True - Discount

1.A man purchased a cow for Rs. 3000 and sold it the same day for Rs. 3600, allowing the buyer a credit of 2 years. If the rate of interest be 10% per annum, then the man has a gain of:

1. 0%
2. 5%
3. 7.5%
4. 10%

Answer:Option 1

Explanation:

Solution:

\begin{aligned} C .P . = R s . 3000. \\ S .P . = R s . [\frac{3600 \times 100}{100 + (10 \times 2)}] \\ = R s . 3000. \\ Gain = 0 % . \end{aligned}

$\eqalign{ & {\text{C}}{\text{.P}}{\text{.}} = Rs.\,3000. \cr & {\text{S}}{\text{.P}}{\text{.}} = Rs.\,\left[ {\frac{{3600 \times 100}}{{100 + \left( {10 \times 2} \right)}}} \right] \cr & = Rs.\,3000. \cr & {\text{Gain}} = 0\% . \cr}$

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2.The true discount on Rs. 2562 due 4 months hence is Rs. 122. The rate percent is:

1. 12%
2. 13%
3. 15%
4. 14%

Answer:Option 1

Explanation:

Solution:

\begin{aligned} P .W . = R s . (2562 - 122) \\ = R s .2440 \\ ∴ S .I . on Rs . 2440 for 4 months is Rs . 122 . \\ ∴ Rate = [\frac{100 \times 122}{2440 \times \frac{1}{3}}] % = 15 % \end{aligned}

$\eqalign{ & {\text{P}}{\text{.W}}{\text{.}} = Rs.\left( {2562 - 122} \right) \cr & = Rs.2440 \cr & \therefore {\text{S}}{\text{.I}}{\text{.}}\,{\text{on}}\,{\text{Rs}}{\text{.}}\,{\text{2440}}\,{\text{for}}\,{\text{4}}\,{\text{months}}\,{\text{is}}\,{\text{Rs}}{\text{.}}\,{\text{122}}{\text{.}} \cr & \therefore {\text{Rate}} = \left[ {\frac{{100 \times 122}}{{2440 \times \frac{1}{3}}}} \right]\% = 15\% \cr}$

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3.A trader owes a merchant Rs. 10,028 due 1 year hence. The trader wants to settle the account after 3 months. If the rate of interest 12% per annum, how much cash should he pay?

1. Rs. 9025.20
2. Rs. 9200
3. Rs. 9600
4. Rs. 9560

Answer:Option 1

Explanation:

Solution:

\begin{aligned} Required Money \\ = P .W . of Rs . 10028 due 9 months hence \\ = R s . [\frac{10028 \times 100}{100 + (12 \times \frac{9}{12})}] \\ = R s . 9200. \end{aligned}

$\eqalign{ & {\text{Required}}\,{\text{Money}} \cr & {\text{ = }}\,{\text{P}}{\text{.W}}{\text{.}}\,{\text{of}}\,{\text{Rs}}{\text{.}}\,{\text{10028}}\,{\text{due}}\,{\text{9}}\,{\text{months}}\,{\text{hence}} \cr & = Rs.\,\left[ {\frac{{10028 \times 100}}{{100 + \left( {12 \times \frac{9}{{12}}} \right)}}} \right] \cr & = Rs.\,9200. \cr}$

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4.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:

1. Rs. 20
2. Rs. 21.81
3. Rs. 22
4. Rs. 18.33

Answer:Option 1

Explanation:

Solution:

S.I. on Rs. (110 - 10) for a certain time = Rs. 10.
S.I. on Rs. 100 for double the time = Rs. 20.
T.D. on Rs. 120 = Rs. (120 - 100) = Rs. 20.

\begin{aligned} T .D . on Rs . 110 \\ = R s . (\frac{20}{120} \times 110) \\ = R s . 18.33 \end{aligned}

