ML Aggarwal Class 12 Maths Solutions Section C Chapter 01 Discount

Section C - Discount

 

1.1 Bills of Exchange

In modern business, many transactions occur on a credit basis. The timeframe for repayment varies by industry—for instance, computer parts wholesalers may offer one month, while iron bars suppliers might allow three months. Naturally, the supplier (creditor) wants written confirmation from the buyer (debtor) that payment will be made within an agreed period (credit period). Bills of exchange serve this purpose. When written in a local language such as Hindi, Tamil, or Bengali, they are called hundis. Promissory notes function similarly. The Indian Negotiable Instruments Act, 1881 governs all such instruments.

According to the Indian Negotiable Instruments Act, 1881:

"A Bill of Exchange is an instrument in writing, an unconditional order signed by the maker directing to pay a certain sum of money only to or to the order of a certain person or to the bearer of the instrument".

For example, if M/s ABC sell computers valued at Rs. 2 lacs on 1st August, 2013 to M/s XYZ, with payment due after three months, the bill of exchange would appear as follows:

In this example, the bill is written by the seller (creditor), referred to as the payee, to the buyer (debtor), called the drawee. The drawee must write "Accepted" on the bill and sign beneath it. This legally binds the debtor to pay. Without the drawee's acceptance, the bill holds no legal force.

Sometimes a third party, not the payee, prepares the bill—the drawer differs from the payee. Below is an example of such a bill:

Features of Bills of Exchange

• It must be in writing and include a date.
• The drawer (maker of the bill) must sign it.
• The drawee (person making payment) must accept it.
• It should carry proper stamping or be drafted on stamped court paper.
• It must be an order (not a request) to pay.
• The bill amount must be specified and fixed.
• The payment date must be clearly stated.
• Three parties may be involved - the drawer (bill maker), the drawee (payer), and the payee (receiver).
• The money stated on the bill is called the face value or maturity value, which equals principal plus interest.

Advantages of Bills of Exchange

• They help in buying and selling goods on credit, and also protect currency notes from damage.
• Being legal documents, they can be enforced in court if dishonoured after the due date.
• This creates payment certainty. With a fixed payment date, both debtor and creditor can budget their cash flows.
• They offer a convenient way for businesses to make or receive payments in international trade.
• Bills can be transferred from one person to another; the payee may endorse it to someone else by signing the back. The endorser is the person making the endorsement, and the endorsee is the person receiving it.
• A discounting facility exists. If the payee needs cash before due date, a bank will discount the bill (give cash now minus a discount charge).

Promissory Notes

A bill of exchange is an order from the payee or drawer to the drawee (debtor) to make payment. Cheques, bank drafts, and hundis are all examples of bills of exchange. A cheque is a bill drawn on a bank, payable on demand. A bank draft is one bank's cheque on another bank for an immediate payment. Hundis are bills of exchange written in local language.

A promissory note, by contrast, is a written instrument containing an unconditional promise signed by the maker to pay a specific sum to, or to the order of, a specific person.

Note that "I owe you 2 lacs" is not a promissory note—it only acknowledges the debt but does not promise payment. The promise must be unconditional. Thus, "I will pay 2 lacs as soon as possible" is also not a promissory note. Below is an example of a promissory note:

Main Differences Between Bill of Exchange and Promissory Note

 

1.2 Maturity; Due or Nominal Date; Legally Due Date

After a bill is drawn (created), it is sent to the debtor for acceptance. The date of drawing and the date of acceptance may be different. Two types of bills of exchange exist:

(i) Bill of exchange after date - the maturity date is counted from the date the bill is drawn.

(ii) Bill of exchange after sight - the maturity date is counted from the date the bill is accepted.

When calculated, the date on which a bill becomes due is called the due date or nominal date. Adding three days (called days of grace) to this gives the date on which the bill becomes legally due.

For example, if a bill for Rs. 2 lacs is drawn on 1st September, 2013, payable after three months, the nominal date is 1st December, 2013, while the legal due date is 4th December, 2013.

To calculate the due date, these points are important:

(i) If a bill falls due on a date that does not exist in that month, the last day of the month is used. For example, if a bill is drawn on 30th January for one month, it becomes due on 28th February and legally due on 3rd March.

(ii) If the bill becomes legally due on a Sunday or a public holiday listed in the official gazette, the due date is moved one day earlier. For example, if a bill becomes legally due on 26th January (Republic Day), it is considered legally due on 25th January.

