435. To find the average or mean time of paying several debts due at different dates. Ex. 1. A owes B $19, $ 5 of which is to be paid in 6 months, $ 6 in 7 months, and $8 in 10 months. average time of paying the whole? What is the Ans. 8 months. The interest of $5 for 6 months is the same as the interest of $1 for 30 months; and of $6 for 7 months, the same as of $1 for 42 months; and of $8 for 10 months, the same as of $1 for 80 months. Hence, the interest of all the sums to the time of pay ment is the same as the interest of $1 for 30mo. +42mo. +80mo. 152 months. Now, if $1 require 152 months to gain a certain sum, $5+$6+$8= $19 will require of 152 months; and 152mo.÷ 19 8 months, the average or mean time for the payment of the whole. = 19 2. Purchased goods of Kendall & White at different times, and on various terms of credit, as by the statement annexed. What is the mean time of payment? January 1, a bill amounting to $375.50 on 4 months. 168.75 on 5 months. 386.25 on 4 months. 144.60 on 5 months. 386.90 on 3 months. 50= 34 = 8437.50 1 3 1 3 2.5 0 38 6.25 X 14 4.60 X 102 14749.20 38 6.9 0 X $146 2.00 67259 2 2.3 0 62241.50 1462.00 = 42187 days. May 143 days We first find the time when each of the bills will become due. Then, since it will shorten the operation and not change the result, we take the first time when any bill becomes due, instead of its date, for the point from which to compute the average time. Now, since May 1 is the period from which the average time is computed, no time will be reckoned on the first bill, but the time for the payment of the second bill extends 50 days beyond May 1, and we multiply its amount by 50. Proceeding in the same manner with the remaining bills, we find the average time of payment to be 43 days, nearly, from May 1, or on June 13. RULE. Multiply each payment by its own term of credit, and divide the sum of the products by the sum of the payments. NOTE 1. When the date of the average time of payment is required, as in Example 2, find the time when each of the sums becomes due. Multiply each sum by the number of days intervening between the date of its becoming due and the earliest date on which any sum becomes due. Then proceed as in the rule, and the quotient will be the average time required, in days forward from the date of the earliest sum becoming due. NOTE 2. - In the result, it is customary, if there be a fraction of a day less than, to reject it; but if more than, to reckon it as 1 day. In practice the work may be somewhat abridged, without varying materially the result, by disregarding, in performing the multiplications, the cents in the several sums, when they are less than 50, and by calling them $1, when more than 50. When a payment is made at the time of purchase, it has no product, but it must be added with the other products in finding the average time. NOTE 3. The method of the rule, or that generally adopted by merchants, as has been intimated (Art. 384), is not perfectly correct. For if I owe a man $ 200, $100 of which I am to pay down, and the other $100 in two years, the equated time for the payment of both sums would be one year. It is evident that, for deferring the payment of the first $100 for one year, I ought to pay the amount of $100 for that time, which is $106; but for the $100 which I pay a year before it is due, I ought to pay the present worth of $100, which is $94.335, and $106 + $94.335 $200.335; whereas, by the mercantile method of equating payments, I only pay $ 200. EXAMPLES. 3. There is owing a merchant $1000; $200 of it is to be paid in 3 months, $300 in 5 months, and the remainder in 10 months. What is the equated time for the payment of the whole sum? Ans. 7mo. 3d. 4. I have bought a farm for $ 6500; $2000 of which is to be paid down, $500 in one year, and the remainder in 2 years. But if a note for the whole amount had been preferred, in what time would it have become due? 5. A owes B $300, of which $50 is to be paid in 2 months, $100 in 5 months, and the remainder in 8 months. What is the equated time for the payment of the whole sum? Ans. 6 months. 6. I have sold H. W. Hathaway several bills of goods, at different times, and on various terms of credit, as by the following statement. What is the average time for the payment of the whole? Jan. 1, a bill amounting to $ 600, on 4 months. 