Q.The banker's discount and the true discount on a certain sum due at the end of a period are ₹100 and ₹80 respectively. Find

When both discounts are known, the face value follows from F = BD × TD / (BD − TD), and the banker's gain is their difference.

The banker's gain is ₹20 and the face value is ₹400.

Given $BD = 100$ and $TD = 80$.

(ii) Banker's gain:

(i) Face value — using the relation $F = \dfrac{BD \times TD}{BD - TD}$:

Cross-check (independent). Present value $PV = F - TD = 400 - 80 = 320$. Then $BG = \dfrac{TD^{2}}{PV} = \dfrac{80^{2}}{320} = \dfrac{6400}{320} = 20$, matching the banker's gain found above. The figures are internally consistent.

Equivalently, $PV = \dfrac{TD \times BD}{BG}$? No — more simply, $PV = TD^2/BG = 6400/20 = 320$, then $F = PV + TD = 320 + 80 = 400$, giving the same face value.

Using $F = BD \times TD/(BD + TD)$ (wrong sign in the denominator), or taking the face value to be $BD + TD$.