Banker’s Discount Questions and Answers

Answer: D

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

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

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

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

\[ 10 = \frac{\text{TD}^{2}}{5760} \quad\Longrightarrow\quad \text{TD}^{2} = 57{,}600 \]

\[ \text{TD} = \sqrt{57600} = \text{Rs. } 240 \]

The identity follows from \( \text{BG} = \text{TD}\,k \) and \( \text{TD} = \text{PW}\,k \): eliminating \( k \) between them gives it directly.

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: B

Correct answer: ₹30.60

The banker’s gain is the simple interest on the true discount, which gives the standard relation \[ \text{BG} = \frac{\text{TD}^{2}}{\text{PW}}. \] Substituting, \[ \text{BG} = \frac{30^{2}}{1500} = \frac{900}{1500} = 0.60. \] The banker’s discount then follows from \( \text{BD} = \text{TD} + \text{BG} \): \[ \text{BD} = 30 + 0.60 = 30.60. \] The relationships worth holding together are these. True discount is the interest on the present worth; banker’s discount is the interest on the face value, which is larger; and the banker’s gain is the difference between them, which is why BD always exceeds TD. The trap is to treat BD as equal to TD, which the small size of BG makes tempting on figures like these.

Answer: B

Correct answer: ₹66

The three quantities are related by \[ \text{BD} = \text{TD} + \text{BG}, \] so \[ \text{BD} = 60 + 6 = 66. \] The relation holds because BD is simple interest on the face value while TD is simple interest on the present worth, and the face value exceeds the present worth by exactly the true discount - so the extra interest is the interest on TD, which is the banker’s gain. That also gives the second expression, \( \text{BG} = \dfrac{\text{TD}^{2}}{\text{PW}} \). From it the present worth here is \( 60^{2}/6 = 600 \), and the face value is \( 600 + 60 = 660 \). BD always exceeds TD, so an answer smaller than the true discount can be rejected without calculation - which disposes of 54 at once.

Answer: D

The discount here is the true discount, so subtract it to get the present worth.

\[ \text{PW} = 498 - 18 = \text{Rs. } 480 \]

True discount is simple interest on that present worth:

\[ 18 = \frac{480 \times 5 \times T}{100} = 24\,T \]

\[ T = \frac{18}{24} = 0.75 \text{ year} = 9 \text{ months} \]

Charging the interest on the Rs. 498 face value instead would give about 8.7 months.

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: D

BD for 3 years = Rs. 1116
BD for 4 years = 1116*4/3 = Rs.1488
TD for 4 year = Rs.1200
F = (BD*TD)/(BD -TD) = (1488*1200)/(1488 -1200) = 124*1200/24 = 124*100/2 = 62*100 = Rs.6200
=> Rs.1488 is the simple interest on Rs. 6200 for 4 years
=> 1488 = (6200*4*R)/100
=> R=(1488*100)/(6200*4) = 372*100/6200 = 372/62 = 6%

Answer: B

Correct answer: 8 months

First find the present worth, since the true discount is interest on that rather than on the face value: \[ \text{PW} = 1860 - 60 = 1800. \] Now apply the simple interest formula to the present worth: \[ 60 = \frac{1800 \times 5 \times T}{100} = 90T \;\Rightarrow\; T = \frac{60}{90} = \frac{2}{3}\text{ year}, \] which is 8 months. The single step that decides this problem is subtracting the discount before applying the interest formula. Using the face value of 1,860 instead gives \( T = 60 \times 100/(1860 \times 5) = 0.645 \) year, about 7.7 months - close enough to look plausible and wrong. The general form is \( \text{TD} = \dfrac{\text{PW} \times R \times T}{100} \), and remembering that PW rather than \( A \) appears in it settles most questions in this topic.

Answer: A

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

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

\[ k = \frac{\text{TD}}{\text{PW}} = \frac{72}{1296} = \frac{1}{18} \]

\[ A = 1296 + 72 = \text{Rs. } 1368 \]

\[ \text{BD} = A\,k = \frac{1368}{18} = \text{Rs. } 76 \]

Equivalently \( \text{BD} = \text{TD}(1+k) = 72 \times \tfrac{19}{18} = 76 \), the banker's gain being the Rs. 4 difference.

Answer: B

Correct answer: ₹264

The banker’s discount is simply the simple interest on the face value for the unexpired time: \[ \text{BD} = \frac{A \times R \times T}{100} = \frac{1650 \times 4 \times 4}{100} = \frac{26400}{100} = 264. \] This is the one quantity in the topic that needs no adjustment for present worth, which is what makes it the easiest to compute and the easiest to confuse with the true discount. For comparison, the true discount here is \( \dfrac{1650 \times 16}{116} \approx 227.59 \), and the banker’s gain is the difference, about 36.41. The distractor 228 is that true discount rounded, placed to catch the confusion. The practical meaning is that a banker discounting a bill charges interest on its full face value for the time it has left to run, and so gains slightly more than the mathematically fair discount.