$\eqalign{ & {\text{T}}{\text{.D}}{\text{.}}\,{\text{on}}\,{\text{Rs}}{\text{.}}\,{\text{110}} \cr & = Rs.\,\left( {\frac{{20}}{{120}} \times 110} \right) \cr & = Rs.\,18.33 \cr}$

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5.Goods were bought for Rs. 600 and sold the same for Rs. 688.50 at a credit of 9 months and thus gaining 2% The rate of interest per annum is:

1. 16²/₃%
2. 14¹/₂%
3. 13¹/₃%
4. 15%

Answer:Option 1

Explanation:

Solution:

\begin{aligned} S .P . = 102 % o f R s . 600 \\ = (\frac{102}{100} \times 600) \\ = R s . 612. \\ Now, P .W . = Rs .612 and sum \\ = R s . 688.50. \\ ∴ T .D . = R s . (688.50 - 612) \\ = R s . 76.50. \\ Thus, S .I . on Rs . 612 for 9 months is Rs . 76 .50 . \\ ∴ Rate = (\frac{100 \times 76.50}{612 \times \frac{3}{4}}) % \\ = 16 \frac{2}{3} % \end{aligned}

$\eqalign{ & {\text{S}}{\text{.P}}{\text{.}} = 102\% \,of\,Rs.\,600 \cr & = \left( {\frac{{102}}{{100}} \times 600} \right) \cr & = Rs.\,612. \cr & {\text{Now,}}\,{\text{P}}{\text{.W}}{\text{. = Rs}}{\text{.612}}\,{\text{and}}\,{\text{sum}} \cr & = Rs.\,688.50. \cr & \therefore {\text{T}}{\text{.D}}{\text{.}} = Rs.\,\left( {688.50 - 612} \right) \cr & = Rs.\,76.50. \cr & {\text{Thus,}}\,{\text{S}}{\text{.I}}{\text{.}}\,{\text{on}}\,{\text{Rs}}{\text{.}}\,{\text{612}}\,{\text{for}}\,{\text{9}}\,{\text{months}}\,{\text{is}}\,{\text{Rs}}{\text{.}}\,{\text{76}}{\text{.50}}{\text{.}} \cr & \therefore {\text{Rate}} = \left( {\frac{{100 \times 76.50}}{{612 \times \frac{3}{4}}}} \right)\% \cr & = 16\frac{2}{3}\% \, \cr}$

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6.The true discount on a bill due 9 months hence at 16% per annum is Rs. 189. The amount of the bill is:

1. Rs. 1386
2. Rs. 1764
3. Rs. 1575
4. Rs. 2268

Answer:Option 1

Explanation:

Solution:

\begin{aligned} Let P .W . be Rs . x . \\ Then, S .I . on Rs . x at 16 % for 9 months \\ = R s . 189. \\ ∴ x \times 16 \times \frac{9}{12} \times \frac{1}{100} \\ = 189 o r x = 1575. \\ ∴ P . W . = R s . 1575. \\ ∴ Sum due = P . W . + T . D . \\ = R s . (1575 + 189) \\ = R s . 1764. \end{aligned}

$\eqalign{ & {\text{Let}}\,{\text{P}}{\text{.W}}{\text{.}}\,{\text{be}}\,{\text{Rs}}{\text{.}}\,{\text{x}}{\text{.}} \cr & {\text{Then,}}\,{\text{S}}{\text{.I}}{\text{.}}\,{\text{on}}\,{\text{Rs}}{\text{.}}\,{\text{x}}\,{\text{at}}\,{\text{16}}\% \,{\text{for}}\,{\text{9}}\,{\text{months}} \cr & = Rs.\,189. \cr & \therefore x \times 16 \times \frac{9}{{12}} \times \frac{1}{{100}} \cr & = 189\,or\,x = 1575. \cr & \therefore P.W. = Rs.\,1575. \cr & \therefore {\text{Sum}}\,{\text{due}} = P.W. + T.D. \cr & = Rs.\left( {1575 + 189} \right) \cr & = Rs.\,1764. \cr}$

