Bankers Discount — Part 1 MCQs with Explanations

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

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

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

$$\eqalign{ & {\text{B}}{\text{.D}}{\text{.}}\,{\text{for}}\frac{3}{2}{\text{years}} = Rs.\,558 \cr & {\text{B}}{\text{.D}}{\text{.}}\,{\text{for 2}}\,{\text{years}} \cr & = Rs.\left( {558 \times \frac{2}{3} \times 2} \right) \cr & = Rs.\,744 \cr & {\text{T}}{\text{.D}}{\text{.}}\,{\text{for}}\,{\text{2}}\,{\text{years}} = Rs.\,600 \cr & \therefore {\text{Sum}} = \frac{{B.D. \times T.D.}}{{B.D. - T.D.}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {\frac{{744 \times 600}}{{144}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,3100 \cr & {\text{Thus,}}\,{\text{Rs}}{\text{.}}\,{\text{744}}\,{\text{is}}\,{\text{S}}{\text{.I}}{\text{.}}\,{\text{on}}\,{\text{Rs}}{\text{.}}\,{\text{3100}}\,{\text{for}}\,{\text{2}}\,{\text{years}}{\text{.}} \cr & \therefore {\text{Rate}} = \left( {\frac{{100 \times 744}}{{3100 \times 2}}} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 12\% \cr} $$

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

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

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

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

$$\eqalign{ & T.D. = \frac{{B.G. \times 100}}{{R \times T}} \cr & = Rs.\,\left( {\frac{{6 \times 100}}{{12 \times 1}}} \right) \cr & = Rs.\,50 \cr} $$

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

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

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

$$\eqalign{ & {\text{Let}}\,{\text{T}}{\text{.D}}{\text{.}}\,{\text{be}}\,{\text{Re}}{\text{.}}\,{\text{1}}{\text{}} \cr & {\text{Then,}}\,{\text{B}}{\text{.D}}. = Rs.\,\frac{{11}}{{10}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,1.10 \cr & \therefore {\text{Sum}} = Rs.\,\left( {\frac{{1.10 \times 1}}{{1.10 - 1}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {\frac{{110}}{{10}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,11 \cr & \therefore {\text{S}}{\text{.I}}{\text{.}}\,{\text{on}}\,{\text{Rs}}{\text{.}}\,{\text{11}}\,{\text{for}}\,{\text{2}}\,{\text{years}}\,{\text{is}}\,{\text{Rs}}{\text{.}}\,{\text{1}}{\text{.10}} \cr & \therefore {\text{Rate}} = \left( {\frac{{100 \times 1.10}}{{11 \times 2}}} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 5\% \, \cr} $$

$$\eqalign{ & B.G. = S.I.\,on\,T.D. \cr & = {\text{Rs}}{\text{.}}\,\left( {120 \times 15 \times \frac{1}{2} \times \frac{1}{{100}}} \right) \cr & = {\text{Rs}}{\text{.}}\,9 \cr & B.D. - T.D. = Rs.\,9 \cr & B.D. = {\text{Rs}}{\text{.}}\,\left( {120 + 9} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,129 \cr} $$

$$\eqalign{ & T.D. = \sqrt {P.W.} \times B.G. \cr & \therefore B.G. = \frac{{{{\left( {T.D.} \right)}^2}}}{{P.W.}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,\frac{{110 \times 110}}{{1100}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,11 \cr & \therefore B.D. = T.D. + B.G. \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,110 + 11 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,121 \cr} $$

S.I. on Rs. 1800 = T.D. on Rs. 1872 P.W. of Rs. 1872 is Rs. 1800 Rs. 72 is S.I. on Rs. 1800 at 12% $$\eqalign{ & {\text{Time}} = \frac{{100 \times 72}}{{12 \times 1800}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{3}\,{\text{years}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4\,{\text{months}} \cr} $$

