Orleans, with an assorted cargo. In consequence of a violent gale in the Gulf of Mexico, the captain was obliged to throw overboard a portion of the cargo, amounting in value to $ 4465.50, and the necessary repairs of the vessel cost $ 423. The contributory interests were as follows : vessel, $ 30000; gross freight, $ 6225; cargo shipped by R. S. Davis & Co., $3650; by Henry Mason, $ 6500; by G. T. Sampson, $2000; by J. Francis & Son, $ 550; by Morton Brothers, $ 5450; and by Sanborn & Carter, $ 8500. Of the cargo thrown overboard, there belonged to Henry Mason the value of $ 3000, and to Morton Brothers the remainder, $ 1465.50. The cost of detention in port in consequence of repairs was $ 116.50. How ought the loss to be apportioned among the contributory interests ? OPERATION CONTRIBUTORY INTERESTS. LOSS FOR GENERAL BENEFIT. Vessel, $ 30000 Thrown overboard, $ 4465.50 Cargo, 26650 Cost of detention, 116.50 Freight, less } 4150 Repairs, less 1 282.00 Entire contrib. int., $ 60800 Entire loss, $ 4864.00 $ 4864.00 ; 60800 .08, the loss per cent. $ 30000 x .08 $ 2400, am't payable by vessel. 4150 x .08 332, am't payable by freight. 550 x .08 44, am't papable by J. Francis & Son. 5450 x .08 436, am't payable by Morton Brothers. 8500 x .08 680, am't payable by Sanborn & Carter. Proof, $ 4864, entire amount payable. $ 2400.00 – $ 398.50 = $ 2001.50, balance payable by vessel. 3000.00 520.00 = 2480.00, bal. rec'ble by H. Mason. 1465.50 436.00 = 1029.50, bal. rec'ble by Morton Bro's. Since the vessel lost $ 116.50 + $ 282, $ 398.50, that amount is deducted in finding the net amount the vessel must contribute to the general loss. Henry Mason lost $ 520; the amount payable by him is therefore made so much less on that account; and Morton Brothers also, having lost $ 436, have the amount of their contribution lessened by that sum RULE. — Multiply each contributory interest by the loss per cent., and the product will be its contribution to the general loss. Ex. 2. The ship Hope, in her passage from Liverpool to New York, was crippled in a storm, in consequence of which the captain had $ 6500 worth of the cargo thrown overboard, and the necessary repairs of the vessel cost $ 1050. The charges for board of seamen, pilotage, and dockage amounted to $ 142. The contributory interests were: vessel, $ 31500; gross amount of freight, $ 4160; cargo shipped by Manning & Brother, $ 2145; by Anderson & Fisk, $ 1460; by Smidt & Huber, $ 960 ; by Greenwood, Laporte, & Co., $ 670 ; and by Allermann, Ritter, & Herr, $ 1000. In adjusting the general average in New York, the deduction made from the .gross amount of freight on account of seamen's wages was one half. Required the several shares of the general loss. EQUATION OF PAYMENTS. 433. EQUATION OF PAYMENTS is the process of finding the average or mean time when the payments of several sums, due at different times, may all be made at one time, without loss either to the debtor or creditor. 434. A strictly accurate method of determining, by Arithmetic, the true average time for the payment of more than two sums, due at different times, it is believed, has not yet been discovered ; and, even when there are only two sums, the accurate method is the translation of an Algebraic Formula. The true equated time, however, may be readily found by Algebra. Mercantile usage, however, gives its sanction to another method, and one which is not entirely correct, though, when the sums are small and the terms short, the variation from the exact truth is practically of no consequence. But this method, being far the most convenient of application, is adopted among business men. OPERATION. = 80 435. 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. What is the average time of paying the whole ? Ans. 8 months. The interest of $ 5 for 6 5 X 6 30 months is the same as the inter6 x 7 42 est of $ 1 for 30 months; and of 8 x 10 $ 6 for 7 months, the same as of $ 1 for 42 months; and of $8 for 19 19) 152 ( 8 months. 10 months, the same as of $ 1 for 1 5 2 80 months. Hence, the interest of all the sums to the time of payment 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. 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. 20, 168.75 on 5 months. February 4, 386.25 on 4 months. March 144.60 on 5 months. April 7, 386.90 on 3 months. 66 11, OPERATION. Due May 1, $3 7 5.5 0 66 June 20, 16 8.7 5 x 50 8 4 3 7.5 0 66 June 4, 38 6.2 5 x 34 131 3 2.5 0 6 Aug. 11, 1 4 4.6 0 X 10 2 1 4 7 4 9.20 66 July 7, 38 6.90 x 67 2 5 9 2 2.30 $ 1 4 6 2.0 0 6 2 2 4 1.5 0 days. 62241.50 : 1462.00 42362 days. May 1 + 43 days = June 13, Ans. 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 ir. 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 1, 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 94.3353 , and $ 106 + $ 94.3351 $ 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? 66 66 66 66 66 66 Jan. 1, a bill amounting to $ 600, on 4 months. Feb. 7, 370, on 5 months. March 15, 560, on 4 months. April 20, 420, on 6 months. Ans. July 11. 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. July 4, 376.18, on 4 months. Sept. 25, 821.75, on 2 months. Oct. 1, 961.25, on 8 months. Jan. 1, 1856, 144.50, on 3 months. Feb. 10, 811.30, on 6 months. Mar. 12, 567.70, on 5 months. April 15, 369.80, on 4 months. Ans. March 16, 1856. 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? Ans. 3mo: 20d. The interest on the $ 300 300 X 5 = 15 00 for 5 months is equal to the 800 X 1 8 0 0 interest of $ 1 for 1500 months, and the interest of $ 800 for 1 1100 2 3 0 0 month is equal to that of $1 $17 28 $1100 = $6 2 8; interest on both partial pay for 800 months; and thus the 2 3 0 0 = 6 2 8 = 3mo. 20d. ments, 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, aig of 2300 months, or 3 months and 20 days. RULE. Multiply each payment by the time it was maile before it becomes due, and divide the sum of the products by the balance remaining unpaid ; and the quotient will be the required time. OPERATION |