What is Average Due Date?
If a party owes a number of payments to another, payable on different dates, it may like to pay a single lump sum payment in lieu of a number of payments such that there is no loss or gain to any party. Thus, if a party, say A, owes another party, say B, a number of payments due on different dates, we have to find a date such that a payment on that date is equivalent to the above payments. The procedure of finding such a date is called equation of payments and the date found is called average due date.
The ideas of simple average and arithmetic weighted average come to our rescue. If there are items, their simple average is given by:
and the weighted average, with respective weights is given by:
Also note that, the average lies between the smallest and the greatest quantities.
Now let us assume that there are payments due to amounts . Fix an arbitrary date called zero date or base date. Let be the number of days between the zero date and the dates of making payments respectively. Let be the number of days between zero date and average due date, so that a single payment of is equivalent to the above payments. Then, using the idea of weighted means,
The value of D (in number of days) is called equated time or equated period; the date found by adding number of days to zero date will give average due date.
Average due date falls between the earliest date of payment and the latest date of payment, so it is useful to take the earliest date of payment as zero date, so that we deal only with positive quantities .
The formula works only if rates of interests for all n payments are the same otherwise those different rates of interest will also creep into the formula. We will not deal with such a situation here.
Remark:
Any fractional number of days (in latex D $) have to be ignored, thus favouring the creditor, as the payment is being made a bit early.
Illustrative Examples:
Example 1: Find the average due date for the following payments:
Rs 1000 due on 1^{st} January 2007
Rs 700 due on 15^{th} February 2007
Rs 500 due on 15^{th} March 2007
Rs 1000 due on 1^{st} April 2007
Mention zero date and equated date.
Solution:
Let the earliest date i.e. 1^{st} January 2007 be the zero date.
Amount(Rs) | Due Date | Number of days from zero date i.e. 1^{st} Jan. ’07 | |
1000 | 1^{st} Jan. | 0 | 0 |
700 | 15^{th} Feb. | 45 | 31500 |
500 | 15^{th} March | 73 | 36500 |
1000 | 1^{st} April | 90 | 90000 |
\sumPi=3200 |
Ignoring the fraction, D=49 days
Here zero date is 1^{st} January 2007, equated time is D=49 days.
average due date is 49 days from 1^{st} January i.e. average due date is 19^{th} February 2007.
Example 2: Mr. Sharma has accepted the following bills drawn by his creditor Mr. Gupta on different dates. Mr. Sharma approaches his creditor to cancel them all and allow him to accept a single bill for the payment of his entire liability on the average due date. Calculate the amount and the date on which Mr. Sharma is required to pay this amount. Take the days of grace into account.
Bill No. | Date of drawing | Date of acceptance | Amount of the bill(Rs) | Tenure |
1 | 16-2-07 | 20-2-07 | 8000 | 50 days after sight |
2 | 6-3-07 | 7-3-07 | 6000 | 2 months |
3 | 24-5-07 | 31-5-07 | 2000 | 4 months |
4 | 1-6-07 | 4-6-07 | 9000 | 30 days |
Solution:
First we calculate the due dates of the bills:
Bill No. | Date of drawing | Date of acceptance | Tenure | Nominal due date | Legally due date |
1 | 16-2-07 | 20-2-07 | 50 days after sight | 11-4-07 | 14-4-07 |
2 | 6-3-07 | 7-3-07 | 2 months | 6-5-07 | 9-5-07 |
3 | 24-5-07 | 31-5-07 | 4 months | 24-9-07 | 27-9-07 |
4 | 1-6-07 | 4-6-07 | 30 days | 1-7-07 | 4-7-07 |
Taking the earliest due date i.e. 14-4-07 as zero date, we construct the following table:
Bill No. | Due date | Amount(Rs)Pi | Number of days from zero date i.e. 14-4-07 | Product |
1 | 14-4-07 | 8000 | 0 | 0 |
2 | 9-5-07 | 6000 | 16+9=25 | 1500000 |
3 | 27-9-07 | 2000 | 16+31+30+31+31+27=166 | 332000 |
4 | 4-7-07 | 9000 | 16+31+30+4=81 | 729000 |
Equated time
Ignoring the fraction, we get Equated time=48 days
average due date
=Zero date+ Equated time
=14-4-07+48 days
=1-6-07
Hence, in lieu of the four bills, Mr. Sharma is required to make a single payment of Rs 25000 on 1^{st} June 2007.
Example 3: A small T.V. is available either for full cash payment of Rs 5000 or for cash down payment of Rs 1000 and five more monthly installments of Rs 1000 each. Find the rate of interest.
Solution:
Let the date of purchase of T.V. be the zero date. If the person buys on installment basis he pays six months of Rs 1000 after 0, 1, 2, 3, 4, 5 months. This is equivalent to paying Rs 6000 after a period D from date of purchase, where
months
Thus an amount of Rs 5000 is equivalent to Rs 6000 payable after 2.5 months. Let r% be rate of interest per annum.
Then,
Thus, the interest is 96% per annum.
Exercise:
1) Find the average due date for the following payments:
Rs 3000 due on 15^{th} March 2007
Rs 6000 due on 20^{th} March 2007
Rs 4000 due on 5^{th} May 2007
Rs 2000 due on 15^{th} May 2007
Clearly mention zero date and equated time.
2) Arko has drawn the following bills on Aishik:
Bill No. | Date of bill | Period of bill | Amount(Rs) |
1 | 15-1-07 | 3 months | 35000 |
2 | 17-2-07 | 2 months | 57500 |
3 | 13-3-07 | 1 month | 7500 |
4 | 19-3-07 | 2 months | 50000 |
Aishik wants to finish his liability by paying a lump sum amount on average due date in lieu of these four bills. Calculate the amount and the date on which Aishik is required to make this payment.
3) Calculate the average due date of the following bills:
Date of acceptance | Amount in Rs | Period |
12 June | 1000 | 2 months |
20 July | 2000 | 2 months |
10 August | 3000 | 3 months |
26 September | 4000 | 3 months |
4) A refrigerator is available for Rs 12000 cash down payment or for cash down payment of Rs 5000 and two more installments of Rs 4000 each payable every six months. Find the simple rate of interest.
5) The average due date of four bills was 8^{th} February 2008. Of these, the first three bills for Rs 4500, Rs 8500 and Rs 7000 were due respectively on 15^{th} January, 4^{th} February and 14^{th} February 2008. The fourth bill was due on 4^{th} March 2008. Find the amount of the fourth bill.
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