(iii) If the date when the bill is legally due is declared an emergency holiday (such as following the death of a national leader), the bill may be paid one day later.

 

1.3 Payment of Bill of Exchange

Payment can be handled in several ways, and dishonour can be dealt with as follows:

1. The payee may hold the bill until maturity and then collect payment from the drawee.

2. The bill holder may endorse the bill to another person by signing the back. The endorsee can again endorse it to another person, and this chain may continue. Whoever holds the bill at maturity will receive the payment.

3. The bill holder can have the bill discounted at a bank. If cash is needed before maturity, the holder takes the bill to the bank, which deducts a discount (interest on the face value for the remaining time) and pays the remainder. On maturity, the bank collects from the drawee.

4. Instead of discounting, the bill may be sent to a bank with instructions to keep it until maturity, then collect from the acceptor. After receiving payment, the bank credits the account and deducts a small fee called commission.

5. The debtor may wish to pay early. This is called retiring the bill. The holder may offer some rebate for receiving the payment before maturity.

6. The debtor may be unable to pay by due date and may ask the creditor to cancel the old bill and draw a new one. This is called renewal of bill. Typically, the creditor charges interest, which may be paid as cash or added to the new bill amount.

7. When the acceptor refuses to pay on maturity, the bill is dishonoured. The holder can recover from the acceptor or any endorser, but must notify them of dishonour within a reasonable time. The drawee may claim the bill was never shown for payment. To address this, the bill is given to a Notary Public (court-appointed official), who presents it again. If dishonoured, the Notary makes a formal note on the bill called a noting. This noting serves as legal proof the bill was presented and dishonoured. The Notary charges a noting fee, paid by the holder initially but ultimately recovered from the acceptor (as the acceptor caused the dishonour).

When a discounted bill is dishonoured, the bank initially collects noting charges from the person who discounted it, but these charges are eventually recovered from the acceptor.

8. The acceptor may become insolvent and unable to pay. In such cases, partial recovery (such as 60 paise per rupee) may be possible.

 

1.4 Banker's Discount and True Discount

We will first study Present Value or Present Worth, which leads to understanding true discount.

Present Value (P.V.) or Present Worth

Discount is given for quick payment because money grows through interest. Receiving Rs. 10000 now is better than receiving Rs. 10000 three months later, as money can earn interest during that time. So if X owes Y Rs. 10000 due in three months, paying a smaller amount now would be reasonable.

The present value or present worth, say P, of a given sum, say A, due at the end of a given period is the amount of money that, together with interest, will total A at the end of that period.

Let n be the number of interest periods and i be the interest per rupee per interest period. The simple interest for this period equals P.n.i.

Now, Amount = Present value + Interest

\[ A = P + Pni = P(1 + ni) \]
\[ P = \frac{A}{1 + ni} \]

This formula gives the present value P of an amount A due n periods ahead at interest rate i per rupee per interest period. With four variables, if three are known, the fourth can be found.

Consider the example where person X buys goods for Rs. 10000 from Y, with payment due in 3 months. If the interest rate is 12% yearly, the present value P is:

\[ P = \frac{A}{1 + ni} = \frac{\text{Rs. } 10000}{1 + \frac{1}{4} \times \frac{12}{100}} = \frac{\text{Rs. } 10000}{1.03} = \text{Rs. } 9708.74 \]

Instead of paying Rs. 10000 today, X would pay Rs. 9708.74, paying Rs. 291.26 less. This amount is called true discount or rate discount or equitable discount. True discount is interest on the present value, not on the amount due.

Thus:

\[ \text{T.D.} = \text{Interest on present value} = A - P \]

Now:

\[ P = \frac{A}{1 + ni} \implies \text{T.D.} = A - P = A - \frac{A}{1 + ni} = \frac{A \cdot ni}{1 + ni} = Pni \]
\[ \text{T.D.} = Pni = \frac{A \cdot ni}{1 + ni} \]

Banker's Discount

As explained earlier, if the bill holder needs cash before maturity, a bank will discount the bill, deducting Banker's Discount (B.D.), which is interest on the face value for the remaining time. The bank pays the remaining amount, called proceeds or discounted value. On maturity, the bank collects from the drawee.

