Q.The banker's discount and the true discount on a certain bill are ₹1,020 and ₹1,000 respectively. Find

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

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

So the difference between face value and present worth is precisely the true discount. (By contrast, the banker's discount is interest computed on the face value itself, and the banker's gain is $\text{BD} - \text{TD}$.)

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 positive difference $\text{BD} - \text{TD}$ is the banker's gain. So the banker's discount is always higher, not lower, than the true discount — the given statement is incorrect.

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:

$\text{B.G.} = \text{B.D.} - \text{T.D.}$

It can also be shown that $\text{B.G.}$ equals the simple interest on the true discount, i.e. $\text{B.G.} = \dfrac{\text{T.D.} \cdot n \cdot r}{100}$.

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:

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

So the amount actually received by the holder after deducting the banker's discount is the cash value of the bill.

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:

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

So the difference between face value and present worth is the true discount. Banker's discount is interest on the full face value, banker's gain is the excess of BD over TD, and cash value relates to the amount actually paid on discounting — none of these equals $\text{FV}-\text{PW}$.

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:

$\text{BD} > \text{TD}$

The positive difference $\text{BD}-\text{TD}$ is called the banker's gain, which is always positive — confirming that BD is always greater than TD.

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.
• The true discount is the interest on the present value, not the sum due itself.

Since the question asks for another name for the sum due — the amount stated on the bill and payable at maturity — this matches the face value.

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) $PV$: $TD = \dfrac{PV \cdot n \cdot r}{100}$.

Because $F > PV$, we always have $BD > TD$; in fact $BD - TD =$ interest on the true discount (the banker's gain). The two are therefore distinct, so calling banker's discount "true discount" is incorrect.

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:

$\text{Legal due date} = \text{Nominal due date} + 3 \text{ days}$

The banker's discount and the period of discount are always calculated up to this legal due date.

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:

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

Comparing the options: Banker's discount is the interest on the face value, and Banker's gain is $\text{BD} - \text{TD}$, so neither matches. "Cash value" is not the standard term here.