Q.State, with reason, whether the following is true or false: "For any bill discounted before maturity, the banker's discount is always greater than the true discount."

Compare what each discount is charged on — the face value versus the smaller present value.

The two discounts differ only in the sum they are charged on, for the same rate $r$ and time $t$:

• Banker's discount is simple interest on the face value: $BD = \dfrac{F\,rt}{100}$.
• True discount is simple interest on the present value: $TD = \dfrac{PV\,rt}{100}$.

Since a bill discounted before maturity always has $PV < F$ (the present value is the face value minus the true discount), and interest increases with the principal, it follows that

$BD - TD = \dfrac{(F - PV)\,rt}{100} = \dfrac{TD \cdot rt}{100} = BG > 0,$ …

Assuming the two discounts are equal (they are only in the no-time / no-rate trivial case), or reasoning that the true discount is larger …