8. If I lend $100 for 4 months, how long am I entitled to the use of $50 in repayment?

ANALYSIS.-The interest of $100 for 4 months is the same as the interest of $1 for 400 months; if I am entitled to the use of $1 for 400 months, I am entitled to the use of $50 for as many months as 50 is contained times in 400=8 months.

9. How long should I lend $100 to pay for the use of $200 for 2 months? For 4 months? For 8 months?

WRITTEN EXERCISES.

1. Samuel Jones bought goods of William Corson, to the amount of $1200, $400 of which is due in 3 months, and $800 in 4 months; what is the average term of credit?

400 × 3 = 1200

800 × 43200

1200

)4400

3 months.

ANALYSIS.-The interest of $400 for 3 months the interest of $1 for 1200 months, and the interest of $800 for 4 months

the interest of $1 for 3200 months; hence the interest on the entire debt = the interest

on $1 for 4400 months, or, the interest on $1200 for as many months as 1200 is contained times in 4400 months=33 months, the average time of credit.

Note. It will be seen in the above example that Jones has the use of $400 for of a month after it becomes due, while Corson, to balance this, has the use of $800 for of a month before it becomes due, that is, the use of twice as much money for half the time; hence the debt can be paid in 3 months, without loss to either party.

RULE.

Multiply each debt by the number denoting its term of credit, and divide the sum of the products by the number denoting the sum of the debts; the quotient will be the average term of credit. The average term of credit, added to the date of the debts, will give the equated time.

2. I buy $300 worth of goods on 3 months' credit, $500 worth on 4 months' credit, and $400 worth on 2 months' credit; what is the average term of credit?

3. What is the average term of credit of $1000 due in 6 months, $1400 in 8 months, and $1500 in 1 year?

4. Find the equated time of three bills of $600 each, bought January 3, 1877, on 2, 3, and 4 months respectively.

5. I owe William Lewis & Co. bills of $200, $600, and $400, for merchandise, bought on 1, 2, and 4 months respectively; when can I pay the entire amount without loss to either party, the goods having been bought May 3d?

6. What is the equated time for the payment of $450 due in 30 days, $400 due in 60 days, and $520 due in 90 days, the date of the bills being March 10th?

CASE II.

To find the equated time when partial payments have been made on a debt.

1. I owe a debt of $800 due in 8 months without interest; I pay $100 at the end of 2 months, and $200 at the end of 4 months; at what time should I pay the balance?

100 × 6= 600
200 × 4= 800

1400

800-300 500

=

=

1400 500 24.

ANALYSIS.-By paying $100 in 2 months, I lose the use of that sum for 6 months, and by paying $200 in 4 months, I lose the use of that sum for 4 months, or, by the two payments I lose the use of $1400 for 1 month; to balance this, I should keep the remainder of the amount due ($500) for as many months after maturity

as 500 is contained times in 1400, or, 24 months.

RULE.

Multiply each payment by the number denoting the time it was paid before becoming due, and divide the sum of the products by the number denoting the balance unpaid; the quotient will be the time the balance should be kept after maturity.

2. I buy a house, January 3, for $4000, to be paid in 6 months without interest; if I pay $1000 March 3, and $1000 May 3, when, in equity, should I pay the balance?

3. I buy a bill of goods, September 5, at 90 days, amounting to $600; if I pay $300 in 30 days, at what date should I pay the balance?

4. A farmer buys a horse at a vendue for $120, on 4 months' credit; he pays, however, $40 cash; what should be the term of credit for the balance?

CASE III.

To find the equated time when the terms of credit begin at different dates.

1. What is the equated time of payment of the following bills: June 22, $500 at 60 days, July 3, $400 at 90 days, and August 5, $300, cash?

$500, June 22, at 60 days, due August 21.
$400, July 3, at 90 days, due October 1.
$300, August 5, cash, due August 5.
From Aug. 21 to Oct. 141 days.
From Oct. 1 to Oct. 10 days.
From Aug. 5 to Oct. 1= 57 days.

500 × 41 = 20500

400 x 0=00000 300 × 57=17100

1200

October 1-31 days August 31.

37600

31

ANALYSIS.-If the entire debt had been paid October 1, the latest date on which any of the items becomes due, we find, by equating, that the debtor would have had the use of $1200 for 31 days after it was due. The date of payment, therefore, must be made 31 days before October 1, or August 31.

RULE.

Multiply each sum by the number denoting the difference in days between the date on which it becomes due, and the latest date on which any sum named in the account becomes due; divide the sum of the products by the sum of the debts, and the quotient will be the number of days to be counted backward from the latest date.

2. William Patterson bought goods of Evans & Co., as follows: May 10, $600, at 30 days, June 3, $400, at 30 days, August 9, $500, at 60 days; what is the equated time of payment?

3. A. W. Hahn bought merchandise of Robert Day, as follows:

Aug. 3, a bill of $100, at 60 days.

What is the equated time of payment?

4. Find the equated time of payment of the following bills: Jan. 3, $80, at 15 days, Jan. 10, $100, at 30 days, Jan. 19, $75, at 30 days, and Feb. 1, $60, at 10 days.

5. Bought the following bills of Hanly Bros., at 60 days: April 5, $150, April 15, $130, May 3, $225, May 20, $175; what is the equated time of payment?

6. Find the equated time of payment of the following bills: July 6, $150, at 90 days, Aug. 1, $200, at 60 days, Sept. 5, $180, at 30 days, and Sept. 25, $166, at 90 days.

AVERAGING OF ACCOUNTS.

To Average an Account is to determine the date when the difference between its two sides should be paid, or, to determine the amount of money which should be paid at a given date, without loss of interest to the debtor or the creditor in either case.

WRITTEN EXERCISES.

1. In the following account, when is the balance due, and what is the cash balance on the 30th of July, 1876?

$2301 — $1680= $621, the balance due by the debtor.

269540 da.

269540-65004 = 204536 days' interest on $1 due to the debtor. 204536 ÷ 621 = = 329, the number of days during which the debtor is entitled to the use of $621, after Nov. 7, or, till Oct. 2, 1877.

ANALYSIS.-We select November 7, the latest date on which any sum in the account becomes due, and reckon the number of days from this date to the date when each of the remaining items becomes cash, or due. We multiply each item by its number of days, and find the sum of the products on each side of the account. The difference of these sums, 204536, shows the `number of days' interest on $1 due to the debtor. If it requires 204536 days for $1 to gain a certain interest, it will require $621, the balance of the account, of 204536 days, or 329 days, to gain an equal amount of interest. The date of payment should therefore be 329 days from November 7, or October 2, 1877.