$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?

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,

$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

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

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

Pond owes Lane

$876.37, due April 5, 1852. 579.48, due May 3, 1852. 487.83, due June 11, 1852.

And that Lane owes Pond $228.13, due April 28, 1852. 347.16, due June 3, 1852.

313.27, due July 28, 1852.

839.42, due Sept. 1, 1852.

$1727.98 amt. due Pond.

$361.08 = balance due Lane.

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, unless it includes February of leap year, when it is reckoned as 366 days.

3.77

$876.37 from April 5 to June 11, 67 da., = $ 9.79
579.48 from May 3 to June 11, 39 da., =
487.83, due June 11,

313.37 from June 11 to July 28, 47 da.,
839.42 from June 11 to Sept. 1, 82 da.,

= 0.00
- 2.45

=

11.47

$27.48

=

$1.41

= 1.67

$145,38 from June 11 to Aug. 8, 58 da.,
228,13 from April 28 to June 11, 44 da.,
347,16 from June 3 to June 11, 8 da., = .46

Sum of losses,
Excess of gain over loss,.

$ 3.54

23.94

As Mr. Pond gains this interest on money which he owes, he ought to pay the debt ($361.08, the balance of the account) as many days before June 11, 1852, as it will take for it to gain $23.94 interest. This gives for the equated time 398 days before June 11, 1852, which is May 10, 1851. The sum necessary to settle the account after the equated time will be the amount of the balance, $361.08, from the equated time to the time of settlement.

3. When was the balance of the following account due?

4. When was the balance of the following account due ? George Black in account with John Brown. Cr.

Dr.

NOTE.

4 mo., 3 mo., &c., means that goods were sold at so many months' credit.

5. What was due on the following account Jan. 1, 1853?

6. What was due on the following account Jan. 1, 1853, interest being 7 per cent, and 4 months' credit being allowed on each entry?

Dr. David H. Daniels in account with George W. Dean. Cr.

195. To find the Principal, or Interest, from the Amount, Rate, and Time.

(a.) When the amount, time, and rate are given to find the principal or interest, we find what part any principal, or (if the interest be required) its interest for the given time, at the given rate, is of its amount, and then take this part of the given amount.

(b.) The first step towards this is to find the fraction expressing what part any interest for the given time, at the

given rate, is of its principal. This fraction will always be the same part of the given annual rate that the given time is of 1 year, or 360 days; or, if the rate is 6 per cent, it will equal the fraction expressing the part which the given time is of 200 months, or 6000 days. The amount, of course, will equal the principal, plus the fractional part of it which the interest equals.

=

200

(c.) Thus, interest for 1 yr. 7 mo., or 19 months, at 6 per cent per year, = = 2% of the principal, and the amount = 208 +20% = 218 of the principal. Hence, zoo of the principal = zł of the amount, and the entire principal 19, and the interest 29, of the amount for 19 mo. at 6 per cent. (The same fractions would have been obtained by considering the interest to be 2 of 18 of the principal.) (d.) Again. Interest for 19 months at 41 per cent per year of the principal, and the amount=

4

of = 100

57

12 of 200

=

800 8571

=

12

800

800

+5 of the principal. Hence, the principal = 889, and the 800 = 857

interest, of the amount for 19 mo. at 42 per cent.

(e.) Again. Interest for 2 yr. 3 mo. 2 da., or 812 days, at 6 per cent

812 6000=

1703
500

203

of the principal, and the amount =

1383 of the principal. Hence, the principal

the interest 203

=

, of the amount for 2 yr. 3 mo. 2 da. at 6 per cent. (The same fractions would have been obtained by considering the interest to be of 100 of the principal.)

(f) Again. Interest for 5 mo. 14 da., or 164 days, at 7 per cent per year, = 364 of 150 = ‡d of 150

1. What principal on interest at 6 per cent per year will amount to $884.125 in 1 yr. 2 mo. 10 da. ?

Solution.

430

- Since, at 6 per cent per year, interest for 1 day = 6000 of the principal, interest for 1 yr. 2 mo. 10 da., or 430 days, must equal , or, of the principal, and the amount must equal 888 + , or 3, of the principal. Hence, do of the principal must equal 6, and the principal itself must equal

43

of the amount.

600

613 of