EQUATIONS. EQUATION is the method of finding the average or mean time at which two or more sums of money, due at different times, may be discharged at one payment, without loss to either the debtor or creditor. I. To find the average or mean time, where there are debits only. RULE.-Multiply each debt by the time it has to run, and divide the sum of the products by the sum of the debts; the quotient will be the average time for the payment of the whole amount of the debts. PROOF.-By Interest. If the interest on the sum of the debts for the mean time, at any rate per cent, equal the interest on the several sums, at their given time, the work is correct: or, find the mean time by the interest on the several sums. (See Supplement to Simple Interest, Rule IV.) EXAMPLES. 1. A. owes B. $100, due in 2 months, and $100 due in 4 months; if he give one note for both amounts, at what time must it be drawn? Ans. 3 months. 2. H. owes L $200, due in 2 months; $200, due in 3 months; and $200, due in 4 months: in what time will the whole be due ? Ans. 3 months. 3. A. owes his friend four equal sums; the first due in 20 days, the second in 30 days, the third in 40 days, and the fourth in 50 days; in what time will the whole be due ? Ans. 35 days. What is Equation? How do you find the average or mean time, when there are debits only? How do you prove the work? 4. M. owes N. four several amounts, as follows: $100, due the 1st of Jan.; $150, due the 10th of Feb.; $100, due the 15th of March; and $120, due the 1st of May; what is the average or mean time at which the whole is due ? Ans. March the 1st. Reckoning from the 1st of Jan. to the 1st of May, That is, 59 days from the 1st of Jan. makes March 1st. Or, reckoning from the 1st of May to the 1st of January, 61 days back from the 1st of May, makes March 1st, the same as before. 5. H. bought goods of I. as follows: Oct. 1st, $50 worth at 2 mo.; Nov. 2d, $100 worth at 3 mo.; Dec. 25th, $200 worth at 3 mo.; and Feb. 15th, $250 worth at 3 mo.: at what date will the whole be due, and how much interest, at 6 per cent per annum must be added, if H. give his note at 2 months from that date? Ans. S The whole will be due on the 28th of March. Interest $6.30 for 2 months and 3 days grace. { First find the time when each amount becomes due, and from these dates make the reckoning. 6. A merchant sold goods to P. as follows: July 1st, $100 worth at 3 mo.; July 20th, $150 worth at 4 mo.; Aug. 16th, $200 worth at 2 mo.; and Oct. 4th, $250 worth at 4 mo.: at what date will the whole be due, and how much interest, at 6 per cent per annum, will be due on the account March the 15th following Ans.{ The whole will be due Nov. 30th. The interest to 40 days from the 1st of January to the 10th of February-73 days to the 15th of March, etc. II. To find the average, or mean time, when there are debits and credits. RULE 1.-Find the time, as in Case L., Ex. 4, for each amount, for both debits and credits. 2.-Multiply each amount by its own time, and divide the difference between the sum of the debit products and the sum of the credit products, by the difference between the sum of the debits and the sum of the credits. PROOF.-As before; or reckon first forward and then backward, as in the 4th and 7th Examples. EXAMPLES. 7. D. owes E. $500, due the 1st of Dec.; $500, due the 1st of Jan.; and $500, due the 1st of May; on which he paid $500 on the 1st of Feb.: at what time was the balance due? 8. A. holds 3 notes, against B., as follows: one dated Oct. 10th, at 2 mo., for $200; another dated Nov. 10th, at 3 mo., for $250; and the other dated Dec. 10th, at 3 mo., for $200; on which he paid, Jan. 10th, $300: from what date should interest be reckoned on the balance? Ans. Feb. 19th. How do you find the average, or mean time, when there are both debits and credits-1si? 2d? How do you prove the work? Find the date for the payment of the above balance. (10) Ans. Aug. 6th. Dr. Mr. E. Small in acet. cur. with Isaac Dayton & Co. Cr. Find the date for the payment of the balance of this account Ans. Sept. 30th. Reckon from the 15th of Jan. to the 10th of May. 11. A man bought goods at 6 months' credit, as follows: Jan. 1, 1848, $400 worth; Feb. 1, $400 worth; March 1, $400 worth. Having some money, for which he had no immediate use, he made the following payments before his account became due Feb. 1, $200; Mar. 1, $200; and Apr. 1, $200 find the equated time for the payment of the balance. Ans. Jan. 1, 1849. 12. Sold to C. W., at 4 months' credit, Feb. 12th, $300 worth of merchandise, and Mar. 20th, $400 worth; on which he paid, July 1st, $200: from what date must I reckon interest on the balance? Ans. July 5th. 13. Sold to J. Hall, Dec. 10, merchandise, $360, at 3 mo.; Jan. 20, do. $400, at 4 mo.; Feb. 20, do. $350, at 4 mo. He has paid me, Mar. 1, $300; June 1, $300: from what date must I reckon interest on the balance, and what will be the amount of interest due on the 1st of Aug. following, at 7 per cent per annum? Ans. From the 31st of May. Interest due on the 1st of Aug., $6.05. * Reckon from or to maturity in all cases. FELLOWSHIP. FELLOWSHIP is a rule used by partners in trade, to find their respective shares of the gain or loss in business, in proportion to each one's share of the capital, and the time his capital has been in use. Fellowship is either Simple or Compound. SIMPLE FELLOWSHIP. SIMPLE FELLOWSHIP is the method of finding the share of the gain or loss, for each partner, in proportion to his share of the stock, or capital, without respect to time. RULE. The proportion the whole capital bears to each partner's share of it, the whole gain or loss will bear to his share of the gain or loss. PROOF.-The sum of the several shares of the gain or oss must equal the whole gain or loss. NOTE. By this rule, creditors may find their several shares in all cases of insolvency. EXAMPLES. 1. Three men engaged in a speculation, with a capital of $1400 A.'s part was $200, B.'s $400, and C.'s $800. They gained $700; how much is each man's share of the gain? Remove all the ciphers, and you will have the work so simplified that you cannot fail to understand it. 2. H., I., and J. formed a copartnership: H.'s share of the capital was $5000, I.'s $6000, J.'s $7000. They gained in one year $4500; how much is each one's share of the gain? Ans. H.'s $1250; I.'s $1500; J.'s $1750. What is Fellowship? How many kinds of Fellowship are there? What is Simple Fellowship? How do you prove your work? What is said in the note? |