Banker’s Discount Questions and Answers

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: 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

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

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 \( k = \dfrac{12 \times 1}{100} = 0.12 \):

\[ \text{BG} = \text{TD} \times k \quad\Longrightarrow\quad \text{TD} = \frac{6}{0.12} = \text{Rs. } 50 \]

Checking the whole chain: PW = \( 50/0.12 = 416.67 \), face value \( 466.67 \), and BD = \( 466.67 \times 0.12 = 56 \), which is indeed Rs. 6 more than the true discount.

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: 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: 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: 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

Correct answer: 10 months

The present worth is \( 1760 - 160 = 1600 \), and the true discount is the simple interest on it: \[ 160 = \frac{1600 \times 12 \times T}{100} = 192T \;\Rightarrow\; T = \frac{160}{192} = \frac{5}{6}\text{ year}. \] Five sixths of a year is 10 months. The single step that decides the problem is subtracting the discount before applying the interest formula - TD is interest on the present worth, not on the face value. Using 1,760 would give \( T = 160 \times 100/(1760 \times 12) \approx 0.758 \) year, about 9.1 months, which is wrong but plausible enough to be offered as a trap in many versions of this question. Converting the fractional year to months at the end is the other place marks are lost: multiply by 12, not by 10.

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.