F = Rs. 2160 TD = Rs. 360 PW = F - TD = 2160 - 360 = Rs. 1800 True Discount is the Simple Interest on the present value for unexpired time ⇒ Simple Interest on Rs. 1800 for unexpired time = Rs. 360 Banker's Discount is the Simple Interest on the face value of the bill for unexpired time = Simple Interest on Rs. 2160 for unexpired time $$\eqalign{ & = \frac{{360}}{{1800}} \times 2160 \cr & = \frac{1}{5} \times 2160 \cr & = {\text{Rs}}{\text{.}}\,432 \cr} $$

$$\eqalign{ & {\text{Sum}} = \frac{{B.D. \times T.D.}}{{B.D. - T.D.}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{120 \times 110}}{{120 - 110}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 1320 \cr} $$

$$\eqalign{ & {\text{Sum}} = \frac{{B.D. \times T.D.}}{{B.D. - T.D.}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{B.D. \times T.D.}}{{B.G.}} \cr & \frac{{T.D.}}{{B.G.}} = \frac{{{\text{Sum}}}}{{B.D.}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{1650}}{{165}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 10 \cr} $$ Thus, if B.G. is Rs. 1, T.D. = Rs. 10 If B.D. is Rs. 11, T.D. = Rs. 10 If B.D. is Rs. 165, $$\eqalign{ & T.D. = {\text{Rs}}{\text{.}}\,\frac{{10}}{{11}} \times 165 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,150 \cr} $$ And, B.G. = Rs. (165 - 150) = Rs. 15

Face value of the bill = Rs. 6000 Date on which the bill was drawn = July 14 at 5 months Nominally due date = December 14 Legally due date = December 17 Date on which the bill was discounted = October 5 Unexpired time : Oct. Nov. Dec. 26 + 30 + 17 = 73 days = $$\frac{1}{5}$$ Years B.D. = S.I. on Rs. 6000 for $$\frac{1}{5}$$ year $$\eqalign{ & = {\text{Rs}}{\text{.}}\,\left( {6000 \times 10 \times \frac{1}{5} \times \frac{1}{{100}}} \right) \cr & = {\text{Rs}}{\text{.}}\,120 \cr & {\text{T}}{\text{.D}}{\text{.}} = {\text{Rs}}{\text{.}}\,\left[ {\frac{{6000 \times 10 \times \frac{1}{5}}}{{100 + \left( {10 \times \frac{1}{5}} \right)}}} \right] \cr & = {\text{Rs}}{\text{.}}\,\frac{{12000}}{{102}} \cr & = {\text{Rs}}{\text{.}}\,117.64 \cr} $$ B.G. = B.D. - T.D. = Rs. 120 - 117.64 = Rs. 2.36 Money received by the holder of the bill = Rs. 6000 - 120 = Rs. 5880

$$\eqalign{ & {\text{Amount}}\,{\text{Due}} \cr & = \frac{{B.D. \times T.D.}}{{B.D. - T.D.}} \cr & = \frac{{80 \times 64}}{{80 - 64}} \cr & = {\text{Rs}}{\text{.}}\,320 \cr} $$

$$\eqalign{ & {\text{Banker's}}\,{\text{Gain}} \cr & = \frac{{{{\left( {T.D.} \right)}^2}}}{{P.W.}} \cr & = \frac{{{{\left( {120} \right)}^2}}}{{2400}} \cr & = \frac{{14400}}{{2400}} \cr & = {\text{Rs}}{\text{.}}\,6 \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{Banker's}}\,{\text{Discount}} \cr & = {\text{Rs}}{\text{.}}\,10 \cr & {\text{Then,}}\,{\text{Banker's}}\,{\text{Gain}} \cr & = {\text{Rs}}{\text{.}}\,\frac{1}{5} \times 10 \cr & = {\text{Rs}}{\text{.}}\,2 \cr & T.D. = \left( {B.D. - B.G.} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {10 - 2} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,8 \cr & {\text{Sum}} = \frac{{10 \times 8}}{{10 - 8}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{80}}{2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,40 \cr} $$ S.I. on Rs. 40 for 2.5 year is Rs. 10 $$\eqalign{ & {\text{Therefore,}} \cr & {\text{Rate}} = \frac{{100 \times 10}}{{40 \times 2.5}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 10\% \cr} $$

$$\eqalign{ & T.D. = \frac{{B.G. \times 100}}{{R \times T}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{10 \times 100}}{{10 \times 2}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,50 \cr} $$

Present Worth (P.W.) = 840 - 105 = Rs. 735 Therefore, S.I. on Rs. 735 = Rs. 105 S.I. on Rs. 8400 = $$\frac{{105 \times 840}}{{735}}$$ = Rs. 120

