Bankers Discount - Aptitude MCQ Questions and Solutions with Explanations

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.
$$\eqalign{ & {\text{Rate}} = \frac{{100 \times 120}}{{1320 \times \frac{2}{3}}}\,\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{150}}{{11}}\,\% \cr} $$

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