# Banker's Discount

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

**Answer**: (a)

F = Rs. 3000, R = 10%

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

Nominally Due Date = 14^{th} December

Legally Due Date = 14^{th} December + 3 days = 17^{th} December

Date on which the bill is discounted = 5th October

Unexpired Time

= [6th to 31^{st} of October] + [30 Days in November] + [1^{st} 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 ?

**Answer**: (a)

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?

**Answer**: (d)

\(TD = \sqrt{PW \times BG}\)

\(TD = \sqrt{5760 \times 10}\)

\(TD = \sqrt{57600}\)

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?

**Answer**: (b)

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?

**Answer**: (b)

\(TD = \frac{BD \times 100}{100 + TR}\)

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

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

\(TD = \frac{420 \times 100}{105}\)

\(TD = \frac{84 \times 100}{21}\) = 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?

**Answer**: (a)

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

\(TD = \frac{20 \times 100}{1 \times 100}\)

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?

**Answer**: (a)

\(F = \frac{BD \times TD}{BD - TD}\)

\(F = \frac{36 \times 30}{36 - 30}\)

\(F = \frac{36 \times 30}{6}\)

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?

**Answer**: (a)

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?

**Answer**: (d)

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

\(TD = \frac{5400 \times 100}{6 \times 12}\)

\(TD = \frac{90 \times 100}{12}\)

\(TD = \frac{15 \times 100}{2}\)

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?

**Answer**: (c)

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

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