$$TD = \frac{20 \times 100}{1 \times 100}$$
TD = Rs. 200

The banker’s discount of a certain sum of money is Rs. 36 and the true discount on the same sum for the same time is Rs. 30. What is the sum due?

$$F = \frac{BD \times TD}{BD - TD}$$
$$F = \frac{36 \times 30}{36 - 30}$$
$$F = \frac{36 \times 30}{6}$$
F = 36×5
F = Rs. 180

The present worth of a certain bill due sometime hence is Rs. 1296 and the true discount is Rs. 72. What is the banker’s discount?

BG = $$\frac{TD^{2}}{PW}$$
BG = $$\frac{72^{2}}{1296}$$
BG = $$\frac{72 \times 72}{1296}$$
BG = $$\frac{12 \times 12}{36} = \frac{12}{3}$$ = Rs. 4

⇒ BG = BD – TD
⇒ 4 = BD - 72
⇒ BD = 72 + 4
= Rs. 76

The banker’s gain on a sum due 6 years hence at 12% per annum is Rs. 540. What is the banker’s discount?

$$TD = \frac{BG \times 100}{TR}$$
$$TD = \frac{5400 \times 100}{6 \times 12}$$
$$TD = \frac{90 \times 100}{12}$$
$$TD = \frac{15 \times 100}{2}$$
TD = Rs. 750

⇒ BG = BD – TD
⇒ 540 = BD – 750
⇒ BD = 540 + 750
= 1290

The banker’s discount and the true discount of a sum at 10% per annum simple interest for the same time are Rs. 100 and Rs. 80 respectively. What is the sum and the time?

BD = Rs.100, TD = Rs.80, R = 10%

$$F = \frac{BD \times TD}{BD - TD}$$
$$F = \frac{100 \times 80}{100 - 80}$$
$$F = \frac{100 \times 80}{20}$$ = Rs. 400

BD = Simple Interest on the face value of the bill for unexpired time = $$\frac{F \times T \times R}{100}$$

⇒ $$100 = \frac{400 \times T \times 10}{100}$$

⇒ 100 = 4 × T × 10

⇒ 10 = 4 × T

⇒ $$T = \frac{10}{4}$$ = 2.5 years