Banker’s Discount Questions and Answers

Answer: C

Correct answer: ₹248

First find the true discount, since the banker’s gain is the simple interest on it: \[ 48 = \frac{\text{TD} \times 8 \times 3}{100} = 0.24\,\text{TD} \;\Rightarrow\; \text{TD} = \frac{4800}{24} = 200. \] Then \[ \text{BD} = \text{TD} + \text{BG} = 200 + 48 = 248. \] The distractor 200 is the true discount, which catches anyone who stops after the first step. For completeness, the present worth is \( \text{TD}^{2}/\text{BG} = 40000/48 \approx 833.33 \) and the face value about 1,033.33. The order of magnitude is always \( \text{PW} > \text{TD} > \text{BG} \) for ordinary rates, and BD lies just above TD - so any answer below the true discount can be rejected without arithmetic.

Answer: D

Correct answer: 10%

The banker’s gain is \( 110 - 100 = 10 \), and it is the simple interest on the true discount: \[ 10 = \frac{100 \times R \times 1}{100} = R \;\Rightarrow\; R = 10. \] The rate is 10% per annum. The same answer follows from the ratio, which depends only on \( RT \): \[ \frac{\text{BD}}{\text{TD}} = \frac{100 + RT}{100} \;\Rightarrow\; \frac{110}{100} = \frac{100 + R}{100} \;\Rightarrow\; R = 10. \] Either route avoids needing the sum, which is not given - though it can be recovered as \( A = \text{BD} \times \text{TD}/\text{BG} = 110 \times 100/10 = 1100 \), with a present worth of 1,000. The distractor 11% comes of dividing 110 by the face value rather than the true discount.

Answer: A

Correct answer: ₹132

The three quantities are related by \[ \text{BD} = \text{TD} + \text{BG} = 120 + 12 = 132. \] The relation holds because the banker’s discount is simple interest on the face value while the true discount is simple interest on the present worth. Since the face value exceeds the present worth by exactly the true discount, the extra interest is the interest on TD - which is precisely what the banker’s gain is. From \( \text{BG} = \text{TD}^{2}/\text{PW} \), the present worth here is \( 120^{2}/12 = 1200 \), and the face value is \( 1200 + 120 = 1320 \). Note the geometric progression 1200, 120, 12 - each a tenth of the previous, the ratio being \( RT/100 = 0.1 \). The banker’s discount always exceeds the true discount, so 108 can be rejected on sight.

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: 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: A

A 2% gain on Rs. 600 means the sale must be worth

\[ 600 \times 1.02 = \text{Rs. } 612 \text{ in today's money} \]

So Rs. 688.50 due in 9 months has a present worth of Rs. 612:

\[ 612\left(1 + \frac{R \times 0.75}{100}\right) = 688.50 \]

\[ 1 + 0.0075R = \frac{688.50}{612} = 1.125 \]

\[ 0.0075R = 0.125 \quad\Longrightarrow\quad R = \frac{50}{3} = 16\tfrac{2}{3}\% \]

The gain is reckoned on the present worth, not on the credit price - comparing 688.50 with 600 directly would suggest a 14.75% profit.

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: D

Correct answer: ₹384

Work outwards from the present worth. The true discount is the simple interest on it: \[ \text{TD} = \frac{1600 \times 10 \times 2}{100} = 320. \] The face value is therefore \( A = 1600 + 320 = 1920 \), and the banker’s discount is the simple interest on that: \[ \text{BD} = \frac{1920 \times 20}{100} = 384. \] The distractor 320 is the true discount, which is what a reader gets by stopping one step early - and it is the commonest wrong answer on this pattern. As a check, the banker’s gain is \( 384 - 320 = 64 \), and the interest on the true discount of 320 at 20% is indeed \( 320 \times 0.2 = 64 \). Note again the geometric progression: 1600, 320, 64.