Question
The banker's discount on Rs. 1650 due a certain time hence is Rs. 165. Find the true discount and the banker's gain
Answer: Option A
$$\eqalign{
& {\text{Sum}} = \frac{{B.D. \times T.D.}}{{B.D. - T.D.}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{B.D. \times T.D.}}{{B.G.}} \cr
& \frac{{T.D.}}{{B.G.}} = \frac{{{\text{Sum}}}}{{B.D.}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{1650}}{{165}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 10 \cr} $$
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
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$$\eqalign{
& {\text{Sum}} = \frac{{B.D. \times T.D.}}{{B.D. - T.D.}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{B.D. \times T.D.}}{{B.G.}} \cr
& \frac{{T.D.}}{{B.G.}} = \frac{{{\text{Sum}}}}{{B.D.}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{1650}}{{165}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 10 \cr} $$
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
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