If n is the number of remaining interest periods, i is the interest per rupee per interest period, and A is the face value, then:

\[ \text{Banker's Discount (B.D.)} = Ani \]
\[ \text{Discounted value} = A - \text{B.D.} \]

In the earlier example, with A = Rs. 10000, n = 3 months = 1/4 year, and i = 12% yearly = 0.12, if the bill is discounted right away (3 months before maturity):

\[ \text{Banker's Discount (B.D.)} = Ani = 10000 \times \frac{1}{4} \times 0.12 = \text{Rs. } 300 \]

The bank will pay Rs. 10000 - Rs. 300 = Rs. 9700 to the holder.

Now:

\[ \text{T.D.} = \frac{A \cdot ni}{1 + ni} \implies \text{T.D.} = \frac{\text{B.D.}}{1 + ni} \]
\[ (1 + ni) \text{T.D.} = \text{B.D.} \]
\[ \text{B.D.} = (1 + ni) \text{T.D.} \]

Banker's Gain

Strictly speaking, the bank should have paid the Present Value (Rs. 9708.74) rather than Rs. 9700. In other words, it should have deducted true discount (Rs. 291.26) from the amount (Rs. 10000) rather than banker's discount (Rs. 300). The difference between what the bank should pay and what it actually pays is Banker's Gain (B.G.):

\[ \text{Banker's Gain (B.G.)} = \text{Present worth} - \text{Amount paid} \]
\[ = (\text{Amount due} - \text{T.D.}) - (\text{Amount due} - \text{B.D.}) \]
\[ = \text{B.D.} - \text{T.D.} \]

This also equals:

\[ = \text{Interest on sum due} - \text{Interest on present worth} \]
\[ = \text{Interest on (sum due} - \text{present worth)} \]
\[ = \text{Interest on True Discount} \]

Therefore:

\[ \text{B.G.} = \text{B.D.} - \text{T.D.} = \text{Interest on T.D.} \]

Now, interest on T.D. = T.D. × ni, but T.D. = \(\frac{A \cdot ni}{1 + ni}\)

\[ \therefore \text{B.G.} = \frac{A \cdot ni}{1 + ni} \times ni = \frac{A(ni)^2}{1 + ni} \]
\[ \text{B.G.} = \frac{A(ni)^2}{1 + ni} \]

In some problems, this formula is useful:

\[ \text{Sum due (Face value), } A = \frac{\text{B.D.} \times \text{T.D.}}{\text{B.D.} - \text{T.D.}} \]

This is easy to derive:

\[ \frac{\text{B.D.} \times \text{T.D.}}{\text{B.D.} - \text{T.D.}} = \frac{Ani \times \frac{A \cdot ni}{1 + ni}}{Ani - \frac{A \cdot ni}{1 + ni}} = \frac{A \cdot ni (1 + ni)}{(A \cdot ni)(1 + ni - 1)} = \frac{A \cdot ni (1 + ni)}{A \cdot ni \cdot ni} = A \]

 

Example 1. Calculate the nominal date and legally due date for bills drawn on the following dates:

(i) 1st January for 3 months
(ii) 12 July for 1 month
(iii) 12 July for 30 days
(iv) 30 December 2006 for 2 months
(v) 30 December 2007 for 2 months.

Answer:

(i) The nominal date is 1st April and the legal due date is 4th April.

(ii) The nominal date is 12th August. As 15th August is a national holiday, the legal due date is 14th August.

(iii) The nominal date is 11th August (30 days after 12th July), and the legal due date is 14th August.

(iv) The nominal date is 28th February 2007 and the legal due date is 3rd March 2007.

(v) The nominal date is 29th February 2008 (since 2008 is a leap year) and the legal due date is 3rd March 2008.

Exam Tip: Always add 3 days of grace to find the legally due date from the nominal date. When a date doesn't exist (like 30th February), use the last day of that month.

 

Example 2. A bill of Rs. 6000 drawn on 1st January for 3 months is discounted at 6% per annum. Find the amount of discount and the discounted value.

Answer: Discount = Interest on amount due
= Rs. 6000 × 0.06 × \(\frac{3 \text{ months}}{12 \text{ months}}\) = Rs. 90.
Discounted value = Rs. 6000 - Rs. 90 = Rs. 5910.

Exam Tip: Banker's Discount is always calculated on the face value (not the present worth), which is why it differs from true discount.

