Banker’s Discount Questions and Answers

Answer: D

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.

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

\[ \text{BG} = \text{TD} \times k \quad\Longrightarrow\quad \text{TD} = \frac{270}{0.36} = \text{Rs. } 750 \]

\[ \text{BD} = \text{TD} + \text{BG} = 750 + 270 = \text{Rs. } 1020 \]

Rs. 750 is the true discount, an intermediate step; the banker's discount is always the larger of the two, exceeding it by exactly the banker's gain.

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.

\[ k = \frac{\text{TD}}{\text{PW}} = \frac{110}{1100} = 0.1 \]

\[ A = \text{PW} + \text{TD} = 1100 + 110 = \text{Rs. } 1210 \]

\[ \text{BD} = A\,k = 1210 \times 0.1 = \text{Rs. } 121 \]

The banker's gain is \( 121 - 110 = \text{Rs. } 11 \), which is also \( \text{TD} \times k = 110 \times 0.1 \) - a useful cross-check.

Answer: C

Correct answer: ₹110

The banker’s gain relates to the present worth and the true discount by \[ \text{BG} = \frac{\text{TD}^{2}}{\text{PW}}. \] Substituting, \[ 11 = \frac{\text{TD}^{2}}{1100} \;\Rightarrow\; \text{TD}^{2} = 12100 \;\Rightarrow\; \text{TD} = 110. \] The relation is derived by combining two facts: BG is the simple interest on TD, and TD is the simple interest on PW. Writing \( \text{TD} = \text{PW} \cdot \frac{RT}{100} \) and \( \text{BG} = \text{TD} \cdot \frac{RT}{100} \), and eliminating \( \frac{RT}{100} \) between them, gives \( \text{BG} = \text{TD}^{2}/\text{PW} \) directly. Equivalently, PW, TD and BG are in geometric progression, each being the previous one multiplied by \( RT/100 \) - here by \( 1/10 \), since \( 1100, 110, 11 \). That observation often solves such problems by inspection.

Answer: A

Amount = Rs. 1860.
True discount = Rs. 60
∴ Present worth = Rs. 1860 - Rs. 60 = Rs. 1800
S.I. on Rs. 1800 for 8 months = Rs. 60
∴ Rate = [(100 * 60) / (1800 * 2/3)]% = 5%
∴ Banker’s discount = (1860 * 5 * 2/3)/100 = Rs. 62

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: D

Correct answer: ₹200

The two discounts are related by \[ \text{TD} = \frac{\text{BD} \times 100}{100 + RT}. \] Here \( RT = 16 \times \tfrac{1}{2} = 8 \), so \[ \text{TD} = \frac{216 \times 100}{108} = 200. \] The banker’s gain is therefore \( 216 - 200 = 16 \), and the check holds: the interest on the true discount of 200 at 16% for six months is \( 200 \times 0.08 = 16 \). The face value follows as \( \text{BD} \times 100/(RT) = 216 \times 100/8 = 2700 \), with a present worth of 2,500. The relation exists because BD is interest on the face value and TD is interest on the present worth, and the two principals stand in the ratio \( (100 + RT) : 100 \).

Answer: D

Correct answer: 6%

The ratio of the two discounts depends only on \( RT \): \[ \frac{\text{BD}}{\text{TD}} = \frac{100 + RT}{100}. \] Setting this equal to \( \frac{11}{10} \), \[ \frac{100 + RT}{100} = \frac{11}{10} \;\Rightarrow\; 100 + RT = 110 \;\Rightarrow\; RT = 10. \] The time is 1 year 8 months, that is \( T = \frac{5}{3} \) years, so \[ R = \frac{10}{5/3} = 10 \times \frac{3}{5} = 6. \] The rate is 6% per annum. The useful point is that the ratio of BD to TD never involves the face value at all - it is fixed entirely by the rate and the time. That is why a problem giving only a ratio is nonetheless solvable, and why the sum due is not needed here.

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.

\[ \text{PW} = 540 - 90 = \text{Rs. } 450 \quad\Longrightarrow\quad k = \frac{\text{TD}}{\text{PW}} = \frac{90}{450} = 0.2 \]

\[ \text{BD} = A\,k = 540 \times 0.2 = \text{Rs. } 108 \]

The banker's discount always exceeds the true discount, here by \( 108 - 90 = \text{Rs. } 18 \), because it charges interest on the larger face value.

Answer: B

Correct answer: ₹650

The true discount relates to the face value by \[ \text{TD} = \frac{A \times RT}{100 + RT}. \] Here \( RT = 10 \times 3 = 30 \), so \[ 150 = \frac{A \times 30}{130} \;\Rightarrow\; A = \frac{150 \times 130}{30} = 650. \] Checking: the present worth is \( 650 - 150 = 500 \), and the simple interest on 500 at 10% for 3 years is \( 500 \times 0.30 = 150 \), which is the true discount as given. The distractor 600 is what comes of treating the discount as interest on the face value - \( 150 \times 100/30 = 500 \) would then be the sum, and 600 is a near miss on the same reasoning. The formula worth carrying is that the present worth is always \( \dfrac{100A}{100+RT} \), and the true discount is what is left of \( A \) after subtracting it.

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

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