Bankers-Discount

1.The banker's discount on a bill due 4 months hence at 15% is Rs. 420. The true discount is:

1. Rs. 400
2. Rs. 360
3. Rs. 480
4. Rs. 320

Answer:Option 1

Explanation:

Solution:

\begin{aligned} T . D . = \frac{B . D . \times 100}{100 + (R \times T)} \\ = R s . [\frac{420 \times 100}{100 + (15 \times \frac{1}{3})}] \\ = R s . (\frac{420 \times 100}{105}) \\ = R s . 400. \end{aligned}

$\eqalign{ & T.D. = \frac{{B.D. \times 100}}{{100 + \left( {R \times T} \right)}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\left[ {\frac{{420 \times 100}}{{100 + \left( {15 \times \frac{1}{3}} \right)}}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {\frac{{420 \times 100}}{{105}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,400. \cr}$

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2.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:

1. 3 months
2. 4 months
3. 6 months
4. 8 months

Answer:Option 1

Explanation:

Solution:

\begin{aligned} S .I . on Rs . 1600 = T .D . on Rs . 1680 . \\ ∴ Rs . 1600 is the P .W . of Rs . 1680, i .e ., \\ Rs . 80 is on Rs . 1600 at 15 % . \\ ∴ Time = (\frac{100 \times 80}{1600 \times 15}) year \\ = \frac{1}{3} year = 4 months . \end{aligned}

$\eqalign{ & {\text{S}}{\text{.I}}{\text{.}}\,{\text{on}}\,{\text{Rs}}{\text{.}}\,{\text{1600 = T}}{\text{.D}}{\text{.}}\,{\text{on}}\,{\text{Rs}}{\text{.}}\,{\text{1680}}{\text{.}} \cr & \therefore {\text{Rs}}{\text{.}}\,{\text{1600}}\,{\text{is}}\,{\text{the}}\,{\text{P}}{\text{.W}}{\text{.}}\,{\text{of}}\,{\text{Rs}}{\text{.}}\,{\text{1680,}}\,{\text{i}}{\text{.e}}{\text{.,}} \cr & {\text{Rs}}{\text{.}}\,{\text{80}}\,{\text{is}}\,{\text{on}}\,{\text{Rs}}{\text{.}}\,{\text{1600}}\,{\text{at}}\,{\text{15}}\% . \cr & \therefore {\text{Time}} = \left( {\frac{{100 \times 80}}{{1600 \times 15}}} \right){\text{year}} \cr & = \frac{1}{3}{\text{year}} = 4\,{\text{months}}{\text{.}} \cr}$

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3.The banker's gain of a certain sum due 2 years hence at 10% per annum is Rs. 24. The present worth is:

1. Rs. 480
2. Rs. 520
3. Rs. 600
4. Rs. 960

Answer:Option 1

Explanation:

Solution:

\begin{aligned} T . D . = (\frac{B . G . \times 100}{R a t e \times T i m e}) \\ = R s . (\frac{24 \times 100}{10 \times 2}) \\ = R s . 120. \\ ∴ P . W . = \frac{100 \times T . D .}{R a t e \times T i m e} \\ = R s . (\frac{100 \times 120}{10 \times 2}) \\ = R s . 600. \end{aligned}

$\eqalign{ & T.D. = \left( {\frac{{B.G. \times 100}}{{Rate \times Time}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {\frac{{24 \times 100}}{{10 \times 2}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,120. \cr & \therefore P.W. = \frac{{100 \times T.D.}}{{Rate \times Time}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {\frac{{100 \times 120}}{{10 \times 2}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,600. \cr}$

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4.The banker's gain on a sum due 3 years hence at 12% per annum is Rs. 270. The banker's discount is:

1. Rs. 960
2. Rs. 840
3. Rs. 1020
4. Rs. 760

Answer:Option 1

Explanation:

Solution:

\begin{aligned} T . D . = (\frac{B . G . \times 100}{R \times T}) \\ = R s . (\frac{270 \times 100}{12 \times 3}) \\ = R s . 750. \\ ∴ B . D . = R s . (750 + 270) \\ = R s . 1020. \end{aligned}

$\eqalign{ & T.D. = \left( {\frac{{B.G. \times 100}}{{R \times T}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {\frac{{270 \times 100}}{{12 \times 3}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,750. \cr & \therefore B.D. = Rs.\left( {750 + 270} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,1020. \cr}$

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5.The banker's discount of 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:

1. Rs. 360
2. Rs. 432
3. Rs. 540
4. Rs. 1080

Answer:Option 1

Explanation:

Solution:

\begin{aligned} \frac{B . D . \times T . D .}{B . D . - T . D .} \\ = R s . (\frac{72 \times 60}{72 - 60}) \\ = R s . (\frac{72 \times 60}{12}) \\ = R s . 360. \end{aligned}

$\eqalign{ & \frac{{B.D. \times T.D.}}{{B.D. - T.D.}} \cr & = Rs.\,\left( {\frac{{72 \times 60}}{{72 - 60}}} \right) \cr & = Rs.\,\left( {\frac{{72 \times 60}}{{12}}} \right) \cr & = Rs.\,360. \cr}$

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6.The certain worth of a certain sum due sometime hence is Rs. 1600 and the true discount is Rs. 160. The banker's gain is:

1. Rs. 20
2. Rs. 24
3. Rs. 16
4. Rs. 12

Answer:Option 1

Explanation:

Solution:

\begin{aligned} B . G . = \frac{{(T . D .)}^{2}}{P . W .} \\ = R s . (\frac{160 \times 160}{1600}) \\ = R s . 16. \end{aligned}

