Banker’s Discount Questions and Answers

Answer: A

Writing \( k = \dfrac{RT}{100} \),

\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{BD} - \text{TD} = \text{TD}\,k \]

With \( k = \dfrac{12 \times 3}{100} = 0.36 \):

\[ \text{TD} = \frac{\text{BG}}{k} = \frac{360}{0.36} = \text{Rs. } 1000 \]

\[ \text{BD} = \text{TD} + \text{BG} = 1000 + 360 = \text{Rs. } 1360 \]

The banker's discount always exceeds the true discount by exactly the banker's gain.

Answer: B

Writing \( k = \dfrac{RT}{100} \),

\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{BD} - \text{TD} = \text{TD}\,k \]

With \( T = \dfrac{6}{12} = 0.5 \) year at 15%:

\[ k = \frac{15 \times 0.5}{100} = 0.075 \]

Since \( \text{BD} = \text{TD}(1+k) \):

\[ \text{BD} = 120 \times 1.075 = \text{Rs. } 129 \]

The Rs. 9 difference is the banker's gain - interest on the Rs. 120 of true discount for the same six months.

Answer: B

Four quantities are related throughout this topic. Writing \( k = \dfrac{RT}{100} \):

\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{BD} - \text{TD} = \text{TD}\,k \]

The banker's discount is interest on the face value; the true discount is interest on the present worth; the banker's gain is the interest on the true discount.

\[ k = \frac{\text{TD}}{\text{PW}} = \frac{36}{800} = 0.045 \]

\[ A = \text{PW} + \text{TD} = 800 + 36 = \text{Rs. } 836 \]

\[ \text{BD} = A\,k = 836 \times 0.045 = \text{Rs. } 37.62 \]

Equivalently \( \text{BD} = \text{TD}(1+k) = 36 \times 1.045 = 37.62 \). The banker's gain is the 62 paise difference.

Answer: D

Writing \( k = \dfrac{RT}{100} \),

\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{BD} - \text{TD} = \text{TD}\,k \]

Neither the rate nor the time is given, so use the form that needs neither:

\[ \text{BG} = \frac{\text{TD}^{2}}{\text{PW}} \]

\[ 10 = \frac{\text{TD}^{2}}{5760} \quad\Longrightarrow\quad \text{TD}^{2} = 57{,}600 \]

\[ \text{TD} = \sqrt{57600} = \text{Rs. } 240 \]

The identity follows from \( \text{BG} = \text{TD}\,k \) and \( \text{TD} = \text{PW}\,k \): eliminating \( k \) between them gives it directly.

Answer: A

Writing \( k = \dfrac{RT}{100} \),

\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{BD} - \text{TD} = \text{TD}\,k \]

Both discounts use the same \( k \), one on the amount and one on the present worth:

\[ \frac{\text{BD}}{\text{TD}} = \frac{A}{A - \text{TD}} \quad\Longrightarrow\quad \frac{36}{30} = \frac{A}{A - 30} \]

\[ 36A - 1080 = 30A \quad\Longrightarrow\quad 6A = 1080 \]

\[ A = \text{Rs. } 180 \]

The shortcut is

\[ \text{Sum} = \frac{\text{BD} \times \text{TD}}{\text{BD} - \text{TD}} = \frac{36 \times 30}{6} = 180 \]

Answer: A

F = Rs. 3000 R = 10%
Date on which the bill is drawn = 14th July at 5 months
Nominally Due Date = 14th December
Legally Due Date = 14th December + 3 days = 17th December
Date on which the bill is discounted = 5th October
Unexpired Time = [6th to 31st of October] + [30 Days in November] + [1st to 17th of December]
= 26 + 30 + 17
= 73 Days =73/365 year=15 year
BD = Simple Interest on the face value of the bill for unexpired time = FTR/ 100 = (3000*1/5*10)/100 = 30*1/5*10 = Rs. 60

Answer: A

Writing \( k = \dfrac{RT}{100} \),

\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{BD} - \text{TD} = \text{TD}\,k \]

\[ k = \frac{\text{TD}}{\text{PW}} = \frac{72}{1296} = \frac{1}{18} \]

\[ A = 1296 + 72 = \text{Rs. } 1368 \]

\[ \text{BD} = A\,k = \frac{1368}{18} = \text{Rs. } 76 \]

Equivalently \( \text{BD} = \text{TD}(1+k) = 72 \times \tfrac{19}{18} = 76 \), the banker's gain being the Rs. 4 difference.

Answer: A

Writing \( k = \dfrac{RT}{100} \),

\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{BD} - \text{TD} = \text{TD}\,k \]

With \( k = \dfrac{10 \times 3}{100} = 0.3 \):

\[ \text{TD} = \frac{\text{BG}}{k} = \frac{36}{0.3} = \text{Rs. } 120 \]

\[ \text{PW} = \frac{\text{TD}}{k} = \frac{120}{0.3} = \text{Rs. } 400 \]

Two divisions by the same \( k \), because the banker's gain is interest on the true discount just as the true discount is interest on the present worth.

Answer: D

BD for 3 years = Rs. 1116
BD for 4 years = 1116*4/3 = Rs.1488
TD for 4 year = Rs.1200
F = (BD*TD)/(BD -TD) = (1488*1200)/(1488 -1200) = 124*1200/24 = 124*100/2 = 62*100 = Rs.6200
=> Rs.1488 is the simple interest on Rs. 6200 for 4 years
=> 1488 = (6200*4*R)/100
=> R=(1488*100)/(6200*4) = 372*100/6200 = 372/62 = 6%

Answer: D

The discount here is the true discount, so subtract it to get the present worth.

\[ \text{PW} = 498 - 18 = \text{Rs. } 480 \]

True discount is simple interest on that present worth:

\[ 18 = \frac{480 \times 5 \times T}{100} = 24\,T \]

\[ T = \frac{18}{24} = 0.75 \text{ year} = 9 \text{ months} \]

Charging the interest on the Rs. 498 face value instead would give about 8.7 months.