The true discount on a bill due 6 months hence at 12% per annum is Rs. 90. What is the amount of the bill ?
Answer: B
Correct answer: Rs. 1590
Here \(RT = 12 \times \tfrac{1}{2} = 6\), so
\[TD = \frac{A \times RT}{100 + RT} \quad\Longrightarrow\quad 90 = \frac{6A}{106}\]
\[A = \frac{90 \times 106}{6} = 1590\]
The present worth is Rs. 1500, on which six months of interest at 12% is Rs. 90.
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A man wants to sell his scooter .There are two offers one at Rs12000 cash and other at a credit of Rs12880 to be paid after 8 months ,money being at 18% per annum which is better offer?
Answer: B
\[ \text{Amount} = \text{PW}\left(1 + \frac{RT}{100}\right), \qquad \text{TD} = \text{Amount} - \text{PW} \]
Convert the credit offer to today's money before comparing.
\[ T = \frac{8}{12} = \frac{2}{3} \text{ year}, \qquad \frac{RT}{100} = \frac{18 \times 2/3}{100} = 0.12 \]
\[ \text{PW of credit offer} = \frac{12880}{1.12} = \text{Rs. } 11500 \]
\[ 12000 > 11500 \]
The cash offer is worth Rs. 500 more today, so Rs. 12,000 cash wins. The Rs. 880 premium fails to cover eight months of interest at 18%.
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If Rs. 10 be allowed as true discount on a bill of Rs. 110 due at the end of a certain time, then the discount allowed on the same sum due at the end of double the time is:
Answer: D
The present worth (PW) is what the sum is really worth today; the true discount (TD) is the interest that PW would earn over the remaining time:
\[ \text{Amount} = \text{PW}\left(1 + \frac{RT}{100}\right), \qquad \text{TD} = \text{Amount} - \text{PW} \]
First find the rate–time product from the original bill.
\[ \text{PW} = 110 - 10 = 100 \quad\Longrightarrow\quad 100 \times \frac{RT}{100} = 10 \quad\Longrightarrow\quad \frac{RT}{100} = 0.1 \]
Doubling the time doubles that product to 0.2, so for the same Rs. 110:
\[ \text{PW}' = \frac{110}{1.2} = 91.67 \]
\[ \text{TD}' = 110 - 91.67 = 18.33 \]
True discount does not double when the time doubles - the Rs. 20 option is exactly that trap. It grows more slowly because the present worth shrinks at the same time.
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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 gain on a bill due 3 year hence at 6% per annum is Rs. 18. The true discount 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.
Here \( k = \dfrac{6 \times 3}{100} = 0.18 \), and the banker's gain is interest on the true discount:
\[ \text{BG} = \text{TD} \times k \quad\Longrightarrow\quad 18 = \text{TD} \times 0.18 \]
\[ \text{TD} = \frac{18}{0.18} = \text{Rs. } 100 \]
The banker's gain is always the smallest of the four quantities, being interest on the discount rather than on either principal.
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Raj got a new chair for 35% discount. Had Raj got no discount, Raj would have had to pay Rs. 224 more. How much did Raj pay for the chair?
Answer: A
The Rs. 224 he did not have to pay is exactly the 35% discount.
\[ 0.35 \times \text{MP} = 224 \quad\Longrightarrow\quad \text{MP} = \frac{224}{0.35} = \text{Rs. } 640 \]
What he actually paid is the remaining 65%:
\[ 0.65 \times 640 = \text{Rs. } 416 \]
Equivalently, \( 640 - 224 = 416 \). Both Rs. 640 and Rs. 224 appear as options because they are the two intermediate values, not the amount paid.
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Find a single discount which is equivalent to three successive discounts of 15%, 16% & 20%.
Answer: C
Multiply the three retained fractions.
\[ (1-0.15)(1-0.16)(1-0.20) = 0.85 \times 0.84 \times 0.80 \]
\[ 0.85 \times 0.84 = 0.714, \qquad 0.714 \times 0.80 = 0.5712 \]
\[ \text{single equivalent discount} = 1 - 0.5712 = 0.4288 = 42.88\% \]
The rates sum to 51%, far above the truth. The order of the three discounts makes no difference, since multiplication is commutative.
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If Rs. 10 be allowed as true discount on a bill of Rs. 110 due at the end of a certain time, then the discount allowed on the same sum due at the end of double the time is:
Answer: D
First recover the rate–time product from the original bill.
\[ \text{PW} = 110 - 10 = 100 \quad\Longrightarrow\quad 100k = 10 \quad\Longrightarrow\quad k = 0.1 \]
Doubling the time doubles \( k \) to 0.2, so for the same Rs. 110:
\[ \text{PW}' = \frac{110}{1.2} = 91.67 \quad\Longrightarrow\quad \text{TD}' = 110 - 91.67 = \text{Rs. } 18.33 \]
True discount does not double when the time doubles - the Rs. 20 option is exactly that assumption. It grows more slowly, because the present worth it is computed on shrinks at the same time.
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The present worth of a sum due 3 years hence at 8% per annum is Rs. 2500. What is the true discount ?
Answer: D
Correct answer: Rs. 600
True discount is simply the simple interest on the present worth.
\[TD = \frac{PW \times R \times T}{100} = \frac{2500 \times 8 \times 3}{100} = 600\]
The bill itself is therefore worth \(2500 + 600 = \) Rs. 3100 at maturity.
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A man buys a watch for Rs. 2825 in cash and sells it for Rs. 3300 at a credit of 1 year. If the rate of interest is 10% per annum, the man gains:
Answer: B
\[ \text{Amount} = \text{PW}\left(1 + \frac{RT}{100}\right), \qquad \text{TD} = \text{Amount} - \text{PW} \]
The Rs. 3300 arrives a year late, so bring it back to today's money before comparing.
\[ \text{PW} = \frac{3300}{1 + \dfrac{10 \times 1}{100}} = \frac{3300}{1.1} = \text{Rs. } 3000 \]
\[ \text{gain} = 3000 - 2825 = \text{Rs. } 175 \]
The raw difference of Rs. 475 looks like the profit, but Rs. 300 of it is merely the price of waiting a year - which is why Rs. 475 is offered as a distractor.
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