Banker’s Discount Questions and Answers

Answer: B

Correct answer: 15%

The present worth is \( 2050 - 50 = 2000 \), and the true discount is the simple interest on that: \[ 50 = \frac{2000 \times R \times \tfrac{1}{3}}{100} = \frac{2000R}{300}. \] Solving, \[ R = \frac{50 \times 300}{2000} = 7.5 \times 2 = 15. \] The rate is 15% per annum. The step that decides the problem is subtracting the discount first: the true discount is interest on the present worth, never on the face value. Using 2,050 instead gives \( R = 50 \times 300/2050 \approx 14.6\% \) - close enough to look right and wrong. Four months is \( \tfrac{1}{3} \) of a year, and converting the period to years before substituting is the other place errors creep in.

Answer: B

Correct answer: 10 months

Write both discounts in terms of the time \( T \) in years. The banker’s discount is simple interest on the face value: \[ \text{BD} = \frac{2200 \times 12 \times T}{100} = 264T. \] The true discount is \[ \text{TD} = \frac{2420 \times 12 \times T}{100 + 12T} = \frac{29040T}{100 + 12T}. \] Setting them equal and cancelling \( T \), \[ 264(100 + 12T) = 29040 \;\Rightarrow\; 26400 + 3168T = 29040 \;\Rightarrow\; T = \frac{2640}{3168} = \frac{5}{6}\text{ year}, \] which is 10 months. The step that matters is remembering that TD is calculated on the present worth, which is why its denominator carries the \( 100 + RT \) rather than 100.

Answer: C

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 two discounts differ exactly by the factor \( 1+k \):

\[ \frac{\text{BD}}{\text{TD}} = 1 + k = \frac{11}{10} \quad\Longrightarrow\quad k = 0.1 \]

With \( T = 2 \) years:

\[ \frac{2R}{100} = 0.1 \quad\Longrightarrow\quad R = 5\% \]

The 11% and 10% on offer come from reading the ratio itself as a rate, which it is not.

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: A

Writing \( k = \dfrac{RT}{100} \), the four quantities are related by

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

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

\[ \text{BD} = A\,k, \qquad \text{TD} = (A - \text{TD})\,k \]

Dividing one by the other:

\[ \frac{72}{60} = \frac{A}{A - 60} \quad\Longrightarrow\quad 72A - 4320 = 60A \]

\[ 12A = 4320 \quad\Longrightarrow\quad A = \text{Rs. } 360 \]

The standard shortcut is

\[ \text{Sum} = \frac{\text{BD} \times \text{TD}}{\text{BD} - \text{TD}} = \frac{72 \times 60}{12} = 360 \]

Answer: A

Correct answer: ₹120

Two steps are needed. First, the banker’s gain is the simple interest on the true discount: \[ 20 = \frac{\text{TD} \times 5 \times 4}{100} = 0.2\,\text{TD} \;\Rightarrow\; \text{TD} = 100. \] Then \[ \text{BD} = \text{TD} + \text{BG} = 100 + 20 = 120. \] The distractor 100 is the true discount, offered for a reader who stops after the first step - which is the usual error on this pattern. The rest of the picture follows: the present worth is \( \text{TD}^{2}/\text{BG} = 10000/20 = 500 \), and the face value is 600. Note the geometric progression 500, 100, 20, each a fifth of the last, the ratio being \( RT/100 = 0.2 \). The order of size is always \( \text{PW} > \text{TD} > \text{BG} \) when \( RT < 100 \), which is a quick check on any answer.

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: D

Correct answer: ₹60

The true discount is the interest on the present worth, not on the face value, which gives \[ \text{TD} = \frac{A \times RT}{100 + RT}. \] Here \( R = 8 \) and \( T = \frac{9}{12} = 0.75 \) year, so \( RT = 6 \). Then \[ \text{TD} = \frac{1060 \times 6}{100 + 6} = \frac{6360}{106} = 60. \] The present worth is therefore \( 1060 - 60 = 1000 \), which checks: the interest on 1,000 at 8% for nine months is \( 1000 \times 0.06 = 60 \). The distractor 63.60 is what comes of taking the interest on the face value instead - \( 1060 \times 0.06 \) - and that figure is in fact the banker’s discount. Confusing the two is the commonest error in this topic.

Answer: C

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.

Here \( k = \dfrac{10 \times 2}{100} = 0.2 \).

\[ \text{BG} = \text{TD} \times k \quad\Longrightarrow\quad \text{TD} = \frac{24}{0.2} = \text{Rs. } 120 \]

\[ \text{TD} = \text{PW} \times k \quad\Longrightarrow\quad \text{PW} = \frac{120}{0.2} = \text{Rs. } 600 \]

The chain runs BG → TD → PW, dividing by \( k \) at each step. Stopping one step early leaves Rs. 120, and the face value would be Rs. 720.

Answer: B

Correct answer: ₹768

Apply \[ \text{TD} = \frac{A \times RT}{100 + RT} \] with \( RT = 14 \times 2 = 28 \): \[ 168 = \frac{A \times 28}{128} \;\Rightarrow\; A = \frac{168 \times 128}{28} = 6 \times 128 = 768. \] The arithmetic simplifies neatly because \( 168/28 = 6 \), which is worth looking for before multiplying out. Checking: the present worth is \( 768 - 168 = 600 \), and the simple interest on 600 at 14% for 2 years is \( 600 \times 0.28 = 168 \) - the true discount as given. For completeness, the banker’s discount would be interest on the face value, \( 768 \times 0.28 = 215.04 \), and the banker’s gain the difference, 47.04 - which is also the interest on the true discount, \( 168 \times 0.28 \). The distractor 800 is the answer obtained by wrongly treating TD as interest on the face value.