Banker’s Discount Questions and Answers

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: 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: C

Correct answer: ₹120

The banker’s discount is simple interest on the face value for the unexpired period: \[ \text{BD} = \frac{3000 \times 8 \times \tfrac{1}{2}}{100} = \frac{12000}{100} = 120. \] The banker hands over \( 3000 - 120 = 2880 \), which is the discounted value of the bill, and collects the full 3,000 at maturity. The mathematically fair charge would be the true discount, \[ \text{TD} = \frac{3000 \times 4}{104} \approx 115.38, \] which is the second option - and the difference of about 4.62 is the banker’s gain, the banker’s profit on the transaction. That gap is the whole point of the topic: a banker charges interest on the amount that will be received rather than on the amount actually advanced, and so always gains a little.

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: 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

\[ \text{Amount} = \text{PW}\left(1 + \frac{RT}{100}\right), \qquad \text{TD} = \text{Amount} - \text{PW} \]

With \( T = 2\tfrac{1}{2} = 2.5 \) years at 15%:

\[ \frac{RT}{100} = \frac{15 \times 2.5}{100} = 0.375 \]

\[ \text{PW} = \frac{2310}{1.375} = \text{Rs. } 1680 \]

Checking: the discount is \( 2310 - 1680 = 630 \), and 37.5% of 1680 is 630 - interest on the present worth, exactly as the definition requires.

Answer: B

Correct answer: ₹2,000

The present worth is the sum which, invested now at simple interest, would amount to the face value at maturity: \[ \text{PW} = \frac{100 \times A}{100 + RT}. \] Here \( R = 12 \) and \( T = 2.5 \) years, so \( RT = 30 \). Then \[ \text{PW} = \frac{100 \times 2600}{130} = \frac{260000}{130} = 2000. \] Checking forwards: 2,000 at 12% for two and a half years earns \( 2000 \times 0.30 = 600 \) in interest, giving exactly 2,600 at maturity. The true discount is that 600, being the difference between the face value and the present worth. The relation \( A = \text{PW} + \text{TD} \) holds always, and it is the quickest check on any answer in this topic.

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: 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.