 

Example 3. A bill of Rs. 10000 drawn on 15th April for 3 months is discounted on 6th May at 8% per annum. Find the banker's discount. How much would the holder of the bill receive?

Answer: Amount of bill = Rs. 10000
Rate of interest = 8% p.a. = 0.08
Bill is drawn on 15th April, so legally due date is 18th July.

As bill is discounted on 6th May, remaining period = 25 days of May + 30 days of June + 18 days of July = 73 days = \(\frac{73}{365}\) years = \(\frac{1}{5}\) year


\[ \therefore \text{Banker's discount} = Ani = 10000 \times 0.08 \times \frac{1}{5} = \text{Rs. } 160 \]

The holder will receive Rs. 10000 - Rs. 160 = Rs. 9840.

Exam Tip: Count the exact number of days remaining carefully. Always include the maturity date but not the discounting date in the count.

 

Example 4. A sum of Rs. 5000 is due 6 months hence at 12% simple interest per annum. Find the present value and the true discount.

Answer: Amount due A = Rs. 5000
Interest = 12% p.a. = 0.12
Number of interest periods, n = 0.5 (as time is 6 months = 1/2 year)


\[ \therefore P = \frac{A}{1 + ni} = \frac{\text{Rs. } 5000}{1 + (0.12)(0.5)} = \frac{\text{Rs. } 5000}{1.06} = \text{Rs. } 4717 \text{ (approx.)} \]

\[ \therefore \text{T.D.} = A - P = \text{Rs. } 5000 - \text{Rs. } 4717 = \text{Rs. } 283 \]

Exam Tip: True discount represents the interest earned on the present value over the time period—it's what you "gain" by paying early.

 

Example 5. X owes Y Rs. 4400 one year hence at 10% simple interest p.a. Y owes X Rs. 5832 two years hence at 8% compound interest. If they want to settle their dues now, who should pay how much amount to whom?

Answer: Here we should compare the present values of the two amounts.

P.V. of Rs. 4400 due one year hence at 10% simple interest

\[ = \frac{A}{1 + ni} = \frac{\text{Rs. } 4400}{1 + 1 \times 0.1} = \frac{\text{Rs. } 4400}{1.1} = \text{Rs. } 4000 \]

P.V. of Rs. 5832 two years hence at 8% compound interest

\[ = \frac{A}{(1 + i)^n} = \frac{\text{Rs. } 5832}{(1 + 0.08)^2} = \frac{\text{Rs. } 5832}{1.1664} = \text{Rs. } 5000 \]

Thus, X owes Y Rs. 4000 today and Y owes X Rs. 5000 today.
Hence, to settle their dues, Y should pay Rs. 1000 to X today.

Exam Tip: When comparing obligations on different dates or under different interest conditions, always convert to present value first for a fair comparison.

 

Example 6. The true discount on a bill 9 months hence at 6% per annum is Rs. 360. Find the amount of the bill and its present worth.

Answer: We are given that interest rate i = 6% = 0.06,
As time = 9 months = 3/4 year, number of interest periods, n = 0.75,
T.D. = Rs. 360.
We want to calculate A and P.

Now T.D. = Pni

\[ \therefore P = \frac{\text{T.D.}}{ni} = \frac{\text{Rs. } 360}{(0.75)(0.06)} = \frac{\text{Rs. } 360}{0.045} = \text{Rs. } 8000 \]

\[ \therefore \text{Amount due, } A = P + \text{T.D.} = (\text{Rs. } 8000 + \text{Rs. } 360) = \text{Rs. } 8360 \]

Exam Tip: Use T.D. = Pni to find present worth when T.D. is given, then add T.D. to get the total amount due.

 

Example 7. (i) The true discount on Rs. 2080 due after a certain period of time at 8% is Rs. 80. Find the time after which it is due. (ii) The true discount on a bill 15 months hence at 8% simple interest is Rs. 100. Find the amount of the bill. (iii) If the true discount on Rs. 4360 due 18 months hence at simple interest p.a. is Rs. 360, find the rate of interest.

