Banker’s Discount Questions and Answers

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: 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.

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: 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.

\[ k = \frac{\text{TD}}{\text{PW}} = \frac{160}{1600} = 0.1 \]

\[ \text{BG} = \text{TD} \times k = 160 \times 0.1 = \text{Rs. } 16 \]

Equivalently \( \text{BG} = \dfrac{\text{TD}^{2}}{\text{PW}} = \dfrac{160^{2}}{1600} = 16 \) - a form worth remembering, since it needs neither the rate nor the time.

Answer: B

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

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

Since \( \text{BG} = \text{BD} - \text{TD} \) and \( \text{TD} = \dfrac{\text{BD}}{1+k} \):

\[ \frac{\text{BG}}{\text{BD}} = 1 - \frac{1}{1+k} = \frac{k}{1+k} \]

\[ \frac{k}{1+k} = \frac{3}{25} \quad\Longrightarrow\quad 25k = 3 + 3k \]

\[ 22k = 3 \quad\Longrightarrow\quad k = \frac{3}{22} \]

With \( k = 1.5r \):

\[ r = \frac{3}{22 \times 1.5} = \frac{1}{11} = \frac{100}{11}\% \]

Neither the face value nor the discount itself is needed - only their ratio.

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

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

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: 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

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.

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

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

\[ 16 = \frac{\text{TD}^{2}}{576} \quad\Longrightarrow\quad \text{TD}^{2} = 9216 \]

\[ \text{TD} = \sqrt{9216} = \text{Rs. } 96 \]

This identity follows from \( \text{BG} = \text{TD}\,k \) and \( \text{TD} = \text{PW}\,k \): substituting \( k = \text{TD}/\text{PW} \) into the first gives it directly.

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.