had ordered remittance to London, and thence to New York, exchange at Amsterdam on London being 11.19 guilders per £1, and at London on New York 4.88, brokerage at 11% in London for remitting. Ans. $94.31+. 5. If at Philadelphia exchange on Liverpool is 4.891, and at Liverpool on Paris 24 francs 96 centimes per £1, what is the arbitrated course of exchange between Philadelphia and Paris, through Liverpool? Ans. 5.10.

6. An American resident of Amsterdam wishing to obtain funds from the U. S. to the amount of $6400, directs his agent in London to draw on the U. S. and remit the proceeds to him in a draft on Amsterdam, exchange on the U. S. being at 4.85 in London, and the course between London and Amsterdam being 18d. per guilder. If the agent charges commission at % both for drawing and remitting, how much better is this arbitration than to draw directly on the U. S. at 41 cents per guilder?

7. A speculator in Pittsburgh, having purchased 58 shares of railroad stock in New Orleans, at 95 %, remits to his agent in New York a draft purchased at 2 % premium, with orders for the agent to remit the sum due in N. O. Now, if exchange on N. O. is at % discount in N. Y., and the agent's commission for remitting is %, how much does the stock cost in Pittsburgh? Ans. $5606.08. 8. A merchant in Boston owes 19570 francs in Paris. Which will be the more advantageous to him, to remit directly to Paris at 5.12 or through London at 4.89, buying there exchange on Paris at 25.19 fr. to £1, and paying % brokerage?

9. If in London exchange on Paris is 25.71, and in New York on Paris it is 5.15, what is the arbitrated course of exchange between New York and London? Ans. 4.987 +.

10. A banker in New York remits $3000 to Liverpool, by arbitration, as follows: first to Paris at 5 francs 16 centimes per $1; thence to Hamburg at 125 francs per 100 marks; thence to Amsterdam at 1.714 marks to the guilder; thence to Liverpool at 11.82 guilders per £1 sterling. How much sterling money will he have in bank at Liverpool, and what will be his gain over direct exchange at 4.91? Proceeds in Liverpool, £610 18s. 3d. Gain by arbitration, 10s. 9d.

Ans.

EQUATION OF PAYMENTS.

609. Equation of Payments is the process of finding the mean or equitable time of payment of several sums, due at different times without interest.

610. The Term of Credit is the time to elapse before a debt becomes due.

611. The Average Term of Credit is the time to elapse before several debts, due at different times, may all be paid at once, without loss to debtor or creditor.

612. The Equated Time is the date at which the several debts may be canceled by one payment.

613. To Average an Account is to find the mean or equitable time of payment of the balance.

614. A Focal Date is a date with which all the others are compared in averaging an account.

NOTE. Each item of a book account draws interest from the time it is due, which may be either at the date of the transaction, or after a specified term of credit.

In averaging, there are two kinds of equations, Simple and Compound.

615. A Simple Equation is the process of finding the average time when the payments or account contains only one side, which may be either a debit or credit.

616. A Compound Equation is the process of averaging when both debts and credits are to be considered.

SIMPLE EQUATIONS.

CASE I.

617. When all the terms of credit begin at the same date.

1. In settling with a creditor on the first day of April, I find that I owe him $12 due in 5 months, $15 due in 2 months, and $18 due in 10 months; at what time may I pay the whole amount?

alent for the various terms of credit on the different items. Now the value of credit on any sum is measured by the product of the money and time. Therefore, the credit on $12 for 5 mo. = the credit on $60 for 1 mo., because 12 x 5 = 60 × 1. In like manner, we have the credit on $15 for 2 mo. = the credit on $30 for 1 mo.; and the credit on $18 for 10 mo. the credit on $180 for 1 mo. Hence, by addition, the value of the several terms of credit on their respective sums equals a credit of 1 month on $270; and this equals a credit of 6 months on $45, because 45 × 6=270 × 1. Hence the following

RULE. 1. Multiply each payment by its term of credit, and divide the sum of the products by the sum of the payments; the quotient will be the average term of credit.

