Banker’s Discount Questions and Answers

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: D

Correct answer: ₹300

The simple interest here is calculated on the face value, so it is the banker’s discount, and the standard relation applies: \[ A = \frac{\text{SI} \times \text{TD}}{\text{SI} - \text{TD}} = \frac{60 \times 50}{60 - 50} = \frac{3000}{10} = 300. \] The derivation is worth seeing. SI is interest on \( A \); TD is interest on \( A - \text{TD} \). Since interest is proportional to principal at a fixed rate and time, \[ \frac{\text{SI}}{\text{TD}} = \frac{A}{A - \text{TD}}, \] and cross-multiplying gives the formula. Checking: the present worth is \( 300 - 50 = 250 \), and \( 50/250 = 60/300 = 0.2 \), so both are computed at \( RT = 20 \). The difference \( \text{SI} - \text{TD} = 10 \) is the banker’s gain, and it is also the interest on the true discount: \( 50 \times 0.2 = 10 \).

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 \( k = \dfrac{12 \times 1}{100} = 0.12 \):

\[ \text{BG} = \text{TD} \times k \quad\Longrightarrow\quad \text{TD} = \frac{6}{0.12} = \text{Rs. } 50 \]

Checking the whole chain: PW = \( 50/0.12 = 416.67 \), face value \( 466.67 \), and BD = \( 466.67 \times 0.12 = 56 \), which is indeed Rs. 6 more than the true discount.

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: B

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

\[ \text{PW} = \frac{2200}{1 + \dfrac{10 \times 1}{100}} = \frac{2200}{1.1} = \text{Rs. } 2000 \]

\[ \text{gain} = 2000 - 1950 = \text{Rs. } 50 \]

The headline difference of Rs. 250 overstates the profit: Rs. 200 of it merely compensates for waiting a year at 10%. Only the Rs. 50 remainder is genuine gain.

Answer: A

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

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

Both discounts use the same \( k \), one on the amount and one on the present worth:

\[ \frac{\text{BD}}{\text{TD}} = \frac{A}{A - \text{TD}} \quad\Longrightarrow\quad \frac{36}{30} = \frac{A}{A - 30} \]

\[ 36A - 1080 = 30A \quad\Longrightarrow\quad 6A = 1080 \]

\[ A = \text{Rs. } 180 \]

The shortcut is

\[ \text{Sum} = \frac{\text{BD} \times \text{TD}}{\text{BD} - \text{TD}} = \frac{36 \times 30}{6} = 180 \]

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.

\[ \text{PW} = 540 - 90 = \text{Rs. } 450 \quad\Longrightarrow\quad k = \frac{\text{TD}}{\text{PW}} = \frac{90}{450} = 0.2 \]

\[ \text{BD} = A\,k = 540 \times 0.2 = \text{Rs. } 108 \]

The banker's discount always exceeds the true discount, here by \( 108 - 90 = \text{Rs. } 18 \), because it charges interest on the larger face value.

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: A

Correct answer: ₹82.50

The banker’s gain is the simple interest on the true discount: \[ \text{BG} = \frac{75 \times 5 \times 2}{100} = \frac{750}{100} = 7.50. \] The banker’s discount follows: \[ \text{BD} = \text{TD} + \text{BG} = 75 + 7.50 = 82.50. \] The same answer comes from the ratio \( \text{BD} = \text{TD} \times \dfrac{100 + RT}{100} = 75 \times \dfrac{110}{100} = 82.50 \), which is quicker when the rate and time are given. For completeness, the present worth is \( \text{TD} \times 100/(RT) = 75 \times 100/10 = 750 \), and the face value is 825. Note the progression 750, 75, 7.50 - present worth, true discount and banker’s gain, each a tenth of the last because \( RT/100 = 0.1 \). Spotting it gives the answer without any formula at all.

Answer: A

Correct answer: 4 months

Write both in terms of the time \( T \) in years. The banker’s discount is interest on the face value: \[ \text{BD} = \frac{1800 \times 12 \times T}{100} = 216T. \] The true discount is \[ \text{TD} = \frac{1872 \times 12 \times T}{100 + 12T} = \frac{22464T}{100 + 12T}. \] Equating and cancelling \( T \), \[ 216(100 + 12T) = 22464 \;\Rightarrow\; 21600 + 2592T = 22464 \;\Rightarrow\; T = \frac{864}{2592} = \frac{1}{3}\text{ year}, \] which is 4 months. Cancelling \( T \) from both sides is the step that makes the algebra manageable; it is legitimate because \( T \) cannot be zero. The pattern - BD on one sum equalling TD on a larger one - always reduces to a linear equation in this way.