Bankers Discount - Aptitude MCQ Questions and Solutions with Explanations

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

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

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

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

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

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