Q.The true discount on a bill is ₹90 and the banker's gain is ₹10. Find

From the true discount and banker's gain, the present value comes from PV = TD²/BG, and the rest follows by addition.

The present value is ₹810, the face value ₹900, and the banker's discount ₹100.

Given $TD = 90$ and $BG = 10$.

1. Present value — from $BG = \dfrac{TD^{2}}{PV}$, rearranged:

   $PV = \dfrac{TD^{2}}{BG} = \dfrac{90^{2}}{10} = \dfrac{8100}{10} = 810.$

2. Face value:

   $F = PV + TD = 810 + 90 = 900.$

3. Banker's discount:

   $BD = TD + BG = 90 + 10 = 100.$

   Cross-check (independent). Verify the banker's discount a second way: $BD = \dfrac{F \times TD}{PV} = \dfrac{900 \times 90}{810} = \dfrac{81000}{810} = 100$, which agrees.
   ✓Final answer

   (i) ₹810; (ii) ₹900; (iii) ₹100.

Writing $PV = BG/TD^2$ (formula upside down), or forgetting that the face value is $PV + TD$, not $PV + BG$.