SECTION XVI. EQUATION OF PAYMENTS. ART. 517. EQUATION OF PAYMENTS is the process of finding the equalized or average time when two or more payments due at different times, may be made at once, without loss to either party OBS. The equalized or average time for the payment of several debts, due at different times, is often called the mean time. 518. From principles already explained, it is manifest, when the rate is fixed, the interest depends both upon the principal and the time. (Art. 404.) Thus, if a given principal produces a certain interest in a given time, Double that principal will produce twice that interest; In double that time the same principal will produce twice that interest; In half that time, half that interest; &c. 519. Hence, it is evident that any given principal will produce the same interest in any given time, ás One half that principal will produce in double One third that principal will Thrice that principal will that time; 66 "thrice that time; 66 half that time; 66 a third of that time; &c. For example, at any given per cent. The int. of $2 for 1 year, is the same as the int. of $1 for 2 yrs. ; $1 for 3 yrs.; &c. QUEST.-517. What is equation of payments? Obs. What is the average time for the payment of several debts sometimes called? 518. When the rate is fixed, upon what does the interest depend? 520. The interest, therefore, of any given principal for 1 year, or 1 month, &c., is the same, as the interest of 1 dollar for as many years, or months, as there are dollars in the given principal. Ex. 1. Suppose you owe a man $15, and are to pay him $5 in 10 months, and $10 in 4 months, at what time may both payments be made without loss to either party? Analysis. Since the interest of $5 for 1 month is the same as the interest of $1 for 5 months, (Art. 519,) the interest of $5 for 10 months must be equal to the interest of $1 for 10 times 5 months. And 5 mo. X 10=50 mo. In like manner the interest of $10 for 4 months is equal to the interest of $1 for 4 times 10 months; and 10 mo. X4-40 months. Now 50 months added to 40 months make 90 months; that is, you are entitled to the use of $1 for 90 months. But $1 is of $15, consequently you are entitled to the use of $15, 1 of 90 months, and 90÷15=6. Ans. 6 months. Proof. The interest of $5 at 6 per cent. for 10 mo. is $5 X.05=$.25 The interest of $10 4 mo. is $10X.02= .20 Sum of both $.45 The interest of $15 at 6 per cent. for 6 mo. is 15.03=$.45. 521. From these principles we derive the following general RULE FOR EQUATION OF PAYMENTS. First multiply each debt by the time before it becomes due; then divide the sum of the products thus obtained by the sum of the debts, and the quotient will be the average time required. OBS. 1. If one of the debts is paid down, its product will be nothing; but in finding the sum of the debts, this payment must be added with the others. 2. When there are months and days, the months must be reduced to days, or the days to the fractional part of a month. 3. This rule is based upon the supposition that discount and interest paid in advance are equal. But this is not exactly true; consequently, the rule, though in general use, is not strictly accurate. (Art. 432. Obs. 1.) QUEST.-521. What is the rule for equation of payments? 2. If you owe a man $60, payable in 4 months, $120 payable in 6 months, and $180 payable in 3 months, at what time may you justly pay the whole at once? Operation. $ 60X4= $240, the same as $1 for 240 months. (Art. 520.) $120X6 = $720, " $1 for 720 $360 debts. $1500, sum of products. Now 1500÷360=44 months. Ans. 3. A merchant bought one lot of goods for $1000 on 5 months; another for $1000 on 4 months; another for $1500 on 8 months: what is the average time of all the payments? 4. If a man has one debt of $150, due in 3 months; another of $200, due in 4 months; another of $500, due in 74 months; what is the average time of the whole ? 5. A man bought a house for $3500, and agreed to pay $500 down, and the balance in 6 equal annual instalments: at what time may he pay the whole? 6. If you owe one bill of $175, due in 30 days; another of $180 due in 60 days; another of $120, due in 65 days, and another of $200, due in 90 days: when may you pay the whole at once? PARTNERSHIP. 