Banker’s Discount Questions and Answers

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: 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.

Answer: B

Correct answer: 8 months

First find the present worth, since the true discount is interest on that rather than on the face value: \[ \text{PW} = 1860 - 60 = 1800. \] Now apply the simple interest formula to the present worth: \[ 60 = \frac{1800 \times 5 \times T}{100} = 90T \;\Rightarrow\; T = \frac{60}{90} = \frac{2}{3}\text{ year}, \] which is 8 months. The single step that decides this problem is subtracting the discount before applying the interest formula. Using the face value of 1,860 instead gives \( T = 60 \times 100/(1860 \times 5) = 0.645 \) year, about 7.7 months - close enough to look plausible and wrong. The general form is \( \text{TD} = \dfrac{\text{PW} \times R \times T}{100} \), and remembering that PW rather than \( A \) appears in it settles most questions in this topic.

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: D

Correct answer: ₹60

The true discount is the interest on the present worth, not on the face value, which gives \[ \text{TD} = \frac{A \times RT}{100 + RT}. \] Here \( R = 8 \) and \( T = \frac{9}{12} = 0.75 \) year, so \( RT = 6 \). Then \[ \text{TD} = \frac{1060 \times 6}{100 + 6} = \frac{6360}{106} = 60. \] The present worth is therefore \( 1060 - 60 = 1000 \), which checks: the interest on 1,000 at 8% for nine months is \( 1000 \times 0.06 = 60 \). The distractor 63.60 is what comes of taking the interest on the face value instead - \( 1060 \times 0.06 \) - and that figure is in fact the banker’s discount. Confusing the two is the commonest error in this topic.

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: B

Correct answer: ₹2,000

The present worth is the sum which, invested now at simple interest, would amount to the face value at maturity: \[ \text{PW} = \frac{100 \times A}{100 + RT}. \] Here \( R = 12 \) and \( T = 2.5 \) years, so \( RT = 30 \). Then \[ \text{PW} = \frac{100 \times 2600}{130} = \frac{260000}{130} = 2000. \] Checking forwards: 2,000 at 12% for two and a half years earns \( 2000 \times 0.30 = 600 \) in interest, giving exactly 2,600 at maturity. The true discount is that 600, being the difference between the face value and the present worth. The relation \( A = \text{PW} + \text{TD} \) holds always, and it is the quickest check on any answer in this topic.

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: 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: B

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

With \( T = 2\tfrac{1}{2} = 2.5 \) years at 15%:

\[ \frac{RT}{100} = \frac{15 \times 2.5}{100} = 0.375 \]

\[ \text{PW} = \frac{2310}{1.375} = \text{Rs. } 1680 \]

Checking: the discount is \( 2310 - 1680 = 630 \), and 37.5% of 1680 is 630 - interest on the present worth, exactly as the definition requires.