Let amount of the bill = Rs. 100. Money deducted = Rs. 10 Money received by the holder of the bill = Rs. (110 - 10) = Rs. 90 ∴ S.I. on Rs. 90 for 10 months = Rs. 10 $$\eqalign{ & \therefore {\text{Rate}} = \left( {\frac{{100 \times 10}}{{90 \times \frac{{10}}{{12}}}}} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 13\frac{1}{3}\,\% \cr} $$

$$\eqalign{ & F = \frac{{BD \times TD}}{{BD - TD}} \cr & \,\,\,\,\,\,\,\, = \frac{{200 \times 100}}{{200 - 100}} \cr & \,\,\,\,\,\,\,\, = \frac{{200 \times 100}}{{100}} \cr & \,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,200 \cr} $$

$$\eqalign{ & TD = \frac{{BG \times 100}}{{TR}} \cr & \,\,\,\,\,\,\,\,\,\,\, = \frac{{20 \times 100}}{{10}} \cr & \,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,200 \cr} $$

$$\eqalign{ & {\text{T}} = 3\,{\text{year}} \cr & {\text{R}} = {\text{10}}\% \cr & {\text{TD = }}\frac{{{\text{BG}} \times 100}}{{{\text{TR}}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = \frac{{36 \times 100}}{{3 \times 10}} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = 12 \times 10 \cr & \,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}120 \cr & {\text{TD = }}\frac{{{\text{PW}} \times {\text{TR}}}}{{100}} \cr & \Rightarrow 120 = \frac{{{\text{PW}} \times 3 \times 10}}{{100}} \cr & \Rightarrow 1200 = {\text{PW}} \times 3 \cr & {\text{PW}} = \frac{{1200}}{3} \cr & \,\,\,\,\,\,\,\,\,\,{\text{ = Rs}}{\text{.}}400 \cr} $$

$$\eqalign{ & BD = {\text{Rs}}.100 \cr & TD = {\text{Rs}}{\text{.}}\,80 \cr & R = 10\% \cr & F{\text{ = }}\frac{{BD \times TD}}{{(BD - TD)}} \cr & \,\,\,\,\,\,\, = \frac{{100 \times 80}}{{(100 - 80)}} \cr & \,\,\,\,\,\,\, = \frac{{100 \times 80}}{{20}} \cr & \,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,400 \cr} $$ BD = Simple Interest on the face value of the bill for unexpired time = $$\frac{{FTR}}{{100}}$$ $$\eqalign{ & \Rightarrow 100 = \frac{{400 \times T \times 10}}{{100}} \cr & \Rightarrow 100 = 4 \times T \times 10 \cr & \Rightarrow 10 = 4 \times T \cr & \Rightarrow T = \frac{{10}}{4} = 2.5{\text{years}} \cr} $$

Let the amount = Rs. 100 Then BD = Rs.10 (∵ banker's discount, BD is the simple Interest on the face value of the bill for unexpired time and bill is discounted at 10% per annum) Proceeds = Rs. 100 - Rs. 10 = Rs. 90 Hence we should get Rs. 10 as the interest of Rs. 90 for 1 year so that nothing will be lost $$\eqalign{ & \Rightarrow 10 = \frac{{90 \times 1 \times R}}{{100}} \cr & \Rightarrow R = \frac{{10 \times 100}}{{90}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{100}}{9} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 11\frac{1}{9}\,\% \cr} $$

F = Rs. 498 TD = Rs. 18 PW = F - TD = 498 - 18 = Rs. 480 R = 5% $$\eqalign{ & TD = \frac{{PW \times TR}}{{100}} \cr & \Rightarrow 18 = \frac{{480 \times T \times 5}}{{100}} \cr & \Rightarrow 18 = 24 \times T{\text{ }} \cr & \Rightarrow {\text{ }}T{\text{ }} = \frac{{18}}{{24}} = \frac{3}{4}{\text{ years}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{12 \times 3}}{4}{\text{ months}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = 9 months}} \cr} $$

$$\eqalign{ & F = \frac{{BD \times TD}}{{(BD - TD)}} \cr & \,\,\,\,\,\,\, = \frac{{36 \times 30}}{{(36 - 30)}} \cr & \,\,\,\,\,\,\, = \frac{{36 \times 30}}{6} \cr & \,\,\,\,\,\,\, = 36 \times 5 \cr & \,\,\,\,\,\,\, = {\text{ Rs}}{\text{. }}180 \cr} $$

