# Banker's Discount

The true discount on a bill of Rs. 2160 is Rs. 360. What is the banker’s discount?

F = Rs. 2160, TD = Rs. 360

⇒ PW = F - TD
= 2160 - 360
= Rs. 1800

True Discount is the Simple Interest on the present value for unexpired time

⇒ Simple Interest on Rs. 1800 for unexpired time = Rs. 360

Banker's Discount is the Simple Interest on the face value of the bill for unexpired time
= Simple Interest on Rs. 2160 for unexpired time
= $$\frac{360}{1800} \times 2160$$
= 15 × 2160
= Rs. 432

The banker’s discount on a sum of money for 3 years is Rs. 1116. The true discount on the same sum for 4 years is Rs. 1200. What is the rate per cent?

BD for 3 years = Rs. 1116
BD for 4 years = 11163 × 4 = Rs. 1488

TD for 4 years = Rs. 1200

$$F = \frac{BD \times TD}{BD - TD}$$

= $$\frac{1488 \times 1200}{1488 - 1200}$$
= $$\frac{1488 \times 1200}{288}$$
= $$\frac{124 \times 1200}{24}$$
= $$\frac{124 \times 100}{2}$$
= 62 × 100 = Rs. 6200

⇒ Rs.1488 is the simple interest on Rs. 6200 for 4 years
⇒ $$1488 = \frac{6200 \times 4 \times R}{100}$$
⇒ $$R = \frac{1488 \times 100}{6200 \times 4}$$

= $$\frac{372 \times 100}{6200}$$
= $$\frac{372}{62}$$
= 6%

A banker paid Rs. 5767.20 for a bill of Rs. 5840, drawn on Apr 4 at 6 months. If the rate of interest was 7%, what was the day on which the bill was discounted?

F = Rs.5840, R = 7%

BD = 5840 - 5767.20 = Rs.72.8
$$BD = \frac{F \times T \times R}{100}$$

⇒ $$72.8 = \frac{5840 \times T \times 7}{100}$$

⇒ $$T = \frac{72.8 \times 100}{7 \times 5840}$$
= $$\frac{10.4 \times 100}{5840}$$
= $$\frac{1040}{5840}$$
= $$\frac{104}{584}$$
= $$\frac{13}{73}$$ years

= $$\frac{13 \times 365}{73}$$ Days = 65 Days

⇒ Unexpired Time = 65 Days
Given that Date of Draw of the bill = 4th April at 6 months
⇒ Nominally Due Date = 4th October
⇒ Legally Due Date = (4th October + 3 days) = 7th October

Hence, The date on which the bill was discounted
= (7th October - 65 days)
= (7th October - 7 days in October - 30 days in September - 28 days in August)
= 3rd August

A bill is discounted at 10% per annum. If banker’s discount is allowed, at what rate percent should the proceeds be invested so that nothing will be lost?

Let the amount = Rs.100

Then BD = Rs.10 (∵ banker's discount, BD is the simple Interest on the face value of the bill for unexpired time and bill is discounted at 10% per annum)

Proceeds = Rs.100 – Rs.10 = Rs.90

Hence we should get Rs. 10 as the interest of Rs. 90 for 1 year so that nothing will be lost
⇒ $$10 = \frac{90 \times 1 \times R}{100}$$

⇒ $$R = \frac{10 \times 100}{90}$$

⇒ $$R = \frac{100}{9}$$

The true discount on a certain sum due 6 months hence at 15% is Rs. 240. What is the banker’s discount on the same sum for the same time at the same rate?

TD = Rs. 240, T = 6 months = $$\frac{1}{2}$$ years, R = 15%

$$TD = \frac{BG \times 100}{TR}$$

⇒ $$240 = \frac{BG \times 100}{\frac{1}{2} \times 15}$$

⇒ $$BG = \frac{240 \times 15}{100 \times 2} = \frac{120 \times 15}{100}$$ = Rs. 18

BG = BD - TD
⇒ 18 = BD - 240
⇒ BD = 18 + 240 = Rs. 258

The present worth of a certain bill due sometime hence is Rs. 400 and the true discount is Rs. 20. What is the banker’s discount?

$$BG = \frac{TD^{2}}{PW} = \frac{202}{400}$$ = Rs. 1

BG = BD – TD
⇒ 1 = BD - 20
⇒ BD = 1 + 20 = Rs. 21

The banker’s discount on a sum of money for $$\frac{3}{2}$$ years is Rs. 120. The true discount on the same sum for 2 years is Rs.150. What is the rate percent?

BD for $$1\frac{1}{2}$$ years = Rs. 120

⇒ BD for 2 years = $$120 \times \frac{2}{3} \times 2$$ = Rs.160

TD for 2 years = Rs. 150
⇒ $$F = \frac{BD \times TD}{BD - TD}$$
⇒ $$F = \frac{160 \times 150}{160 - 150}$$
⇒ $$F = \frac{160 \times 150}{10}$$
⇒ F = Rs. 2400

Rs.160 is the simple interest on Rs. 2400 for 2 years
⇒ $$160 = \frac{2400 \times 2 \times R}{100}$$
⇒ $$R = \frac{160 \times 100}{2400 \times 2}$$
⇒ $$R = \frac{160}{48}$$
⇒ $$R = \frac{10}{3}$$
⇒ $$R = 3\frac{1}{3}$$ %

The banker’s discount on a bill due 6 months hence at 6% is Rs. 18.54. What is the true discount?

T = 6 months = $$\frac{1}{2}$$ year
R = 6%

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

⇒ $$TD = \frac{18.54 \times 100}{100 + \left ( \frac{1}{2} \times 6 \right )}$$

⇒ $$TD = \frac{18.54 \times 100}{103}$$

⇒ $$TD = \frac{1854}{103}$$

⇒ TD = Rs. 18

The B.D. and T.D. on a certain sum is Rs.200 and Rs.100 respectively. Find out the sum.

⇒ $$F = \frac{BD \times TD}{BD - TD}$$

⇒ $$F = \frac{200 \times 100}{200 - 100}$$

⇒ $$F = \frac{200 \times 100}{100}$$

⇒ F = Rs. 200

The B.G. on a certain sum 4 years hence at 5% is Rs. 200. What is the present worth?

T = 4 years

R = 5%

Banker's Gain, BG = Rs.200

$$TD = \frac{BG \times 100}{TR} = \frac{200 \times 100}{4 \times 5}$$

$$TD = \sqrt{PW \times 200}$$

$$1000 = \sqrt{PW \times 200}$$

1000000 = PW × 200

$$PW = \frac{1000000}{200}$$ = Rs. 5000