$\eqalign{ & B.G. = \frac{{{{\left( {T.D.} \right)}^2}}}{{P.W.}} \cr & = Rs.\,\left( {\frac{{160 \times 160}}{{1600}}} \right) \cr & = Rs.\,16. \cr}$

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7.The present worth of a certain bill due sometime hence is Rs. 800 and the true discount is Rs. 36. The banker's discount is:

1. Rs. 37
2. Rs. 37.62
3. Rs. 34.38
4. Rs. 38.98

Answer:Option 1

Explanation:

Solution:

\begin{aligned} B . G . = \frac{{(T . D .)}^{2}}{P . W .} \\ = R s . (\frac{36 \times 36}{800}) \\ = R s . 1.62. \\ ∴ B . D . = (T . D . + B . G .) \\ = R s . (36 + 1.62) \\ = R s . 37.62 \end{aligned}

$\eqalign{ & B.G. = \frac{{{{\left( {T.D.} \right)}^2}}}{{P.W.}} \cr & = Rs.\,\left( {\frac{{36 \times 36}}{{800}}} \right) \cr & = Rs.\,1.62. \cr & \therefore B.D. = \left( {T.D. + B.G.} \right) \cr & = Rs.\,\left( {36 + 1.62} \right) \cr & = Rs.\,37.62 \cr}$

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8.The banker's gain on a certain sum due 1¹/₂ years hence is ³/₂₅ of the banker's discount. The rate percent is:

1. 5¹/₅ %
2. 9¹/₁₁ %
3. 8¹/₈ %
4. 6¹/₆ %

Answer:Option 1

Explanation:

Solution:

\begin{aligned} Let,B .D . = Re . 1. \\ Then, B .G . = Re . \frac{3}{25} . \\ ∴ T .D . = (B .D . - B .G .) \\ = Re . (1 - \frac{3}{25}) \\ = Re . \frac{22}{25} . \\ Sum = (\frac{1 \times (22 / 25)}{1 - (22 / 25)}) \\ = R s . \frac{22}{3} . \\ S .I . on R s . \frac{22}{3} f o r 1 \frac{1}{2} years is Re . 1. \\ ∴ Rate = (\frac{100 \times 1}{\frac{22}{3} \times \frac{3}{2}}) % \\ = \frac{100}{11} % \\ = 9 \frac{1}{11} % \end{aligned}

$\eqalign{ & {\text{Let,B}}{\text{.D}}{\text{.}} = \operatorname{Re} .\,1. \cr & {\text{Then,}}\,{\text{B}}{\text{.G}}{\text{.}} = \operatorname{Re} .\,\frac{3}{{25}}. \cr & \therefore {\text{T}}{\text{.D}}{\text{. = }}\left( {{\text{B}}{\text{.D}}{\text{. - B}}{\text{.G}}{\text{.}}} \right) \cr & = \operatorname{Re} .\,\left( {1 - \frac{3}{{25}}} \right) \cr & = \operatorname{Re} .\,\frac{{22}}{{25}}. \cr & {\text{Sum}} = \left( {\frac{{1 \times \left( {22/25} \right)}}{{1 - \left( {22/25} \right)}}} \right) \cr & = Rs.\,\frac{{22}}{3}. \cr & {\text{S}}{\text{.I}}{\text{.}}\,{\text{on}}\,Rs.\,\frac{{22}}{3}\,for\,1\frac{1}{2}{\text{years}}\,{\text{is}}\,\operatorname{Re} .\,1. \cr & \therefore {\text{Rate}} = \left( {\frac{{100 \times 1}}{{\frac{{22}}{3} \times \frac{3}{2}}}} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{100}}{{11}}\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 9\frac{1}{{11}}\% \cr}$

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9.The present worth of a sum due sometime hence is Rs. 576 and the banker's gain is Rs. 16. The true discount is:

1. Rs. 36
2. Rs. 72
3. Rs. 48
4. Rs. 96

Answer:Option 1

Explanation:

Solution:

\begin{aligned} T . D . = \sqrt{P . W . \times B . G .} \\ = \sqrt{576 \times 16} \\ = 96 \end{aligned}

$\eqalign{ & T.D. = \sqrt {P.W. \times B.G.} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \sqrt {576 \times 16} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 96 \cr}$

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10.The true discount on a bill of Rs. 540 is Rs. 90. The banker's discount is:

1. Rs. 60
2. Rs. 108
3. Rs. 110
4. Rs. 112

Answer:Option 1

Explanation:

Solution:

\begin{aligned} P .W . = R s . (540 - 90) \\ = R s . 450. \\ ∴ S .I . on R s . 450 = R s . 90. \\ S .I . on R s . 540 \\ = R s . (\frac{90}{450} \times 540) \\ = R s . 108 \\ ∴ B .D . = R s . 108. \end{aligned}

$\eqalign{ & {\text{P}}{\text{.W}}{\text{.}} = Rs.\,\left( {540 - 90} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,450. \cr & \therefore {\text{S}}{\text{.I}}{\text{.}}\,{\text{on}}\,Rs.\,450 = Rs.\,90. \cr & {\text{S}}{\text{.I}}{\text{.}}\,{\text{on}}\,Rs.\,540 \cr & = Rs.\,\left( {\frac{{90}}{{450}} \times 540} \right) \cr & = Rs.\,108 \cr & \therefore {\text{B}}{\text{.D}}. = Rs.\,108. \cr}$

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