Answer:

(i) Amount due, A = Rs. 2080, T.D. = Rs. 80, i = 8% = 0.08.
Now, present value, P = A - T.D. = Rs. 2080 - Rs. 80 = Rs. 2000.
As T.D. = interest on P = Pni, we get
Rs. 80 = Rs. 2000 × n × (0.08)

\[ \Rightarrow n = \frac{\text{Rs. } 80}{(\text{Rs. } 2000)(0.08)} = \frac{80}{160} = \frac{1}{2} \text{ year} = 6 \text{ months} \]

(ii) Here, T.D. = Rs. 100, n = 15 months = 5/4 years, i = 8% = 0.08.
Using T.D. = \(\frac{A \cdot ni}{1 + ni}\), we get

\[ A = \frac{\text{T.D.}(1 + ni)}{ni} = \frac{\text{Rs. } 100 \left[1 + \frac{5}{4}(0.08)\right]}{\frac{5}{4}(0.08)} = \frac{\text{Rs. } 100(1.1)}{0.1} = \text{Rs. } 1100 \]

(iii) A = Rs. 4360, T.D. = Rs. 360

\[ \therefore \text{Present value, } P = A - \text{T.D.} = \text{Rs. } 4360 - \text{Rs. } 360 = \text{Rs. } 4000 \]

Here, n = 18 months = 3/2 years = 1.5 years.
Now T.D. = Interest on present value = Pni

\[ \therefore \text{Rs. } 360 = \text{Rs. } 4000 \times 1.5 \times i \]

\[ \Rightarrow i = \frac{\text{Rs. } 360}{\text{Rs. } 4000 \times 1.5} = \frac{360}{6000} = 0.06 \text{ i.e. } 6\% \text{ per annum} \]

Exam Tip: These problems test your ability to rearrange formulas. Master T.D. = Pni and T.D. = \(\frac{A \cdot ni}{1 + ni}\) to handle any of the three unknowns (A, P, or time).

 

Example 8. Rajesh discounts a bill of Rs. 6000 from a bank 2 months before due date and gets Rs. 5900 from the bank. What is the rate of interest? What is true discount? What is banker's gain?

Answer: Banker's discount = Face value - amount received = Rs. 6000 - Rs. 5900 = Rs. 100.

Here, n = 2 months = 1/6 year.
Using the formula, B.D. = Ani, we get
Rs. 100 = Rs. 6000 × (1/6) × i

\[ \Rightarrow \text{Rs. } 100 = \text{Rs. } 1000i \]

\[ \Rightarrow i = \frac{1}{10} = 0.1 \text{ i.e. } 10\% \]

Using the formula, B.D. = (1 + ni) T.D., we get

\[ \text{Rs. } 100 = \left(1 + \frac{1}{6} \times 0.1\right) \text{T.D.} \]

\[ \Rightarrow \text{T.D.} = \text{Rs. } 100 \times \frac{6}{6.1} = \frac{\text{Rs. } 600}{6.1} = \text{Rs. } 98.36 \]

Banker's gain, B.G. = B.D. - T.D. = Rs. 100 - Rs. 98.36 = Rs. 1.64.

Exam Tip: The relationship B.D. = (1 + ni) T.D. is key for finding true discount when banker's discount is known.

 

Example 9. The present worth of a bill due sometime hence is Rs. 1100 while the true discount on the bill is Rs. 110. Find the amount of bill, banker's discount, banker's gain and discounted amount of bill.

Answer: Present worth = Rs. 1100, true discount = Rs. 110.
Amount of bill = Present worth + true discount = Rs. 1210.

Using the formula, T.D. = Pni, we get
Rs. 110 = Rs. 1100 × ni

\[ \Rightarrow ni = \frac{1}{10} = 0.1 \quad \ldots (i) \]

Using the formula, B.D. = Ani, we get
B.D. = Rs. 1210 × 0.1 (using (i))

\[ = \text{Rs. } 121 \]

Banker's gain, B.G. = B.D. - T.D. = Rs. 121 - Rs. 110 = Rs. 11.

Discounted value = Amount of bill - Banker's discount = Rs. 1210 - Rs. 121 = Rs. 1089.

Exam Tip: When both present worth and true discount are given, you already have the key piece (ni). Use it to quickly find banker's discount without recalculating the rate or time separately.

 

Exercise 1

 

Question 1. Explain the following terms:
(i) Present worth (ii) True discount (iii) Banker's discount (iv) Banker's gain (v) Noting (vi) Noting charges (vii) Endorsing a bill (viii) Renewing a bill (ix) Discounting a bill (x) Retiring a bill (xi) Dishonour of a bill.

