The bankers discount and the true discount of a sum at 10% per annum simple interest for the same time are Rs. 100 and Rs. 80 respectively. What is the sum and the time?

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