Banker’s Discount Questions and Answers

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

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: 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: C

Correct answer: ₹37.62

The banker’s gain is given by \[ \text{BG} = \frac{\text{TD}^{2}}{\text{PW}} = \frac{36^{2}}{800} = \frac{1296}{800} = 1.62. \] The banker’s discount is then \[ \text{BD} = \text{TD} + \text{BG} = 36 + 1.62 = 37.62. \] The face value is \( 800 + 36 = 836 \), and as a check the ratio \( \text{BD}/\text{TD} = 37.62/36 = 1.045 \) equals \( 836/800 \), confirming that BD is interest on the face value while TD is interest on the present worth. The formula \( \text{BG} = \text{TD}^{2}/\text{PW} \) is the one to reach for whenever the present worth and true discount are given and the rate and time are not - it needs neither. Note that PW, TD and BG form a geometric progression, here 800, 36, 1.62.

Answer: C

Correct answer: ₹2,500

The banker’s gain is the simple interest on the true discount, so \[ \text{BG} = \frac{\text{TD} \times R \times T}{100} \;\Rightarrow\; 100 = \frac{\text{TD} \times 10 \times 2}{100} = 0.2\,\text{TD}, \] giving \( \text{TD} = 500 \). Now use \( \text{BG} = \dfrac{\text{TD}^{2}}{\text{PW}} \): \[ \text{PW} = \frac{\text{TD}^{2}}{\text{BG}} = \frac{500^{2}}{100} = \frac{250000}{100} = 2500. \] The answer can be checked directly: the true discount on a present worth of 2,500 at 10% for 2 years is \( 2500 \times 0.2 = 500 \), and the interest on that 500 is 100 - which is the banker’s gain as given. The two formulas for BG - interest on TD, and \( \text{TD}^{2}/\text{PW} \) - are the pair that solves most problems of this kind.

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: ₹80

The banker’s gain is the simple interest on the true discount: \[ \text{BG} = \frac{\text{TD} \times R \times T}{100}. \] Here \( R = 5 \) and \( T = 2 \), so \( RT = 10 \) and \[ 8 = \frac{\text{TD} \times 10}{100} = 0.1\,\text{TD} \;\Rightarrow\; \text{TD} = 80. \] The whole set follows from that. The present worth is \( \text{TD}^{2}/\text{BG} = 6400/8 = 800 \); the face value is \( 800 + 80 = 880 \); and the banker’s discount is \( 80 + 8 = 88 \). Note the geometric progression: PW, TD and BG stand as 800, 80, 8 - each a tenth of the previous, the ratio being \( RT/100 = 0.1 \). Recognising that pattern often gives the answer without any algebra at all. The distractor 88 is the banker’s discount, offered to catch the confusion between the two.

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: ₹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.

Answer: B

Correct answer: 10 months

Write both discounts in terms of the time \( T \) in years. The banker’s discount is simple interest on the face value: \[ \text{BD} = \frac{2200 \times 12 \times T}{100} = 264T. \] The true discount is \[ \text{TD} = \frac{2420 \times 12 \times T}{100 + 12T} = \frac{29040T}{100 + 12T}. \] Setting them equal and cancelling \( T \), \[ 264(100 + 12T) = 29040 \;\Rightarrow\; 26400 + 3168T = 29040 \;\Rightarrow\; T = \frac{2640}{3168} = \frac{5}{6}\text{ year}, \] which is 10 months. The step that matters is remembering that TD is calculated on the present worth, which is why its denominator carries the \( 100 + RT \) rather than 100.