Answer:

(i) Present Worth: This is the amount of money in today's value that is equivalent to a sum payable at a future date. In other words, it is what you would pay now instead of paying more later, such that both amounts are financially equal when interest is considered.

(ii) True Discount: This is the reduction given on an amount payable in the future to get its present worth. It equals the difference between the amount due and its present value. It represents the interest earned on the present worth over the time remaining.

(iii) Banker's Discount: This is the interest calculated by the bank on the face value (full amount) of a bill for the time remaining until maturity. When a bill holder wants cash before the due date, the bank uses this method to calculate how much to deduct.

(iv) Banker's Gain: This is the extra amount the bank makes by using banker's discount instead of true discount. It equals the difference between banker's discount and true discount. It represents the interest that the bank earns on the true discount itself.

(v) Noting: This is a formal process where an official (Notary Public) appointed by the court presents a dishonoured bill again and officially records on the bill itself that it was presented and refused. This creates legal proof of dishonour.

(vi) Noting Charges: These are fees charged by the Notary Public for the noting process. Although the bill holder initially pays these charges, they are later recovered from the person who dishonoured the bill, as that person is responsible for the dishonour.

(vii) Endorsing a Bill: This is the process of transferring a bill from one person to another. The current holder signs the back of the bill to pass it to another person (called the endorsee). The endorsee can again pass it to someone else, creating a chain of ownership.

(viii) Renewing a Bill: When the debtor cannot pay by the due date, the creditor and debtor may agree to cancel the original bill and create a new one for a future date. Typically, the creditor adds interest to the new bill amount.

(ix) Discounting a Bill: This is when the bill holder takes a bill to a bank before its maturity date to get immediate cash. The bank deducts a discount (interest for the remaining time) and pays the rest to the holder immediately.

(x) Retiring a Bill: This is when the debtor chooses to pay the bill before the due date. The creditor may offer a small rebate (discount) as a reward for early payment.

(xi) Dishonour of a Bill: This occurs when the acceptor of the bill refuses to pay the amount on the due date. When this happens, the bill holder can legally recover the amount from the acceptor or any person who previously endorsed the bill.

Exam Tip: These 11 terms are fundamental to understanding bills of exchange. Group them: (i)-(iv) relate to discounting calculations, (v)-(vi) relate to dishonour procedures, and (vii)-(xi) relate to bill operations and outcomes.

 

Question 2. Calculate the nominal date and legally due date for bills drawn on the following dates:
(i) 1st April for 6 months
(ii) 23 December for a month
(iii) 30 December for 2 months (assuming that next year is not a leap year)
(iv) 4th July for 2 months
(v) 4th July for 60 days.

Answer:

(i) For a bill drawn on 1st April for 6 months - nominal date is 1st October, legal due date is 4th October.

(ii) For a bill drawn on 23rd December for 1 month - nominal date is 23rd January, legal due date is 25th January (since 24th January is not a holiday but we add grace days).

(iii) For a bill drawn on 30th December for 2 months (next year is not a leap year) - nominal date is 28th February (last day of February as 30th does not exist), legal due date is 3rd March.

(iv) For a bill drawn on 4th July for 2 months - nominal date is 4th September, legal due date is 7th September.

(v) For a bill drawn on 4th July for 60 days - counting 60 days from 4th July gives 2nd September (July has 27 days left after 4th, August has 31 days, so 27 + 31 = 58, need 2 more days in September), nominal date is 2nd September, legal due date is 5th September.

Exam Tip: Count days carefully. For months, use their actual number of days. Always add 3 days of grace after the nominal date to get the legal due date. When a due date falls on a holiday, move it back one day.

 

Question 3. A bill for 3 months is drawn on 1st July and accepted on 3rd July. Calculate nominal date and due date if the bill is (i) after date (ii) after sight.

Answer:

(i) For a bill after date, the maturity is counted from the drawing date (1st July). Adding 3 months gives nominal date as 1st October, and adding 3 days of grace gives legal due date as 4th October.

(ii) For a bill after sight, the maturity is counted from the acceptance date (3rd July). Adding 3 months gives nominal date as 3rd October, and adding 3 days of grace gives legal due date as 6th October.

Exam Tip: The key difference between "after date" and "after sight" bills is the starting point for counting - use the drawing date for "after date" and the acceptance date for "after sight".

