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in 7 mo., and the rest in 11 mo. without gain or loss?

When can I

pay

the whole

4. I owe $960, payable as follows: $180 in 4 mo. 20 da., $348 in 6 mo. 15 da., $234 in 8 mo. 5 da., and the rest in 10 mo. 13 da. Required the equated time of payment.

5. A trader bought $1800 worth of goods, agreeing to pay of the money down, & of it in 5 mo., of it in 6 mo., of it in 9 mo., and the rest in 12 mo. At what time may the

whole be paid?

6. Bought a lot of goods, for which I agreed to pay $437.75 in 3 mo., $394.25 in 6 mo., and $628.19 in 8 mo When may the whole be paid without gain or loss?

7. A owes B $800, payable in 10 mo. ; but to accommodate B, he pays $250 down. When ought the remainder to be paid?

$250

$550,

Solution. After paying $250, he will owe $800 which he ought to keep till its interest shall equal the interest of $800 for 10 months. But the interest of $800 for 10 mo., equals the interest of one dollar for 800 times 10 mo., or 8000 mo. equals the interest of $550 for of 8000 mo., which is 14 mo. Hence it ought to be paid in 14 mo.

6

8. I owe $1000, payable in 9 mo.; but to accommodate my creditor, I pay $300 down, and agree to pay $300 more in 2 mo. How long ought I, in justice, to keep the remainder? 9. I owe $600, payable in 8 mo. 15 da., and $400, payable in 12 mo.; but afterwards agree to pay $400 down, and $300 in 2 mo. 20 da., on condition that I may keep the remainder enough longer to compensate for my loss. When will the remainder become due?

10. A owes B $480, due in 1 yr., and B owes A $720, due in 1 yr. 6 mo. If A should pay his debt at once, when

ought B to pay his?

193. To find Date of Equated Time.

(a.) The best method of solving such examples as the following is to see how much interest will be gained or lost by paying the sum of the debts at any assumed time.

(b.) It will be well as a general thing, to select for the assumed time a date on which one of the debts becomes due, as by that means we shall avoid the necessity of reckoning interest on that debt. Reference should also be had to the probable equated time.

(c.) The time is reckoned by counting the days between the dates considered, as in the English method of computing interest.

1. James Brown owes William Greene the following debts, viz. $534.83, due Jan. 7, 1855; $285, due April 4, 1855; $327.38, due July 3, 1855; and $438,75, due Aug. 17, 1855. When may the whole be paid without gain or loss?

Solution. Suppose that April 4, 1855, be selected as the assumed time. Then Mr. Brown would gain interest on

$534.83 from Jan. 7 to April 4, 88 da.
$285.00 due at assumed time,

=

$7.84

$0.00

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Showing that Mr. Brown is entitled to keep $1585.96, the entire debt due, as many days after April 4 as it will take it to gain $6.94 interest. This, found by 191, is 26 days, plus a fraction less than .

Therefore the equated time is 26 days after April 4, which is April 30.

NOTE. The above shows that on April 4th Mr. Brown could justly have settled the account by paying $1585.96 - $6.94 = $1579.02.

Again. Suppose that July 3 be selected as the assumed time. Then Mr. Brown would gain interest on

$534.83 from Jan. 7 to July 3, 178 da., = $15.86
$285.00 from April 4 to July 3, 90 da., =
$327.38 due at assumed time,

Giving for sum of gains,

4.27

00.00

$20.13

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Showing that Mr. Brown ought to pay $1585.96, the entire debt due, as many days before July 3 as it will take it to gain $16.84 interest. This, found as before, is 64 days nearly. Therefore the equated time is 64 days before July 3, which is April 30, as before.

=

NOTE. The above shows that if the account should not be settled till July 3, Mr. Brown ought justly to pay $1585.96 + $16.84 $1602.80.

Proof. By paying the debt on April 30, Mr. Brown will gain in

terest on

=

$534.83 from Jan. 7 to April 30, 114 da.: $10.16
$285.00 from April 4 to April 30, 26 da.: = 1.23

Making sum of gains

=

$11.39

He will lose interest on

$327.38 from April 30 to July 3, 64 da. :
$438.75 from April 30 to Aug. 17, 109 da.= $7.97

= $3.49

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which, being less than the interest of $1585.96 for a half day, shows that April 30 is the correct equated time.

2. I owe $387.53, due Nov. 7, 1851; $467.81, due Dec. 21, 1851; $256.19, due Feb. 11, 1852; $136.43, due March 1, 1852; and $387.59, due May 3, 1852. What is the equated time of payment?