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7.A man buys a watch for Rs. 1950 in cash and sells it for Rs. 2200 at a credit of 1 year. If the rate of interest is 10% per annum, the man:

1. gains Rs. 55
2. gains Rs. 50
3. loses Rs. 30
4. gains Rs. 30

Answer:Option 1

Explanation:

Solution:

\begin{aligned} S .P . = P .W . of Rs . 2200 due 1 year hence \\ = R s . [\frac{2200 \times 100}{100 + (10 \times 1)}] \\ = R s . 2000. \\ ∴ Gain = R s . (2000 - 1950) \\ = R s . 50. \end{aligned}

$\eqalign{ & {\text{S}}{\text{.P}}{\text{. = P}}{\text{.W}}{\text{.}}\,{\text{of}}\,{\text{Rs}}{\text{.}}\,{\text{2200}}\,{\text{due}}\,{\text{1}}\,{\text{year}}\,{\text{hence}} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left[ {\frac{{2200 \times 100}}{{100 + \left( {10 \times 1} \right)}}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,2000. \cr & \therefore {\text{Gain}} = Rs.\left( {2000 - 1950} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,50. \cr}$

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8.The present worth of Rs. 2310 due 2¹/₂ years hence, the rate of interest being 15% per annum, is:

1. Rs. 1750
2. Rs. 1680
3. Rs. 1840
4. Rs. 1443.75

Answer:Option 1

Explanation:

Solution:

\begin{aligned} P . W . = R s . [\frac{100 \times 2310}{100 + (15 \times \frac{5}{2})}] \\ = R s . 1680. \end{aligned}

$\eqalign{ & P.W. = Rs.\,\left[ {\frac{{100 \times 2310}}{{100 + \left( {15 \times \frac{5}{2}} \right)}}} \right] \cr & = Rs.\,1680. \cr}$

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9.Rs. 20 is the true discount on Rs. 260 due after a certain time. What will be the true discount on the same sum due after half of the former time, the rate of interest being the same?

1. Rs. 10
2. Rs. 10.40
3. Rs. 15.20
4. Rs. 13

Answer:Option 1

Explanation:

Solution:

\begin{aligned} S .I . on Rs . (260 - 20) for a given time \\ = R s . 20. \\ S .I . on Rs . 240 for half time \\ = R s .10 . \\ T . D . o n R s . 250 = R s .10 . \\ ∴ T . D . o n R s . 260 \\ = R s . (\frac{10}{250} \times 260) \\ = R s . 10.40 \end{aligned}

$\eqalign{ & {\text{S}}{\text{.I}}{\text{.}}\,{\text{on}}\,{\text{Rs}}{\text{.}}\,\left( {{\text{260 - 20}}} \right)\,{\text{for}}\,{\text{a}}\,{\text{given}}\,{\text{time}} \cr & = Rs.\,20. \cr & {\text{S}}{\text{.I}}{\text{.}}\,{\text{on}}\,{\text{Rs}}{\text{.}}\,{\text{240}}\,{\text{for}}\,{\text{half}}\,{\text{time}} \cr & = Rs.10. \cr & T.D.\,on\,Rs.\,250 = Rs.10. \cr & \therefore T.D.\,on\,Rs.\,260 \cr & = Rs.\,\left( {\frac{{10}}{{250}} \times 260} \right) \cr & = Rs.\,10.40 \cr}$

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10.The interest on Rs. 750 for 2 years is the same as the true discount on Rs. 960 due 2 years hence. If the rate of interest is the same in both cases, it is:

1. 12%
2. 14%
3. 15%
4. 16%

Answer:Option 1

Explanation:

Solution:

S.I. on Rs. 750 = T.D. on Rs. 960.
This means P.W. of Rs. 960 due 2 years hence is Rs. 750.
∴ T.D. = Rs. (960 - 750) = Rs. 210.
Thus, S.I. on R.s 750 for 2 years is Rs. 210.