# 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:

**Answer**: (a)

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?

**Answer**: (a)

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}

⇒ [latex]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}[/latex] 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:

**Answer**: (d)

\(BG = \frac{TD^{2}}{PW}\)

\(BG = \frac{340^{2}}{3400}\)

\(BG = \frac{340 \times 340}{3400} = \frac{340}{10}\) = 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 ?

**Answer**: (a)

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%

**Answer**: (b)

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?

**Answer**: (d)

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

**Answer**: (a)

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

**Answer**: (a)

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?

**Answer**: (a)

T = 2 years, R = 10%

\(TD = \frac{BG \times 100}{TR} \)

\(TD = \frac{24 \times 100}{2 \times 100}\)

\(TD = 12 \times 10\) = Rs. 120

\(TD = \frac{PW \times TR}{100} \)

⇒ \(120 = \frac{PW \times 2 \times 10}{100} \)

⇒ 1200 = PW × 2

⇒ \(PW = \frac{1200}{2}\)

= Rs. 600

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