522. PARTNERSHIP is the associating of two or more individ uals together for the transaction of business. (Art. 464.) The persons thus associated are called partners; and the association itself, a company or firm. The money employed is called the capital or stock; and the profit or loss to be shared among the partners, the dividend. CASE I. When stock is employed an equal length of time. Ex. 1. A and B formed a partnership; A furnished $600 capital, and B $900; they gained $300: what was each partner's share of the gain? QUEST.-522. What is partnership? What are the persons thus associated called? What is the association itself called? What is the money employed called? What the profit or loss? Analysis. Since the whole stock is $600+$900=$1500, A's part of it was , and B's part was 3. Now since 900 A put in of the stock, he must have of the gain; and $300 X=$120. For the same reason B must have 3 of the gain; and $300-$180. Or, we may reason thus: As the whole stock is to the whole gain or loss, so is each man's particular stock to his share of the gain or loss. That is, $1500 $300 :: $600 : A's gain; or $120. And $1500: $300 :: $900: B's gain; or $180. Proof.-$120+$180=$300, the whole gain. (Art. 21. Ax. 11.) 523. Hence, to find each partner's share of the gain or loss, when the stock of each is employed for the same time. Multiply each man's stock by the whole gain or loss; divide the product by the whole stock, and the quotient will be his share of the gain or loss. Or, make each man's stock the numerator, and the whole stock the denominator of a common fraction; multiply the gain or loss by the fraction which expresses each man's share of the stock, and the product will be his share of the gain or loss. PROOF.-Add the several shares of the gain or loss together, and if the sum is equal to the whole gain or loss, the work is right. (Art. 21. Ax. 11.) OBS. 1. The preceding case is often called Single Fellowship. But since a partnership is necessarily composed of two or more individuals, it is somewhat difficult to see the propriety of calling it single. 2. This rule is applicable to questions in Bankruptcy, and all other operations in which there is to be a division of property in specified proportions. (Arts. 465, 466.) 2. A, B, and C formed a partnership; A put in $1200 of the capital, B $1600, and C $2000; they gained $960: what was each man's share of the gain? QUEST.-523. How is each man's share of the gain or loss found, when the stock of each is employed for the same time? How is the operation proved? Obs. What is it sometimes called? To what is this rule applicable ? 3. A, B, and C entered into partnership; A furnished $2350, B $3200, and C $1820; they lost $860: what was each man's share of the loss? 4. A bankrupt owes A $2400, B $4600, C $6800, and D $9000;" his whole effects are worth $11200: how much will each creditor receive? 5. A, B, C, and D, engaged in an adventure; A put in $170, B $160, C $140, and D $130; they made $3000: what was each man's share? CASE II. When the stocks are employed unequal lengths of time. 6. A and B formed a partnership; A put in $900 for 4 months, and B put in $400 for 12 months; they gained $763: what was each man's share of the gain? Note.-It is obvious that the gain of each depends both upon the capital he furnished, and the time it was employed. (Art. 518.) Analysis. Since A's capital $900, was employed 4 months, his share of the gain is the same as if he had put in $3600 for 1 month; (Art. 519;) for $900X4=$3600. Also B's capital $400, being employed 12 months, his share of the gain is the same as if he had put in $4800 for 1 month; for $400×12= $4800. The sum of $3600 and $4800 is $8400. Therefore, A's share of the gain must be 3489=4. B's 66 Now $763X=$327, A's share. And $763X+ $436, B's share. Hence, 524. To find each partner's share of the gain or loss, when the stock of each is employed unequal lengths of time. Multiply each partner's stock by the time it is employed; make each man's product the numerator, and the sum of the products the denominator of a common fraction; then multiply the whole gain or loss by each man's fractional share of the stock, and the produci will be his share of the gain or loss. OBS. This case is often called Compound or Double Fellowship. QUEST.-524. When the stock of each partner is employed unequal lengths of time how is each man's share found? Obs. What is this case sometimes called? |