BG = Rs. 360 T = 3 years R = 12% $$\eqalign{ & TD = \frac{{BG \times 100}}{{TR}} \cr & \,\,\,\,\,\,\,\,\,\,\, = \frac{{360 \times 100}}{{3 \times 12}} \cr & \,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}1000 \cr} $$ BG = BD - TD ⇒ BD = BG + TD = 360 + 1000 = Rs. 1360

T = 4 years R = 5% Banker's Gain, BG = Rs. 200 $$\eqalign{ & TD = \frac{{BG \times 100}}{{TR}} \cr & \,\,\,\,\,\,\,\,\,\, = \frac{{200 \times 100}}{{4 \times 5}} \cr & \,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,1000 \cr & \cr & TD = \sqrt {PW \times BG} \cr & 1000 = \sqrt {PW \times 200} \cr & 1000000 = PW \times 200 \cr & PW = \frac{{1000000}}{{200}} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}5000 \cr} $$

$$\eqalign{ & BD\,{\text{for}}\,3\,{\text{years}} = {\text{Rs}}{\text{.}}\,1116 \cr & BD\,{\text{for}}\,4\,{\text{years}} = \frac{{1116}}{3} \times 4 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}1488 \cr & TD\,{\text{for}}\,4\,{\text{years}} = {\text{Rs}}{\text{.}}\,1200 \cr & F = \frac{{BD \times TD}}{{BD - TD}} \cr & \,\,\,\,\,\,\, = \frac{{1488 \times 1200}}{{1488 - 1200}} \cr & \,\,\,\,\,\,\, = \frac{{1488 \times 1200}}{{288}} \cr & \,\,\,\,\,\,\, = \frac{{124 \times 1200}}{{24}} \cr & \,\,\,\,\,\,\, = \frac{{124 \times 100}}{2} \cr & \,\,\,\,\,\,\, = 62 \times 100 \cr & \,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,6200 \cr} $$ ⇒ Rs. 1488 is the simple interest on Rs. 6200 for 4 years $$\eqalign{ & \Rightarrow 1488 = \frac{{6200 \times 4 \times R}}{{100}} \cr & \Rightarrow R = \frac{{1488 \times 100}}{{6200 \times 4}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{372 \times 100}}{{6200}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{372}}{{62}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 6\% \cr} $$

$$\eqalign{ & TD = \sqrt {PW \times BG} \cr & \,\,\,\,\,\,\,\,\,\, = \sqrt {5760 \times 10} \cr & \,\,\,\,\,\,\,\,\,\, = \sqrt {57600} \cr & \,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}240 \cr} $$

$$\eqalign{ & T = 6{\text{ months}} = \frac{1}{2}{\text{year}} \cr & R = 6\% \cr & TD = \frac{{BD \times 100}}{{100 + TR}} \cr & \,\,\,\,\,\,\,\,\,\,\, = \frac{{18.54 \times 100}}{{100 + \left( {\frac{1}{2} \times 6} \right)}} \cr & \,\,\,\,\,\,\,\,\,\,\, = \frac{{18.54 \times 100}}{{103}} \cr & \,\,\,\,\,\,\,\,\,\,\, = \frac{{1854}}{{103}} \cr & \,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,18 \cr} $$

$$\eqalign{ & {\text{Sum}} = \frac{{B.D. \times T.D.}}{{B.D. - T.D.}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,\frac{{120 \times 110}}{{120 - 110}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,1320 \cr} $$ Since B.D. is S.I. on sum due, so S.I. on Rs. 1320 for 8 months is Rs. 120. $$\eqalign{ & {\text{Rate}} = \frac{{100 \times 120}}{{1320 \times \frac{2}{3}}}\,\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{150}}{{11}}\,\% \cr} $$

$$\eqalign{ & BG = \frac{{{{(TD)}^2}}}{{PW}} = \frac{{{{20}^2}}}{{400}} = {\text{Rs}}{\text{. }}1 \cr & BG = BD - TD \cr & \Rightarrow 1 = BD - 20 \cr & \Rightarrow BD = 1 + 21 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,21 \cr} $$