# Banker's Discount

The banker’s gain on a sum due 3 years hence at 12% per annum is Rs. 360. The banker’s discount is:

BG = Rs. 360
T = 3 years
R = 12%

⇒ TD = $$\frac{BG \times 100}{TR}$$
⇒ TD = $$\frac{360 \times 100}{3 \times 2}$$
⇒ TD = Rs. 1000

BG = BD – TD

⇒ BD = BG + TD
⇒ BD = 360 + 1000
⇒ BD = Rs. 1360

The banker’s gain on a certain sum due $$\frac{5}{2}$$ years hence is $$\frac{9}{25}$$ of the banker’s discount. What is the rate percent?

T = $$2\frac{1}{2}$$ years =$$\frac{5}{2}$$ years

Let the banker's discount, BD = Rs. 1

Then, banker's gain, BG = $$\frac{9}{25} \times 1$$ = Rs. $$\frac{9}{25}$$

BG = BD - TD

⇒ $$\frac{9}{25} = 1 - TD$$

⇒ $$TD = 1 - \frac{9}{25} = \frac{16}{25}$$

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

$$F = \frac{1 \times \frac{16}{25}}{1 - \frac{16}{25}}$$

$$F = \frac{\frac{16}{25}}{\frac{9}{25}} = Rs. \frac{16}{9}$$

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

⇒ $$1 = \frac{\frac{16}{9} \times \frac{5}{2} \times R}{100}$$

⇒ $$100 = \frac{16}{9} \times \frac{5}{2} \times R$$

⇒ $$100 = \frac{16 \times 5 \times R}{9 \times 2}$$

⇒ $$100 = \frac{8 \times 5 \times R}{9}$$

$$R = \frac{100 \times 9}{8 \times 5} ⇒ $R = \frac{100 \times 9}{40} = \frac{5 \times 9}{2} = \frac{45}{2} = 22\frac{1}{2}$$% The banker’s discount on Rs. 1600 at 15% per annum is the same as true discount on Rs. 1680 for the same time and at the same rate. What is the time? Answer: (a) Bankers Discount, BD = Simple Interest on the face value of the bill for unexpired time. True Discount, TD = Simple Interest on the present value for unexpired time. Simple Interest on Rs. 1600 = True Discount on Rs.1680 ⇒ Rs. 1600 is the Present Worth (PW) of Rs. 1680 ⇒ Rs. 80 is the Simple Interest of Rs.1600 at 15% ⇒ 80 = $$\frac{1600 \times T \times 15}{100}$$ ⇒ 80 = 16 × T × 15 ⇒ 5 = T × 15 ⇒ 1 = T × 3 ⇒ T = $$\frac{1}{3}$$ year = \frac{12}{3}$ months = 4 months

The present worth of a certain sum due sometime hence is Rs. 3400 and the true discount is Rs. 340. The banker’s gain is:

$$BG = \frac{TD^{2}}{PW}$$

$$BG = \frac{340^{2}}{3400}$$

$$BG = \frac{340 \times 340}{3400} = \frac{340}{10}$$ = Rs. 34

The banker’s gain of a certain sum due 3 years hence at 10% per annum is Rs. 36. What is the present worth ?

T = 3 years

R = 10%

TD = $$\frac{BG \times 100}{TR} = \frac{36 \times 100}{3 \times 10}$$ = 12 × 10 = Rs. 120

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

⇒ 120 = $$\frac{PW \times 3 \times 10}{100}$$

⇒ 1200 = PW × 3

⇒ PW = $$\frac{1200}{3}$$ = Rs. 400

What is the difference between the banker’s discount and the true discount on Rs. 8100 for 3 months at 5%

F = Rs. 8100
R = 5%
T = 3 months = $$\frac{1}{4}$$ years

⇒ BD = $$\frac{F \times T \times R}{100}$$ = $$\frac{8100 \times \left ( \frac{1}{4} \right ) \times 5}{100}$$ = $$\frac{2025}{20}$$ = $$\frac{405}{4}$$ = Rs. 101.25

TD = $$\frac{F \times T \times R}{100 + TR}$$
TD = $$\frac{8100 \times \left ( \frac{1}{4} \right ) \times 5}{100 + \left ( \frac{1}{4} \times 5 \right )}$$
TD = $$\frac{2025 \times 5}{100 + \frac{5}{4}}$$

= $$\frac{2025 \times 5 \times 4}{400 + 5}$$
= $$\frac{2025 \times 5 \times 4}{405}$$
= $$\frac{405 \times 5 \times 4}{81}$$
= $$\frac{45 \times 5 \times 4}{9}$$
= 5 × 5 × 4
= Rs. 100

⇒ BD − TD = Rs. 101.25 − Rs. 100 = Rs. 1.25

If the discount on Rs. 498 at 5% simple interest is Rs.18, when is the sum due?

F = Rs. 498
TD = Rs. 18
PW = F − TD = 498 − 18 = Rs. 480
R = 5%

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

⇒ $$18 = \frac{480 \times T \times 5}{100} ⇒ [latex]18 = 24 × T ⇒ [latex]T = [latex]\frac{18}{24}$$ = $$\frac{3}{4}$$ years = $$\frac{12 \times 4}{4}$$ months = 9 months

What is the present worth of a bill of Rs. 1764 due 2 years hence at 5% compound interest is

Since the compound interest is taken here,

$$PW\left ( 1 + \frac{5}{100} \right )^{2} = 1764$$

$$PW\left ( 1 + \frac{1}{20} \right )^{2} = 1764$$

$$PW\left ( 1 + \frac{5}{100} \right )^{2} = 1764$$

$$PW\left (\frac{21}{20} \right )^{2} = 1764$$

$$PW \times \frac{441}{400} = 1764$$

$$\Rightarrow PW = \frac{1764 \times 100}{441} = 4 \times 400 = Rs.1600$$

The true discount on a bill for Rs. 2520 due 6 months hence at 10% per annum is

F = Rs. 2520
T = 6 months = $$\frac{1}{2}$$ year
R = 10%

$$TD = \frac{FTR}{100 + TR} = \frac{2520 \times \frac{1}{2} \times 10}{100 + \left ( \frac{1}{2} \times 10 \right )}$$

$$TD = \frac{1260 \times 10}{100 + 5} = \frac{12600}{105} = \frac{2520}{21} = Rs.120$$

The banker’s gain of a certain sum due 2 years hence at 10% per annum is Rs. 24. What is the present worth?

$$TD = \frac{BG \times 100}{TR}$$
$$TD = \frac{24 \times 100}{2 \times 100}$$
$$TD = 12 \times 10$$ = Rs. 120
$$TD = \frac{PW \times TR}{100}$$
⇒ $$120 = \frac{PW \times 2 \times 10}{100}$$
⇒ $$PW = \frac{1200}{2}$$