II. Add the average term of credit to the date at which all the credits begin; the result will be the equated time of payment. NOTES. - 1. The periods of time used as multipliers must all be of the same denomination, and the quotient will be of the same denomination as the terms of credit; if these be months, and there be a remainder after the division, continue the division to days by reduction, always taking the nearest unit in the last result.

2. The several rules in equation of payments are based upon the principle of bank discount; for they imply that the discount of a sum paid before it is due equals the interest of the same amount paid after it is due.

EXAMPLES FOR PRACTICE.

1. On the first day of January, 1860, a man gave 3 notes, the first for $500 payable in 30 days; the second for $400 payable in 60 days; the third for $600 payable in 90 days. What was the average term of credit, and what the equated time of payment? Ans Term of credit, 62 da. ; time of payment, Mar. 3, 1860. 2. A man purchased real estate, and agreed to pay of the price in 3 mo., in 8 mo., and the remainder in 1 year. Wishing to cancel the whole obligation at a single payment, how long shall this payment be deferred?

8 mo.

3. I owe $480 payable in 90 days, and $320 payable in 60 days. My creditor consents to an extension of time to 1 year, and offers to take my note for the whole amount on interest at 6 per cent. from the equated time, or a note for the true present worth of both debts, on interest from date. How much will I gain if I choose the latter condition? Ans. $1.14.

4. Bought merchandise April 1, as follows: $280 on 3 mo., $300 on 4 mo., $200 on 5 mo., $560 on 6 mo.; what is the equated time of payment? Ans. Aug. 24.

CASE II.

618. When the terms of credit begin at different dates.

1. When does the amount of the following bill become due, per average?

ANALYSIS. The three items of the bill are due Jan. 12, Mar. 16, and Apr. 20, respectively. In the first operation we use the earliest maturity, Jan. 12, for a focal date, and find the difference in days between this date and each of the others; thus, from Jan. 12 to Mar.

16 is 64 da.; from Jan. 12 to Apr. 20 is 99 da. Hence, from Jan. 12 the first item has no credit, the second has 64 days' credit, and the third 99 days' credit, as appears in the column marked da. We now proceed to find the products as in Case I, whence we obtain the average credit, 55 da., and the equated time, Mar. 7.

=

But interest

The second

In the second operation, the latest maturity, Apr. 20, is taken for a focal date, and the work may be explained thus: Suppose the account to be settled Apr. 20. At that time the first item has been due 99 days, and must therefore draw interest for this time. on $400 for 99 days the interest on $39600 for 1 day. item must draw interest 35 days; but interest on $600 for 35 days = interest on $21000 for 1 day. Taking the sum of the products, we find that the whole amount of interest due Apr. 20 equals the interest on $60600 for 1 day; and this is found, by division, equal to the interest on $1375 for 44 da., which is the average term of interest. Hence the account would be settled Apr. 20, by paying $1375, with interest on the same for 44 days, This shows that the $1375 has been used 44 days, that is, it falls due Mar 7, without interest, Hence we have the following.

RULE. I. Find the time at which each item becomes due, by adding to the date of each transaction the term of credit, if any be specified, and write these dates in a column

II. Assume either the earliest or the latest date for a focal date, and find the difference in days between the focal date and each of the other dates, and write the results in a second column,

III. Write the items of the account in a third column, and multiply each by the corresponding number of days in the preceding column, writing the products in a fourth column,

IV. Divide the sum of the products by the sum of the items, The quotient will be the average term of credit or interest, and must be reckoned from the focal date TOWARD the other dates, to find the equated time of payment,

NOTES. 1, When dollars and cents are given, it is generally sufficient to take only dollars in the multiplicand, rejecting the cents when less than 50, and carrying 1 to the dollars, if the cents are more than 50,

2, Months in any terms of credit are understood to be calendar months; the time must therefore be carried forward to the same day of the month in which the term of credit expires,