Banker’s Discount Questions and Answers

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: C

Correct answer: ₹50

The banker’s gain is the simple interest on the true discount, so \[ \text{BG} = \frac{\text{TD} \times R \times T}{100}. \] Substituting, \[ 6 = \frac{\text{TD} \times 12 \times 1}{100} = 0.12\,\text{TD} \;\Rightarrow\; \text{TD} = \frac{600}{12} = 50. \] That identity is the most useful single fact in this topic. It follows from BD being interest on the face value and TD being interest on the present worth: the difference in principal is exactly TD, so the difference in interest is the interest on TD. From it the rest follows here. The present worth is \( \text{TD}^{2}/\text{BG} = 2500/6 \approx 416.67 \), the face value is about 466.67, and the banker’s discount is \( 50 + 6 = 56 \). Note that BG is always the smallest of the three quantities.

Answer: A

Correct answer: 4 months

Write both in terms of the time \( T \) in years. The banker’s discount is interest on the face value: \[ \text{BD} = \frac{1800 \times 12 \times T}{100} = 216T. \] The true discount is \[ \text{TD} = \frac{1872 \times 12 \times T}{100 + 12T} = \frac{22464T}{100 + 12T}. \] Equating and cancelling \( T \), \[ 216(100 + 12T) = 22464 \;\Rightarrow\; 21600 + 2592T = 22464 \;\Rightarrow\; T = \frac{864}{2592} = \frac{1}{3}\text{ year}, \] which is 4 months. Cancelling \( T \) from both sides is the step that makes the algebra manageable; it is legitimate because \( T \) cannot be zero. The pattern - BD on one sum equalling TD on a larger one - always reduces to a linear equation in this way.

Answer: C

Correct answer: 12%

Let the face value be \( A \) and the rate \( R \). The banker’s discount is interest on the face value, so for 1.5 years \[ \frac{A \times R \times 1.5}{100} = 558 \;\Rightarrow\; AR = 37200. \] The true discount for 2 years is \[ \frac{A \times R \times 2}{100 + 2R} = 600 \;\Rightarrow\; \frac{2 \times 37200}{100 + 2R} = 600. \] Solving, \[ 74400 = 600(100 + 2R) = 60000 + 1200R \;\Rightarrow\; 1200R = 14400 \;\Rightarrow\; R = 12. \] The rate is 12% per annum, and the face value follows as \( A = 37200/12 = 3100 \). The step that makes this tractable is treating \( AR \) as a single unknown, which the first equation supplies directly - the two quantities never need to be separated until the end.

Answer: B

Correct answer: ₹50

The two discounts are related by \[ \text{TD} = \frac{\text{BD} \times 100}{100 + RT}. \] Here \( RT = 10 \times 2 = 20 \), so \[ \text{TD} = \frac{60 \times 100}{120} = 50. \] The relation follows from what each discount is interest on. BD is interest on the face value \( A \); TD is interest on the present worth. Since \( A = \text{PW}\left(1 + \frac{RT}{100}\right) \), the two discounts stand in the ratio \( (100+RT) : 100 \). As a check, the banker’s gain is \( 60 - 50 = 10 \), and the interest on the true discount of 50 at 10% for 2 years is indeed \( 50 \times 0.2 = 10 \). The face value here is \( 60 \times 100/20 = 300 \) and the present worth 250.

Answer: D

Correct answer: ₹1,320

The face value follows from \[ A = \frac{\text{BD} \times \text{TD}}{\text{BD} - \text{TD}}. \] The denominator is the banker’s gain, \( 120 - 110 = 10 \), so \[ A = \frac{120 \times 110}{10} = 1320. \] Checking against the definitions: the present worth is \( 1320 - 110 = 1210 \), and the ratio \( 110/1210 = 120/1320 \), both equal to \( 1/11 \) - confirming that the two discounts are computed at the same rate and time, with \( RT/100 = 1/11 \). The distractor 1,210 is that present worth, and 1,100 is a plausible-looking round figure with no basis; both are placed to catch a reader who stops one step early. Note also the geometric progression PW, TD, BG - here 1210, 110, 10 - each a tenth of the previous, which is a useful sanity check.

Answer: D

Correct answer: ₹840

The face value is given by \[ A = \frac{\text{BD} \times \text{TD}}{\text{BD} - \text{TD}}. \] Substituting, \[ A = \frac{42 \times 40}{42 - 40} = \frac{1680}{2} = 840. \] The formula comes from the definitions. BD is interest on \( A \) and TD is interest on \( \text{PW} = A - \text{TD} \), so \[ \frac{\text{BD}}{\text{TD}} = \frac{A}{A - \text{TD}}, \] and rearranging gives the result. The denominator \( \text{BD} - \text{TD} \) is the banker’s gain, here 2, so the formula can also be read as \( A = \text{BD} \times \text{TD} / \text{BG} \). As a check, the present worth is \( 840 - 40 = 800 \), and \( 40/800 = 42/840 \), confirming that both discounts are computed at the same rate and time.

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.