Banker’s Discount Questions and Answers

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.

\[ k = \frac{\text{TD}}{\text{PW}} = \frac{36}{800} = 0.045 \]

\[ A = \text{PW} + \text{TD} = 800 + 36 = \text{Rs. } 836 \]

\[ \text{BD} = A\,k = 836 \times 0.045 = \text{Rs. } 37.62 \]

Equivalently \( \text{BD} = \text{TD}(1+k) = 36 \times 1.045 = 37.62 \). The banker's gain is the 62 paise difference.

Answer: D

Correct answer: ₹840

The face value is given by \[ A = \frac{\text{BD} \times \text{TD}}{\text{BD} - \text{TD}}. \] Substituting, \[ A = \frac{42 \times 40}{42 - 40} = \frac{1680}{2} = 840. \] The formula comes from the definitions. BD is interest on \( A \) and TD is interest on \( \text{PW} = A - \text{TD} \), so \[ \frac{\text{BD}}{\text{TD}} = \frac{A}{A - \text{TD}}, \] and rearranging gives the result. The denominator \( \text{BD} - \text{TD} \) is the banker’s gain, here 2, so the formula can also be read as \( A = \text{BD} \times \text{TD} / \text{BG} \). As a check, the present worth is \( 840 - 40 = 800 \), and \( 40/800 = 42/840 \), confirming that both discounts are computed at the same rate and time.

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: 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: 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: D

Correct answer: ₹200

The two discounts are related by \[ \text{TD} = \frac{\text{BD} \times 100}{100 + RT}. \] Here \( RT = 16 \times \tfrac{1}{2} = 8 \), so \[ \text{TD} = \frac{216 \times 100}{108} = 200. \] The banker’s gain is therefore \( 216 - 200 = 16 \), and the check holds: the interest on the true discount of 200 at 16% for six months is \( 200 \times 0.08 = 16 \). The face value follows as \( \text{BD} \times 100/(RT) = 216 \times 100/8 = 2700 \), with a present worth of 2,500. The relation exists because BD is interest on the face value and TD is interest on the present worth, and the two principals stand in the ratio \( (100 + RT) : 100 \).

Answer: C

Correct answer: 12%

Let the face value be \( A \) and the rate \( R \). The banker’s discount is interest on the face value, so for 1.5 years \[ \frac{A \times R \times 1.5}{100} = 558 \;\Rightarrow\; AR = 37200. \] The true discount for 2 years is \[ \frac{A \times R \times 2}{100 + 2R} = 600 \;\Rightarrow\; \frac{2 \times 37200}{100 + 2R} = 600. \] Solving, \[ 74400 = 600(100 + 2R) = 60000 + 1200R \;\Rightarrow\; 1200R = 14400 \;\Rightarrow\; R = 12. \] The rate is 12% per annum, and the face value follows as \( A = 37200/12 = 3100 \). The step that makes this tractable is treating \( AR \) as a single unknown, which the first equation supplies directly - the two quantities never need to be separated until the end.

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

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

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

With \( k = \dfrac{10 \times 3}{100} = 0.3 \):

\[ \text{TD} = \frac{\text{BG}}{k} = \frac{36}{0.3} = \text{Rs. } 120 \]

\[ \text{PW} = \frac{\text{TD}}{k} = \frac{120}{0.3} = \text{Rs. } 400 \]

Two divisions by the same \( k \), because the banker's gain is interest on the true discount just as the true discount is interest on the present worth.

Answer: D

\[ \text{PW} = 2562 - 122 = \text{Rs. } 2440 \]

True discount is simple interest on the present worth over \( T = \dfrac{4}{12} = \dfrac{1}{3} \) year:

\[ 122 = \frac{2440 \times R \times \tfrac{1}{3}}{100} \]

\[ R = \frac{122 \times 300}{2440} = 15\% \]

Using the Rs. 2562 face value instead of the present worth gives about 14.3% - which is why 14% is on the list.