7. Purchased goods of J. D. Martin, at different times, and on various terms of credit, as by the statement annexed. What is the equated time of paying for the same? March 1, 1855, a bill amounting to $675.25, on 3 months. 436. To find the time of paying the balance of a debt, when partial payments have been made before the debt is due. Ex. 1. I have bought of Leonard Johnson goods to the amount of $1728, on 6 months' credit. At the end of one month I pay him $300, and at the end of 5 months, $800. How long, in equity, after the expiration of 6 months, should the balance remain unpaid? OPERATION. 300 X 5 = 1500 800 × 1 = 800 1100 $1728 2300 $1100 = $628; Ans. 3mo. 20d. The interest on the $300 for 5 months is equal to the interest of $1 for 1500 months, and the interest of $800 for 1 month is equal to that of $1 for 800 months; and thus the interest on both partial payments, at the expiration of the 6 months, is equal to the interest of $1 for 1500+800 =2300 months. To equal this credit of interest, the balance of the debt, which we find to be $628, should remain unpaid, after the 6 months, 2 of 2300 months, or 3 months and 20 days. 23006283mo. 20d. RULE. - Multiply each payment by the time it was made before it becomes due, and divide the sum of the products by the balance remaining unpaid; and the quotient will be the required time. EXAMPLES. 2. A merchant has $144 due him, to be paid in 7 months, but the debtor agrees to pay one half ready money, and two thirds of the remainder in 4 months. What time should be allowed for paying the balance? Ans. 2y. 10mo. 3. March 23, 1856, I sold John Morse goods to the amount of $8000 on a credit of 8 months. April 5, he paid me $1200; July 4, $1500; September 25, $1800; October 1, $1000; November 20, $500. When, in equity, should I receive the balance? 4. There is due to a merchant $ 800, one sixth of which is to be paid in 2 months, one third in 3 months, and the remainder in 6 months; but the debtor agrees to pay one half down. How long may the debtor retain the other half so that neither party may sustain loss? Ans. 83 months. 5. I have sold Charles Fox goods to the amount of $3051, on a credit of 6 months, from September 25, 1856. October 4, he paid $476; November 12, $375; December 5, $800; January 1, 1857, $ 200. When, in equity, ought I to receive the balance? Ans. October 8, 1857. AVERAGING OF ACCOUNTS. 437. AN Account Current is a statement of the mercantile transactions of one person with another, when immediate payments are not made. An account is marked Dr. on the left, to indicate that the person with whom the account is kept is debtor for the items on that side; and is marked Cr. on the right, to indicate that he is creditor for the items on that side. Accounts current are generally made up or settled at the end of every six months or year. 438. To find the equated time when the balance of an account current will be due. Ex. 1. In the following account when did the balance become due, the merchandise articles being on 6 months' credit? Ans. December 22, 1856. Dr. Messrs. James Dutton & Co. in account with David Hale. 66 9. "cash paid draft, Mar. 3." merchandise, 24. 1856. $96.51 Jan.30. By cash, Cr. $240.00 Apr. 9." 66 23.60 May 15. Errors excepted. Settled as above, Boston, July 7, 1856. $748.89 On equating each side of the account (Art. 435), we find the debits became due 186 days from February 4, or on August 8; and the credits became due 26 days from January 30, or on February 25. If the account had been settled on February 25, it is evident the debits would have been paid 165 days, or the time from February 25 to August 8, before having become due. This would have been a loss of interest to the debit side of the account, and a corresponding gain to the credit side. Now, as the settlement should be one of equity, we find how long it will take the balance, $ 410, to gain the same interest that $ 339 would gain in the 165 days. If it take $339 to gain a certain interest in 165 days, it would take $1 to gain the same interest 339 times 165 days: =55935 days; and it would take $410 to gain the same amount of interest of 55935 days *Included only to illustrate the manner of settling an account. = = 136 |