3. I owe $2867, due April 15, 1850; $1642, due July 27, 1850; $4371, due Oct. 8, 1850; and $5940, due Jan. 1, 1851. What is the equated time of payment?

4. I owe $628.13, due Dec. 17, 1852; $427.19, due Dec. 23, 1852; $371.16, due Dec. 30, 1852; $587.83, due Jan. 3, 1853; $987.62, due Jan. 7, 1853; and $843.28, due Jan. 14, 1853. What is the equated time of payment?

How much is due on the above Jan. 1, 1853 ?

5. I owe $543.28, due April 24, 1855; $723.13, due May 11, 1855; $484, due Sept. 3, 1855; $426.18, due Oct. 10, 1855; $236, due Nov. 10, 1855. What is due on the above Sept. 1, 1855, interest being reckoned at 5 per cent ?

6. What is the equated time for paying the following debts: $600, due March 7, 1850; $400, due June 11, 1850;

$800, due Aug. 17, 1850; $500, due Oct. 3, 1850; and $1000, due Nov. 27, 1850?

194. Equation of Accounts.

(a.) The method of finding the equated time when each party owes the other, that is, when there are entries on both the debit and credit side of an account, does not differ in principle from that in which there are entries only on one side. The following example and solution will illustrate it:

1. The account books of A and B show that

A owes B

$426.70, due Jan. 6, 1855. $413.65, due Feb. 2, 1855.

$169.28, due April 13, 1855. $328.57, due Aug. 29, 1855.

And that B owes A
$148.37, due Dec. 22, 1854.
$173.19, due Jan. 29, 1855.
$587,23, due May 7, 1855.
$658.45, due Sept. 30, 1855.

When ought the balance to be paid?

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Solution. Suppose that April 13, 1855, be the assumed time of payment. Then A will gain interest on each of his debts which becomes due to B before that time, and on each of B's debts which become due to him after that time; for he will have the use of each for a longer time than he is justly entitled to. He will lose interest on each of his debts which becomes due to B after that time, and on each of B's debts which becomes due to him before that time; for he will not have the use of them for so long a time as he is justly entitled to. Hence A will gain the interest of

$426.70 from Jan. 6 to April 13, 97 da.,
$413.65 from Feb. 2 to April 13, 70 da.,
$169.28 from Feb. 13,

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$587.23 from April 13 to May 7, 24 da.,

$658.45 from April 13 to Sept. 30, 170 da.,

= $ 6.90

$ 4.83

$ 0.00

= $ 2.35

= $18.65

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$148.37 from Dec. 22, 1854, to April 13, 1855, 112 da., $173.19 from Jan. 29, to April 13, 74 da.,

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=

$2.14

Sum of losses,

$12.46

Excess of A's gain over his loss, or of B's loss over his gain, $20.27

But the sum of A's debts is $1338.20, and of B's is $1567.24. $1567.24 $1338.20= $229.04, the balance which B owes A.

The question now resolves itself into this: If by B's paying A $229.04 April 13, 1855, A gains and B loses $20.27 interest, when can he pay it without any gain or loss of interest? The answer evidently is, As many days after April 13, 1852, as it will take $229.04, or, disregarding the cents, $229, to gain $20.27 interest. This, found by methods before explained, is 531 days 1 yr.* 166 da., and shows the equated time to be Sept. 26, 1853, which may be proved as were the former examples.

=

In all such Thus, if the

NOTE. Although accounts like the above are sometimes settled by notes payable at the equated time, they are more frequently settled by notes payable at some more convenient time, or by cash. cases, allowance is made for the interest gained or lost. above account should be settled by cash April 13, 1852, $20.27 would be deducted from the balance due from B to A, in order to compensate B for the interest he would lose; that is, B would pay A $229.04 $20.27 = $208.77. If it should be paid May 1, 1852, B would have to pay A $.69 (the interest of the balance due A from April 13 to May 1) more than if he had paid it April 13; or, which is the same thing, he would have to pay the balance $229.04, minus its interest $19.58, from May 1, 1852, to the equated time. If the balance due at any given time had been originally required, it should have been found directly by making the given time the "assumed time."

2. By the respective accounts of Henry Lane and William Pond, it appears that

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When can this balance be paid without gain or loss to either party?

Solution. Suppose it to be paid June 11, 1852. Then will Mr Pond gain the interest of

* Reckoning the year as 365 days, as is always done in such cases, un less it includes February of leap year, when it is reckoned as 366 days.

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