Q.The banker's gain on a bill is equal to which of the following?
(A) Banker's discount − True discount
(B) Banker's discount + True discount
(C) True discount − Banker's discount
(D) Face value − Banker's discount

By definition the true discount is the difference between the face value (amount due at maturity) and the present worth, so the correct option is True discount.

For a bill, the present worth (PW) is the amount that, invested now at the given rate, accumulates to the face value $F$ on the due date. The true discount (TD) is defined as the interest on this present worth for the unexpired period, and equivalently as

$\text{TD} = F - \text{PW}$

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Banker's discount is computed on the face value and true discount on the present worth; since face value exceeds present worth, $\text{BD} > \text{TD}$, making the statement False.

For the same bill, rate $r$ and unexpired time $n$:

$\text{Banker's Discount (BD)} = F \cdot n \cdot r \quad(\text{interest on face value } F)$

$\text{True Discount (TD)} = \text{PW} \cdot n \cdot r \quad(\text{interest on present worth PW})$

Because the present worth is always less than the face value ($\text{PW} < F$), it follows that

$\text{BD} > \text{TD}$ …

The gap between banker's discount and true discount is the Banker's Gain, i.e. $\text{B.G.} = \text{B.D.} - \text{T.D.}$

For a bill of face value (sum due) $A$ discounted for $n$ years at rate $r\%$:

• Banker's Discount $\text{B.D.} = \dfrac{A \cdot n \cdot r}{100}$ — simple interest charged on the face value.
• True Discount $\text{T.D.} = \dfrac{A \cdot n \cdot r}{100 + nr}$ — interest reckoned on the present worth.

Because the banker charges interest on the larger amount (the face value) rather than on the present worth, $\text{B.D.} > \text{T.D.}$. The excess is a standard defined term:

…

The amount paid to the holder after deducting the banker's discount from the face value is called the cash value of the bill: $\text{Cash Value} = \text{Face Value} - \text{Banker's Discount}$.

When a bill of exchange is discounted before its due date, the banker charges interest on the face value (FV) for the unexpired period at the given rate; this interest is the banker's discount (BD). The banker then pays the holder the remaining amount, which is called the cash value (CV) of the bill:

$\text{Cash Value} = \text{Face Value} - \text{Banker's Discount}.$ …

True discount (TD) is defined as the interest on the present worth for the unexpired period, and it equals $\text{Face Value}-\text{Present Worth}$. So the difference between face value and present worth is the true discount — option (b).

For a bill of exchange the key quantities are:

• Face Value (FV) — the amount written on the bill, payable on the due date.
• Present Worth (PW) — the amount which, if invested now at the given rate, would amount to the face value by the due date.
• True Discount (TD) — the true interest lost by receiving the money early, defined as $\text{TD}=\text{FV}-\text{PW}$ (equivalently, the interest on the present worth).
• Banker's Discount (BD) — interest charged on the face value.
• Banker's Gain (BG) — $\text{BD}-\text{TD}$.

By definition:

$\text{TD} = \text{FV} - \text{PW}$ …

Banker's discount (BD) is interest on the face value, while true discount (TD) is interest on the present worth. Because the face value is larger than the present worth, $\text{BD}>\text{TD}$ — so the blank is greater.

For a bill of a given face value FV with present worth PW (where $\text{FV}>\text{PW}$), for the same rate and time:

$\text{Banker's Discount} = \text{interest on FV}, \qquad \text{True Discount} = \text{interest on PW}$

Since interest is directly proportional to the principal and $\text{FV}>\text{PW}$, it follows that: …

The sum due is the amount payable on the maturity date of a bill, i.e. the amount printed on its face; hence it is called the Face value — option (A).

In bill-discounting terminology:

• The face value (sum due) is the amount written on the bill, payable in full on the due date.
• The present value / cash value is what the holder actually receives today after deducting discount, so it is smaller than the sum due. …

Banker's discount $=$ interest on the face value, while true discount $=$ interest on the present value; since face value $>$ present value, banker's discount $>$ true discount, so the statement is False.

For a bill discounted before maturity:

• Banker's discount (BD) is the simple interest charged on the face value (sum due) $F$ for the unexpired time: $BD = \dfrac{F \cdot n \cdot r}{100}$.
• True discount (TD) is the interest on the present value (cash value) $PV$: $TD = \dfrac{PV \cdot n \cdot r}{100}$. …

The date up to which the drawee is legally permitted to pay a bill is the legal due date — the nominal due date plus $3$ days of grace.

When a bill of exchange is drawn, it states a period after which it is payable. The end of that stated period is the nominal due date. However, by legal custom the drawee (buyer) is allowed an additional 3 days of grace to make the payment.

Hence the actual date by which the buyer is legally allowed to pay the amount is:

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Face value $-$ present worth $=$ true discount, the interest on the present worth for the unexpired period.

For a bill of exchange, the present worth (PW) is the amount which, if invested now at the given rate for the unexpired time, would grow to the face value (FV) on the due date.

The true discount (TD) is defined as the interest on the present worth for the unexpired period, and it equals the shortfall between face value and present worth:

$\text{TD} = \text{FV} - \text{PW}$.

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