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NOTE I. In accounts current the details of each transaction should be drawn from the book of original entry and presented with the several items. To save space, some details are here omitted.

NOTE II. In the settlement of mercantile accounts, interest is calculated or not, according to custom or the agreement of the parties, and 3 days of grace are generally allowed on items of merchandise on credit.

NOTE TO THE TEACHER. - In interest accounts and in average, accountants generally, though not universally, reckon the time in days. To avoid confusing the pupil with various methods, time is here reckoned exclusively in days.

EXAMPLES.

What is the balance due on the following accounts Jan. 1, 1870, reckoning interest at 6% on each item from the time when due?

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July 15. To Mdse. 3 m. 794 82 Oct. 18 Sept. 25. By Mdse. 3 m. 200 00 Dec. 28.

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Aug. 27.
Sept. 25.

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do. 3 m. 408 33 Nov. 30. Nov. 8. Cash,
do. 3 m. 217 90 Dec. 28.

NOTE. To find the time when due, add 3 m.

item of merchandise.

287 00 Nov. 8.

+3 d. to the date of each Ans. Bal. due from E. R. R. Co., $ 943.61.

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483. ILL. Ex. A owes B several sums, as follows: $60 due Jan. 21, 1869, $ 80 due Feb. 2, $100 due Mar. 10. At what date may A pay all these sums at once without loss of interest to either party?

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Explanation. — If all the sums are to be paid at once, it should be at such a time after or before some selected date as will be required for the total, $ 240, to gain as much interest as the several sums would gain between the selected date and the times they severally become due. For convenience, the selected date may be the carliest date at which any sum becomes due, which, in this example, is Jan. 21.

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It will require as many days for $ 240 to gain $.96 as there are .04's in .96, which is 24. A date 24 days after Jan. 21 is Feb. 14, the time when the whole may be paid in one sum.

Ans. Feb. 14.

484. The process of finding the time when the payment of several items, due at different times, may be made at once without loss of interest to either party is average, or equation of payments.

485. The date at which several sums due at different times may be paid at once, as Feb. 14 above, is the average date or equated time of payment.

486. From the operation above may be derived the following

RULE. To find the average date for the payment of several sums due at different dates: 1. Select some convenient date as the earliest date at which any item matures.

2. Compute the interest on each item from the selected date to the date of its maturity.

3. Divide the sum of the interests thus found by the interest of the sum

of the items for 1 day. The quotient will express the number of days from the selected date to the average date of payment.

4. Add this number to the selected date; the result will be the average

date required.

NOTE I. If any item contains a number of cents, disregard them if less than 50, call them $ 1 if 50 or more.

NOTE II.

If a result contains a fraction of a day, disregard it if less than, call it 1 day if or more.

NOTE III.
NOTE IV.

The interest may be reckoned at any rate per cent. Instead of dividing by the interest for 1 day, we may divide by the interest for 1 month (at 6%, of of principal), in which case the time to elapse before the average date will be in months. Though this method is sometimes more convenient, it will not always give an accurate result.

487. PRODUCT METHOD.

The following is the operation of ILL. Ex. Art. 483 by the product method:

Days.

OPERATION.

Products.

0 × 60 = 0

12 X 80 960 48 X 100 = 4800

240) 5760

24 Jan. 2124 Feb. 14.

Explanation. The interest of $80 for 12 days equals the interest of $1 for 960 days; the interest of $100 for 48 days equals the interest of $1 for 4800 days.

Adding the several products, we find that the total of interest of the several sums for their respective times equals the interest of $1 for 5760 days. But the interest of $1 for 5760 days equals the interest of $240 for of 5760 days, or 24 days. Jan. 21+ 24 days— Feb. 14, Ans. Hence the following

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RULE. To find the average date for the payment of several sums due at different dates: 1. Select some convenient date, as the earliest date at which any item matures.

2. Multiply the time each item has to run by the number of units in the item.

3. Divide the sum of the products thus obtained by the number of units in the sum of the items; the quotient will express the time from the selected date to the average date of payment.

4. Add this time to the selected date; the result will be the average date required.

NOTE. The examples in this book are performed by the interest method, which, by the use of Interest Tables, has the advantage of brevity.

488. Proof. - Find the sum of the interests on all items due before the average date, from the date at which they are due to the average date; also find the sum of the interests on all items due after the average date from that date to the date at which they are due. If these sums are equal, the work is correct.

489. EXAMPLES.

1. What is the average date for paying $162 due Oct. 1, $120 due Oct. 10, and $ 150 due Nov. 6? Ans. Oct. 16.

2. I owe three notes as follows: a $300 note, payable March 31; a $ 200 note, payable April 30; a $100 note, payable May 30: what is the equated time for paying all at once? Ans. April 20.

3. A purchased goods August 5, amounting to $ 2100; $ 700 to be paid Nov. 5, $ 700 Dec. 5, and $ 700 Feb. 5; when may the whole be paid without loss to either party? Ans. Dec. 16.

4. I purchased goods of Eben Sutton to the amount of $3000; $800 to be paid Sept. 10, $1250 to be paid Oct. 1, and $ 950 to be paid Oct. 28; what is the average date of payment, and what must be the date of a note payable in 2 months, with grace, that it may become due at the average date of payment? Ans. Oct. 4; Aug. 1. 5. Bought goods of George Peirce & Co. as follows: Jan. 5, 1868, a bill of $ 180.20 on 2 months' credit.*

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What is the average date for paying the whole? Ans. Mar. 15, '68.

6. July 10, 1869, my ledger contains items of account against John Cahill, as follows:

1869. Jan. 25. To Mdse. on 6 mo. credit, $199.84.

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What is the average date for paying the whole? Ans. Aug. 9, '69.

* Add 3 days of grace to times of credit. (Art. 482. Note II.)

7. When shall a note to settle the following account be made paya

ble?

E. E. WHITE

To WICKERSHAM & Co., Dr.

1869. April 10. To Mdse. on 30 days' cr. $100

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8. Find the average date of payment of the following account: —

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490. TO FIND THE AVERAGE TIME FOR THE SETTLEMENT OF AN

ACCOUNT WHEN THERE ARE BOTH DEBIT AND CREDIT ITEMS.

ILLUSTRATIVE EXAMPLE I.

My ledger contains the following account:

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The above is an account of merchandise sold on my account by Henry B. Dyer, commission merchant, with his charges for the sale.

At what date should Dyer pay the balance of the account?

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