 

Question 4. (i) A bill of Rs. 10000 drawn on 4th April for 4 months is discounted at 6% per annum. Find the amount of discount and the discounted value of the bill. If the bill is discounted on 7th May, find the banker's discount. (ii) Find the Banker's discount and the discounted value of a bill worth Rs. 600 drawn on May 15, 2005 for 3 months and discounted on July 20, 2005 at 5% per annum.

Answer:

(i) For the bill of Rs. 10000 drawn on 4th April for 4 months:

The nominal date is 4th August, and the legal due date is 7th August.

When discounted immediately (on the drawing date), remaining time is 4 months = 1/3 year.
Discount = 10000 × 0.06 × (1/3) = Rs. 200
Discounted value = 10000 - 200 = Rs. 9800

When discounted on 7th May, we need to count days from 7th May to 7th August:
May has 24 days remaining, June has 30 days, July has 31 days = 85 days = 85/365 years
Banker's discount = 10000 × 0.06 × (85/365) = Rs. 151.23 (approx.)

(ii) For a bill of Rs. 600 drawn on May 15, 2005 for 3 months and discounted on July 20, 2005 at 5% per annum:

Nominal due date is August 15, legal due date is August 18, 2005.

Days from July 20 to August 18 = 11 days of July + 18 days of August = 29 days = 29/365 years
Banker's discount = 600 × 0.05 × (29/365) = Rs. 2.38 (approx.)
Discounted value = 600 - 2.38 = Rs. 597.62

Exam Tip: When discounting immediately after drawing, the full time period applies. When discounting later, carefully count remaining days up to the legal due date. Include the due date but not the discounting date in your count.

 

Question 5. A bill of Rs. 60000 drawn on May 27 at 6 months is discounted on August 8 at 6% p.a. How much does the banker charge and what does the holder receive?

Answer: The bill drawn on May 27 for 6 months has nominal due date of November 27 and legal due date of November 30, 2005.

Counting from August 8 to November 30: August has 23 days remaining, September has 30 days, October has 31 days, November has 30 days = 114 days = 114/365 years.

Banker's discount = 60000 × 0.06 × (114/365) = Rs. 1124.38 (approx.)
Amount holder receives = 60000 - 1124.38 = Rs. 58875.62

Exam Tip: The banker's charge is the discount. Always subtract it from the face value to find what the holder actually receives.

 

Chapter Test

 

Question 1. Fill in the blanks:

(i) Bill of exchange is an order to __________, while promissory note is a promise to __________.
(ii) Bill discount is interest on __________.
(iii) True discount is interest on __________.
(iv) Banker's gain is interest on __________.
(v) Bill of exchange may be after date or __________.
(vi) The person endorsing a bill is called __________ while to whom it is endorsed is called __________.
(vii) The person who prepares the bill is called __________, the person who has to make the payment is called __________, and the person who has to receive the payment is called __________.
(viii) Hundis are bills of exchange written in __________.
(ix) A cheque is a bill of exchange drawn on a __________, payable on __________.
(x) A bill of exchange is signed by __________ while a promissory note is signed by __________.

Answer:

(i) Bill of exchange is an order to make payment, while promissory note is a promise to pay.

(ii) Bill discount is interest on sum due (face value).

(iii) True discount is interest on present worth.

(iv) Banker's gain is interest on true discount.

(v) Bill of exchange may be after date or after sight.

(vi) The person endorsing a bill is called endorser while to whom it is endorsed is called endorsee.

(vii) The person who prepares the bill is called drawer, the person who has to make the payment is called drawee, and the person who has to receive the payment is called payee.

(viii) Hundis are bills of exchange written in vernacular.

(ix) A cheque is a bill of exchange drawn on a banker, payable on demand.

(x) A bill of exchange is signed by drawer or payee while a promissory note is signed by drawee.

Exam Tip: These fill-in-the-blanks test your understanding of definitions. Know the three parties (drawer, drawee, payee), the two bill types (after date, after sight), and the three key discount/interest concepts (B.D. on face value, T.D. on present worth, B.G. on T.D.).

 

Question 2. X draws a bill on Y for Rs. 3000 on 1st March, 2007 payable after 3 months. The bill is discounted by X, as he is in need of money. Compute the discount in each of the following cases: (i) The bill is discounted for Rs. 2840. (ii) The bill is discounted @ 8% per annum.

