The true discount on a bill of Rs. 1860 due after 8 months is Rs. 60. Find the banker’s discount ?
Answer: A
Amount = Rs. 1860.
True discount = Rs. 60
∴ Present worth = Rs. 1860 - Rs. 60 = Rs. 1800
S.I. on Rs. 1800 for 8 months = Rs. 60
∴ Rate = [(100 * 60) / (1800 * 2/3)]% = 5%
∴ Banker’s discount = (1860 * 5 * 2/3)/100 = Rs. 62
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The banker’s gain of a certain sum due 2 years hence at 5% per annum is Rs. 80. The present worth is ?
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 \( k = \dfrac{5 \times 2}{100} = 0.1 \):
\[ \text{TD} = \frac{\text{BG}}{k} = \frac{80}{0.1} = \text{Rs. } 800 \]
\[ \text{PW} = \frac{\text{TD}}{k} = \frac{800}{0.1} = \text{Rs. } 8000 \]
Rs. 800 is only the true discount, one step short of the answer. As a check, the face value is \( 8000 + 800 = 8800 \) and the banker's discount is \( 8800 \times 0.1 = 880 \), exactly Rs. 80 above the true discount.
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The banker’s discount on a bill due 6 months hence at 6% is Rs. 18.54. What is the true discount?
Answer: D
Writing \( k = \dfrac{RT}{100} \), the four quantities are related by
\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{TD}\,k \]
With \( T = \dfrac{6}{12} = 0.5 \) year at 6%:
\[ k = \frac{6 \times 0.5}{100} = 0.03 \]
Since \( \text{BD} = A\,k \) and \( \text{TD} = \dfrac{A\,k}{1+k} \), the two differ only by the factor \( 1+k \):
\[ \text{TD} = \frac{\text{BD}}{1+k} = \frac{18.54}{1.03} = \text{Rs. } 18 \]
The banker's gain is the 54 paise difference - small here because the period is short and the rate low.
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The banker’s gain on a certain sum due 5/2 years hence is 9/25 of the banker’s discount. What is the rate percent?
Answer: D
T = 5/2 year
Let the banker’s discount, BD = Rs.1
Then, banker’s gain, BG = (9/25)*1 = Rs. 9/25
BG = BD-TD
=> 9/25 = 1-TD
=> TD = 1-9/25 = 16/25
F = BD*TD/(BD-TD)
= (1*16/25)/(1-(16/25)) (16/25)/(9/25) = Rs. 16/9
BD = simple interest on the face value of the bill for unexpired time = FTR/100
=> 1 = (16/9)*(5/2)*R/100
=> 100 = (16/9)*(5/2)*R
=> 100 = 16*5*R/9*2
=> 100 = 8*5*R/9
=> R = 100*9/8*5
= 100*9/40 = 5*9/2 = 45/2 = 22 1/2 %
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The true discount on a bill of Rs. 540 is Rs. 90. The banker’s discount is
Answer: B
Writing \( k = \dfrac{RT}{100} \), the four quantities are related by
\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{TD}\,k \]
\[ \text{PW} = 540 - 90 = \text{Rs. } 450 \quad\Longrightarrow\quad k = \frac{90}{450} = 0.2 \]
\[ \text{BD} = A\,k = 540 \times 0.2 = \text{Rs. } 108 \]
The banker's discount charges interest on the Rs. 540 face value, the true discount only on the Rs. 450 present worth - hence the Rs. 18 gap.
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The banker’s discount on Rs. 1600 at 15% per annum is the same as true discount on Rs. 1680 for the same time and at the same rate. The time is
Answer: B
S.I. on Rs. 1600 = T.D. on Rs. 1680.
=> Rs. 1600 is the P.W. of Rs. 1680,
=> Rs. 80 is S.I. on Rs. 1600 at 15%.
=> 80 = 1600*T*15/100
=> 80 = 16*T*15
=> 5 = T*15
=> T = 1/3 year = 12/3 months
= 4 months
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The banker's discount on a certain sum of money is Rs. 72 and the true discount on the same sum for the same time is Rs. 60. The sum due is:
Answer: A
Writing \( k = \dfrac{RT}{100} \), the four quantities are related by
\[ \text{BD} = A\,k, \qquad \text{TD} = \frac{A\,k}{1+k} = \text{PW}\,k, \qquad \text{BG} = \text{TD}\,k \]
Both discounts use the same \( k \), one on the amount and one on the present worth:
\[ \text{BD} = A\,k, \qquad \text{TD} = (A - \text{TD})\,k \]
Dividing one by the other:
\[ \frac{72}{60} = \frac{A}{A - 60} \quad\Longrightarrow\quad 72A - 4320 = 60A \]
\[ 12A = 4320 \quad\Longrightarrow\quad A = \text{Rs. } 360 \]
The standard shortcut is
\[ \text{Sum} = \frac{\text{BD} \times \text{TD}}{\text{BD} - \text{TD}} = \frac{72 \times 60}{12} = 360 \]
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The profit earned by selling an article for Rs. 900 is double the loss incurred when the same article is sold for Rs. 450. At what price should the article be sold to make a 25% profit?
Answer: D
Let the cost price be \( C \). Selling at Rs. 900 gives a profit of \( 900 - C \); selling at Rs. 450 gives a loss of \( C - 450 \).
\[ 900 - C = 2(C - 450) \]
\[ 900 - C = 2C - 900 \quad\Longrightarrow\quad 3C = 1800 \quad\Longrightarrow\quad C = \text{Rs. } 600 \]
Checking: the profit is Rs. 300 and the loss Rs. 150 - double, as required.
\[ \text{SP for 25\% profit} = 600 \times 1.25 = \text{Rs. } 750 \]
Averaging the two selling prices would give Rs. 675, which is the cost price only when profit and loss are equal - not the case here.
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The present worth of Rs. 1320 due 2 years hence at 10% per annum is:
Answer: B
Correct answer: Rs. 1100
\[PW = \frac{A \times 100}{100 + RT} = \frac{1320 \times 100}{100 + 20} = \frac{132000}{120} = 1100\]
Checking: the simple interest on Rs. 1100 for 2 years at 10% is Rs. 220, and \(1100 + 220 = 1320\).
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The present worth of a bill of Rs. 2650 due 9 months hence at 8% per annum is:
Answer: A
Correct answer: Rs. 2500
With \(RT = 8 \times \tfrac{3}{4} = 6\):
\[PW = \frac{2650 \times 100}{106} = 2500\]
The true discount is the remaining Rs. 150, which is nine months of interest on Rs. 2500 at 8%.
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