Banker’s Discount Questions and Answers

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.

Answer: B

Write \( k = \dfrac{R \times 2}{100} \) for the rate–time product over the two years. Simple interest is charged on the full sum, true discount on the present worth:

\[ \text{SI} = 750\,k, \qquad \text{TD} = \frac{960\,k}{1+k} \]

Setting them equal and cancelling \( k \):

\[ 750(1+k) = 960 \quad\Longrightarrow\quad 1 + k = 1.28 \quad\Longrightarrow\quad k = 0.28 \]

\[ \frac{2R}{100} = 0.28 \quad\Longrightarrow\quad R = 14\% \]

Cancelling \( k \) from both sides is what makes the problem solvable in one line; the two years then divide out at the very end.

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: A

Correct answer: ₹965

The present worth is \[ \text{PW} = \frac{100 \times A}{100 + RT}, \] with \( RT = 8 \times 3 = 24 \). So \[ \text{PW} = \frac{100 \times 1196}{124} = \frac{119600}{124} = 964.52, \] which to the nearest rupee is 965. Checking forwards: 965 at 8% for three years earns about 231.60 in interest, giving 1,196.60 - the small excess being the effect of rounding. The true discount is the remainder, \( 1196 - 965 = 231 \) approximately. The distractor 1,000 is what a reader arrives at by treating the discount as interest on the face value: \( 1196 \times 0.24 = 287 \), leaving about 909, or by rounding carelessly. Applying the formula rather than working backwards from a guess avoids it.

Answer: A

Correct answer: 9%

The banker’s discount is the amount deducted: \[ \text{BD} = 3000 - 2910 = 90. \] It is simple interest on the face value for the unexpired time, so \[ 90 = \frac{3000 \times R \times \tfrac{1}{3}}{100} = 10R \;\Rightarrow\; R = 9. \] The rate is 9% per annum. The point of the question is the phrase a banker pays: the sum handed over is the discounted value, and the difference from the face value is the banker’s discount rather than the true discount. Computing on the 2,910 instead would give \( R = 90 \times 300/2910 \approx 9.28\% \) - which would be the rate implied by the true discount, a different quantity. Four months must be converted to \( \tfrac{1}{3} \) of a year before substituting, which is where the other common slip occurs.

Answer: A

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 \( T = \dfrac{4}{12} = \dfrac{1}{3} \) year at 15%:

\[ k = \frac{15 \times 1/3}{100} = 0.05 \]

Since \( \text{BD} = A\,k \) and \( \text{TD} = \dfrac{A\,k}{1+k} \), the two differ only by the factor \( 1+k \):

\[ \text{TD} = \frac{\text{BD}}{1+k} = \frac{420}{1.05} = \text{Rs. } 400 \]

The face value is \( 420/0.05 = \text{Rs. } 8400 \), and the banker's gain is Rs. 20 - but neither is needed once the \( 1+k \) relation is used.

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.

Let \( k \) be the rate–time product. The banker's discount on Rs. 1600 equals the true discount on Rs. 1680:

\[ 1600\,k = \frac{1680\,k}{1+k} \]

Cancelling \( k \):

\[ 1600(1+k) = 1680 \quad\Longrightarrow\quad 1+k = 1.05 \quad\Longrightarrow\quad k = 0.05 \]

\[ \frac{15\,T}{100} = 0.05 \quad\Longrightarrow\quad T = \frac{1}{3} \text{ year} = 4 \text{ months} \]

The rate enters only at the final step; everything before it depends on the ratio \( 1680/1600 \) alone.

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

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

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

Write \( r \) for the annual rate as a decimal. The banker's discount for \( \tfrac{3}{2} \) years is interest on the face value:

\[ A \times r \times 1.5 = 558 \quad\Longrightarrow\quad Ar = 372 \]

The true discount for 2 years is

\[ \frac{A \times 2r}{1 + 2r} = 600 \quad\Longrightarrow\quad \frac{744}{1 + 2r} = 600 \]

\[ 744 = 600 + 1200r \quad\Longrightarrow\quad r = 0.12 = 12\% \]

Substituting \( Ar = 372 \) into the second equation is what eliminates the unknown face value.

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.