# Banker's Discount

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

**Answer**: (a)

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?

**Answer**: (d)

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?

**Answer**: (d)

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 = 4^{th} April at 6 months

⇒ Nominally Due Date = 4^{th} October

⇒ Legally Due Date = (4^{th} October + 3 days) = 7^{th} October

Hence, The date on which the bill was discounted

= (7^{th} October - 65 days)

= (7^{th} October - 7 days in October - 30 days in September - 28 days in August)

= 3^{rd} 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?

**Answer**: (b)

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?

**Answer**: (d)

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?

**Answer**: (d)

\(BG = \frac{TD^{2}}{PW} = \frac{202}{400}\) = 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?

**Answer**: (a)

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?

**Answer**: (d)

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.

**Answer**: (d)

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

⇒ \(F = \frac{200 \times 100}{200 - 100}\)

⇒ \(F = \frac{200 \times 100}{100}\)

⇒ F = Rs. 200

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

**Answer**: (c)

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

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