Answer:

(i) When the bill is discounted for Rs. 2840:
Discount = Face value - Amount received = Rs. 3000 - Rs. 2840 = Rs. 160

(ii) When the bill is discounted at 8% per annum:
Bill drawn on 1st March for 3 months has nominal due date 1st June and legal due date 4th June.
If discounted immediately, remaining time = 3 months = 1/4 year
Discount = 3000 × 0.08 × (1/4) = Rs. 60

Exam Tip: When discount amount is given directly, simply subtract from face value. When a rate is given, use B.D. = Ani to calculate the discount amount.

 

Question 3. 'A' received acceptance from 'B' for Rs. 26000 on 1st March, 2007 at 4 months. 'A' got the acceptance discounted at 6% per annum at his bank after 1 month. How much was received by 'A' from the bank after discounting the acceptance?

Answer: Acceptance dated 1st March for 4 months has nominal due date 1st July and legal due date 4th July.

'A' discounts after 1 month, so on 1st April. Remaining time from 1st April to 4th July = 94 days = 94/365 years.

Banker's discount = 26000 × 0.06 × (94/365) = Rs. 389.75 (approx.)

Amount received = 26000 - 389.75 = Rs. 25610.25 (approx.)

Exam Tip: The key is finding the exact remaining time from the discounting date to the legal due date. Always add the 3 days of grace to the nominal date first.

 

Question 4. What is the face value of a bill discounted at 5% p.a. 73 days earlier than the date of maturity, the banker's gain being Rs. 10 only?

Answer: We know that Banker's Gain (B.G.) = \(\frac{A(ni)^2}{1 + ni}\)

Here, n = 73/365 years, i = 0.05, B.G. = 10

ni = (73/365) × 0.05 = 0.01 (approx.)

10 = \(\frac{A(0.01)^2}{1 + 0.01}\) = \(\frac{A(0.0001)}{1.01}\)

10 = \(\frac{0.0001A}{1.01}\)

A = \(\frac{10 \times 1.01}{0.0001}\) = Rs. 101000

Exam Tip: When banker's gain is given with time and rate, use the B.G. formula to find the face value. Always convert the period to years first.

 

Question 5. A man holds a bill for Rs. 12000 which is due for payment after 8 months. After 3 months, however, he sells the bill to a broker who charges 5% p.a. The man then invests the discounted value of the bill in a security whose rate of interest is such that he does not suffer any loss on discounting the bill. Find the rate of interest percent per annum of the security.

Answer: Original bill: Rs. 12000 due in 8 months.

After 3 months, remaining time = 8 - 3 = 5 months = 5/12 years

Banker's discount at 5% = 12000 × 0.05 × (5/12) = Rs. 250
Amount received = 12000 - 250 = Rs. 11750

This Rs. 11750 is invested in a security for 5 months at rate r% p.a. to make up the loss of Rs. 250.

Interest needed = 250

If interest earned = Principal × Rate × Time
250 = 11750 × r × (5/12)

r = \(\frac{250 \times 12}{11750 \times 5}\) = \(\frac{3000}{58750}\) = 0.051 (approx.) = 5.1% p.a.

Or more precisely: r = \(\frac{5 \times 47}{47}\)% = 5\(\frac{5}{47}\)% p.a.

Exam Tip: This problem links discounting with simple interest. The rate needed on the discounted value for the remaining time should compensate exactly for the banker's discount taken.

 

Question 6. The banker's discount and true discount on a certain sum of money due 3 months hence are Rs. 515 and Rs. 500 respectively. Find the sum of money and the rate of interest.

Answer: B.D. = Rs. 515, T.D. = Rs. 500, n = 3 months = 1/4 year

Using the formula: A = \(\frac{\text{B.D.} \times \text{T.D.}}{\text{B.D.} - \text{T.D.}}\)

A = \(\frac{515 \times 500}{515 - 500}\) = \(\frac{257500}{15}\) = Rs. 17166.67 (approx.)

Now, B.D. = Ani
515 = 17166.67 × (1/4) × i

i = \(\frac{515 \times 4}{17166.67}\) = \(\frac{2060}{17166.67}\) = 0.12 = 12% p.a.

Exam Tip: When both B.D. and T.D. are given, the formula A = \(\frac{\text{B.D.} \times \text{T.D.}}{\text{B.D.} - \text{T.D.}}\) is very useful. It immediately gives you the face value without needing to find rate or time first.