True Discount Questions and Answers

Q: The True discount on Rs1760 due after a certain time at 12% per annum is Rs160. The time after which it is due is?

\[ \text{PW} = 1760 - 160 = \text{Rs. } 1600 \]\[ 160 = \frac{1600 \times 12 \times T}{100} = 192\,T \]\[ T = \frac{160}{192} = \frac{5}{6} \text{ year} = 10 \text{ months} \]The interest runs on the Rs. 1600 present worth, not the Rs. 1760 face value; using the latter would give about 9 months.

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101.

The True discount on Rs1760 due after a certain time at 12% per annum is Rs160. The time after which it is due is?

A.15months
B.5months
C.12months
D.10months

Answer: D

\[ \text{PW} = 1760 - 160 = \text{Rs. } 1600 \]

\[ 160 = \frac{1600 \times 12 \times T}{100} = 192\,T \]

\[ T = \frac{160}{192} = \frac{5}{6} \text{ year} = 10 \text{ months} \]

The interest runs on the Rs. 1600 present worth, not the Rs. 1760 face value; using the latter would give about 9 months.

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102.

A bill falls due in 1 year.The creditor agrees to accept immediate payment of the half and to defer the payment of the other half for 2 years.By this arrangement ins Rs.40.what is the amount of the bill,if he money be worth 12.5%?

A.1200
B.3000
C.3600
D.1300

Answer: C

Let the bill be \( A \), due in 1 year, with money worth \( 12.5\% \).

Original arrangement - the whole bill in 1 year:

\[ \text{PW}_{\text{old}} = \frac{A}{1 + 0.125} = \frac{A}{1.125} = 0.8889\,A \]

New arrangement - half now (no discount) and half in 2 years:

\[ \text{PW}_{\text{new}} = \frac{A}{2} + \frac{A/2}{1 + 0.25} = 0.5A + 0.4A = 0.9A \]

The creditor gains the difference:

\[ 0.9A - 0.8889A = 0.01111\,A = 40 \]

\[ A = \frac{40}{0.01111} = \text{Rs. } 3600 \]

Deferring half by a further year costs less than being paid the other half a year early gains - because discounting bites harder on the nearer payment.

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103.

The true discount on a bill due 9 months hence at 12% per annum is Rs. 540. Find the present worth of the bill.

A.1500
B.1600
C.6200
D.6000

Answer: D

True discount is simple interest on the present worth, so the present worth follows directly.

With \( T = \dfrac{9}{12} = 0.75 \) year at 12%:

\[ k = \frac{12 \times 0.75}{100} = 0.09 \]

\[ \text{PW} = \frac{\text{TD}}{k} = \frac{540}{0.09} = \text{Rs. } 6000 \]

The bill itself is \( 6000 + 540 = \text{Rs. } 6540 \). Dividing the discount by \( k \) always lands on the present worth; the face value needs the discount added back.

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104.

If Rs.20 is the True Discount on Rs260due after a certain time.What wil be true discount on the same sum due after half of the former time, the rate of interest being the same?

A.Rs.9.40
B.Rs.10.14
C.Rs.10.40
D.Rs.9.14

Answer: C

Simple Interest on Rs. (260-20) for a gven time= Rs.20

Simple Interest on Rs. (260-20) for a half time = Rs.10

True Discount on Rs.250 = Rs.10

True Discount on Rs.260 = Rs.[(10/250) x 260] = Rs. 10.40

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105.

The true discount on a certain sum of money due 3 year hence is Rs.100 and the S.I. on the same sum for the same time and at the same rate is Rs.120. Find the sum and the rate percent.

A.(6 + 2/3)%
B.(5+ 2/3)%
C.(2+ 2/3)%
D.(4+ 2/3)%

Answer: A

Simple interest is charged on the whole sum, true discount only on the present worth, so

\[ \frac{\text{SI}}{\text{TD}} = \frac{A}{A - \text{TD}} \quad\Longrightarrow\quad \frac{120}{100} = \frac{A}{A - 100} \]

\[ 120A - 12000 = 100A \quad\Longrightarrow\quad A = \text{Rs. } 600 \]

Now use the simple-interest formula on that sum over 3 years:

\[ 120 = \frac{600 \times R \times 3}{100} = 18R \]

\[ R = \frac{120}{18} = \frac{20}{3} = 6\tfrac{2}{3}\% \]

The present worth is \( 600 - 100 = \text{Rs. } 500 \), and 20% of 500 is indeed Rs. 100 - a useful check on both answers at once.

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106.

The present value of a bill due at the end of 2 years is Rs.1250. If the bill were due at the end of 2 years and 11 months, its present worth would be Rs.1200. Find the rate of interest and the sum.

A.rs.1175
B.rs.1375
C.rs.1475
D.rs.1575

Answer: B

The same bill \( A \) is being discounted over two different periods, so write each present worth in terms of \( A \) and the rate \( r \) (as a decimal).

\[ A = 1250(1 + 2r) \]

\[ A = 1200\left(1 + \tfrac{35}{12}r\right) \qquad \left(2 \text{ years } 11 \text{ months} = \tfrac{35}{12} \text{ years}\right) \]

Equating the two:

\[ 1250 + 2500r = 1200 + 3500r \]

\[ 50 = 1000r \quad\Longrightarrow\quad r = 0.05 = 5\% \]

\[ A = 1250(1 + 0.10) = \text{Rs. } 1375 \]

The longer the wait, the smaller the present worth - and the size of that drop over the extra 11 months is exactly what pins the rate down.

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