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AVERAGE, OR EQUATION OF PAYMENTS.

308. Equation of Payments is the process of finding an average time for the equitable payment of several sums due at different times.

309. The Equated Time is the date at which all the items may be paid without loss to either party.

310. The Term of Credit is the time from the contracting of a debt to the date of its becoming due.

311. TO FIND THE EQUATED TIME, WHEN THE TERMS OF CREDIT BEGIN AT THE SAME DATE.

ILL. Ex. I owe P. Benjamin two notes dated March 1, one for $80, to be paid in 12 months, the other for $40, to be paid in 3 months. When, without loss to either Benjamin or myself, can I pay both notes at once?

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I am entitled to keep these two notes till their interest equals $5.40; if I pay them both at once, it should be at such time after Mar. 1 as will be required for $120 to gain $5.40; $120 gains 60 cts. a month; .. to gain $5.40, it will require as many

months as 60 cts. is contained times in $5.409 months, which added to Mar. 1, is Dec. 1. Hence

RULE I. To find the equated time when all the terms of credit begin at the same date: Find the interest on each item for its time of credit; divide the sum of the interests by the interest of the sum of the items for one month. The quotient will be the equated time in months.

Add the equated time to the date.

NOTE I.

To obtain the interest for 1 month, remove the decimal point two places to the left, and divide by 2.~

NOTE II. If any item contains cents, reject them if they are less than 50, and increase the dollars by one if they equal or exceed 50.

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RULE II. Multiply each payment by the number of days or months to elapse before it becomes due; divide the sum of the prod ucts by the sum of the payments, and add the quotient to the date.

NOTE.-The examples in this book are performed by the Interest

method.

EXAMPLES.

1. What is the equated time for paying $50 due in 5 m. from May 14, 1863, $35 due in 4 m., and $25 due in 2 m. from the same date? Ans. Sept. 14, 1863.

2. B. Frank holds five notes against me, dated June 7, 1864; one for $500 on 4 m., one for $750 on 5 m., one for $200 on 12 m., one for $400 on 2 m., and one for $400 on 17 months' credit; what is the time at which all should be paid if paid in one sum ? Ans. Jan. 7, 1865.

3. Having sold Samuel Bond real estate to the amount of $2000, he gave me four equal notes for it, dated Oct. 4, and payable in 5, 6, 9, and 12 months; what is the average time for the payment of all the notes? Ans. 8 m.

4. What is the average time for paying $20 due in 20 days, $20 due in 100 days, $70 due in 30 days, $20 due in 60 days, and $40 in 70 days? Ans. 1 m. 20 d.

5. April 1, C. A. Brown purchased coal to $5000, † of which was to be paid in 6 months,

the amount of in 12 months,

and the remainder in 9 months; for what time should a note without interest, dated April 1, 1865, in payment of all the sums, be allowed to run, and when should the note be paid?

Ans. Jan. 28, 1866.

6. A owes B $360.25 (Note 2, Art. 311), of which is to be paid in 7 months, of the remainder in 8 months, of what then remains in 10 months, and the balance in 4 months; in how many months and days should the whole be paid? Ans. 6 m. 22 d.

7. Sept. 25, bought 3 parcels of goods, as follows: a bill amounting to $225.25 on 12 months' credit, a bill amounting to $125.25 on 8 months' credit, and a bill amounting to $40 on 5 months' credit; what was the mean time for paying all?

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Ans. 10 m.

NOTE. When a sum is paid immediately, the term of credit expires instantly, and it will have no corresponding interest or product in time.

8. A person promised to pay $7000, of it immediately, of the remainder in 8 months, of what then remained in 22 months, and the balance in 16 months; what is the equated time for paying the whole? Ans. 12 m.

9. A merchant tailor finds, on examining his account with Jones & Co., May 5, that he owes them for 150 yds. of silk, at $.50 per yd., which is due that day; for 2339 yds. of cambric, at $.10, which will be due in 6 days; for 12 yds. broadcloth, at $5.00 per yd., which will be due in 16 days; for 50 yds. of doeskin at $3.75 per yd., which will be due in 20 days; what is the average time for paying the whole? If the tailor settles the account by giving his note, when should the note begin to bear interest?

312. TO FIND THE EQUATED TIME, WHEN THE TERMS OF CREDIT BEGIN WITH DIFFERENT DATES.

ILL. EX. J. Rives bought of A. Ainger the following bills of goods: -

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Sept. 15, a bill amounting to $100,

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" $250,

What is the equated time for paying the whole?

To equate the above bills, it is necessary to assume a date from which to compute the interest on the several items; any date may be assumed, but the most convenient date for examples generally, on account of reckoning the time, will be found to be the last day of the month before the earliest date at which any item becomes due; this in the above example is Aug. 31.

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RULE. To find the equated time when the terms of credit begin with different dates: Assume that the time for paying all the items is on the last day of the month previous to the earliest day at which any item is due; find the interest on each item from the assumed date to the date when it is due, and divide the sum of the interests by of To of the sum of the items; the quotient will be the time after the assumed date, in months, when all should be paid.

NOTE I.

.1 month 3 d. ; .03 month = 1 day nearly, etc. NOTE II. Reject the fraction of a day when it is less than ; reckon it 1 day when it is or more.

1. Find the equitable time for the payment of the following: $300, due April 1, 1869; $450, due Dec. 1, 1869; and $600, due March 10, 1870. (Assumed date, March 31, 1869.)

Ans. Nov. 25, 1869

2. Find the mean or average time for paying the following: $12.45, due Feb. 10, 1860; $24.17, due Mar. 1, 1860; $15, due Mar. 14, 1860; $30, due Mar. 16, 1860; and $12.70, due Mar. 7, 1860. Ans. Mar. 5, 1860.

Edwin Foote's ledger contains the following account.

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NOTE. This account shows that Wing bought of Foote merchandise at the times and to the amount indicated.

3. Allowing interest on each item from its date, what is the time from which a note should draw interest in payment of all of the above items? Ans. May 9, 1862. 4. What is the equated time for paying the following bill?

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The above items will be due as follows: April 20, May 15, and

June 12. Equate from these dates.

5. Equate the following:

Assumed date, Mar. 31.

MR. J. STOW

1864.

Ans. May 16, 1860.

ROXBURY, Jan. 1, 1865.

Bo't of Z. CHURCHILL,

$400.

600.

200.

Ans. Aug. 8.

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