And since a': c':: a" : c'"' :: a""' : c'"' . . . . it follows that The required division is made by means of these proportions; for the divided number, A, is to the number which is to be divided, C, as any part of the divided number is to a fourth proportional, which is the corresponding part of the number which is to be divided. In practice, it is not necessary to form the proportions, but merely to multiply successively the parts of the divided number by the ratio of the undivided to the divided number; the results are the corresponding parts of the undivided number. 1st Example. Given A=12; C=20; a'=2; a'=3; a""'=7; divide 20 into parts proportional to 2, 3, 7. C 20 5 The ratio == A 12-3 c"=7x3=3=11}; Whence the parts required are 3, 5, 11. The sum of these parts ought to be equal to C. In this example 33+5+11=20=C. When the sum of the parts is not equal to C, the calculation must be revised. 2d Example. Divide 100 into parts which shall be in the proportion ,,, and . 1 30 260 1 1 15 12 +60+ + 77 60 60 60° When a', a", a"", .... are given, and not A, it is necessary to add together the proportional parts a', a", a""...; their sum is A. When the given proportional parts a', a", a"", . are fractions, the common denominators of these fractions reduced for addition, and the denominator of their sum, A, must be the same. The ratios of fractions which have the same denominator being equal to the ratios of the numerators, the denominators may be suppressed; in the preceding calculation they are cancelled against each other. 363. Proportion, when employed to divide a number into parts proportional to other given numbers, is named Distributive Proportion, Partnership, Fellowship, &c. The terms partnership and fellowship are used when the rule is employed to determine the particular gain or loss of the individuals engaged in any joint undertaking. The principle upon which the apportionment of gain or loss is made in such cases is that the gain or loss of each individual is proportional to that individual's share of the joint undertaking, or of the money expended upon it, when all the shares have been the same time embarked in the undertaking: and to the product of the share by the time during which it has been so embarked when the shares have been acquired or their values contributed at different times. 1st Example. Three merchants engage in a joint speculation, towards which the first contributes 750 £., the second 1127 £., and the third 1280 £.; the gain being 600 £. it is required to find the share of each partner? C 600 A=750+1127+1280=3157 £.; C=600.·. A 3157 2d Example. A tradesman begins business with a capital of 1250£.; 5 months afterwards he borrows 2000 £. from one person, and 6 months after obtaining the first loan he borrows 3000 £. from another; at the end of two years the gain on the business is 4000 £. If the tradesman is allowed 5 per cent. upon the whole gain for conducting the business, and the balance is divided between him and his creditors in proportion to the value of the contribution of each to the common stock, what is the share which each receives of the gain? 5 100 The allowance for management being 4000 £. × =200 £., 4000 £.—200£. or 3800 £. is the sum which is to be shared among the partners in proportion to the value of their contributions, that is, in proportion to the product of the contribution of each by the time during which it has been embarked in the business. Now 1250 £. for 24 months=1250×24=30000 £. for 1 month; 2000 £. for 24-5 or 19 months=2000 × 19=38000 £. for 1 month; 3000 £. for 19-6 or 13 months 3000 x 13=39000 £. for 1 month. Hence the question is reduced to the division of 3800 £. into parts proportional to the three numbers, 30000, 38000, 39000, or to the three numbers 30, 38, 39; ... a=30; a"=38; a""=39; A=30+38+39=107. 364. To divide a given number into parts proportional to other given numbers. Rule. Form the sum of the given proportional numbers, and the ratio of the given undivided number to this sum; multiply the given proportional numbers successively by this ratio; the products are the proportional parts required. When the proportional numbers (as in the 2d Ex. of Art. 363) are contributions, for different periods of time, to a common stock, multiply each contribution by the time of its continuance; take these products for the proportional numbers, and find the parts of the undivided number as before. 365. Exercises in Distributive Proportion and Partnership. 1st. Divide 240 into parts proportional to the numbers 1, 2, 3 ? 8 4th. Two persons, A and B, made a stock of 120 £., of which A contributed 75 £. and B the rest; by trading they gain 30£. What is the share of each? Ans. A's share is 18 £. 15 sh.; B's, 11 £. 5 sh. 5th. Three persons, A, B, and C, freighted a ship with 340 tuns of wine, of which A supplied 110 tuns, B 97, and C the rest; in a storm the seamen threw overboard 85 tuns. How much must each person sustain of the loss? Ans. A, 27 tuns; B, 244; and C, 334. 6th. A field containing 37 ac. 2 ro. 14 po. is to be divided among three persons, A, B, C, in proportion to the values of their estates; now if A's estate is worth 500 £. a year, B's 320 £., and C's 75 £., what quantity of land must each have? Ans. A, 20 ac. 3 ro. 3913% po.; B, 13 ac. 1 ro. 46 30,4% po.; C, 3 ac. 0 ro. 23173 po. 179 7th. A general imposed a contribution of 700 £. on four villages, to be paid in proportion to the number of inhabitants contained in each; the first contained 250, the second 350, the third 400, and the fourth 500 persons. What part had each village to pay? Ans. First, 116 £. 13 sh. 4d.; second, 163 £. 6 sh. 8d.; third, 186£. 13 s. 4 d.; fourth, 233 £. 6sh. 8d. 8th. Five companies consisting the first of 54 men, the second of 51 men, the third of 48 men, the fourth of 39 men, and the fifth of 36 men, are detached to a post, the duty of which requires 76 men a day; if the companies are required to contribute in proportion to their strength, what number of men ought to be furnished by each company daily? Ans. First, 18 men; second, 17 men; third, 16 men ; fourth, 13 men; fifth, 12 men. 9th. A and B traded in partnership and gained 100£.; A con- 10th. A sum of money is to be divided among A, B, C, D, in such 11th. A had, in company, 50£. for 4 months, and B had 60 £. Ans. A's share, 9 £. 12 sh.; B's share, 14£. 8 sh. 12th. Two troops of cavalry rent a field in common, for which they are to pay 36£.; one company put into the field 23 horses for 27 days, the other 21 horses for 39 days. How much ought each to pay of the rent? Ans. The first comp., 15 £. 10 sh. 6 d.; the second, 20 £. 9 sh. 6 d. 13th. A began business on the 1st of January with a capital of 1000 £.; on the 1st of March following he took B as a partner, with a capital of 1500 £.; on the 1st of June A and B admit a third partner, C, who has a capital of 2800 £.; after trading together till the end of the year they find that their gain amounts to 1776 £. 10 sh. How must this be divided amongst the partners? Ans. A's share, 457 £. 9 sh. 44 d.; B's, 571 £. 16 sh. 81 d.; C's, 747 £. 3 sh. 114 d. 14th. A and B entered into partnership for a year; A contributed 500 £. on the 1st of January, and B on the 1st of May as much as entitled him to half the gain at the end of the year. What was B's contribution? Ans. 750 £. 366. Many treatises of commercial arithmetic contain a number of rules applicable to the business of bankers and merchants. Generally, each rule embodies instructions for the application of proportion to some particular class of questions; but as the application may be made without the help of a rule by any one who understands the principles of proportion and the meaning of such questions, it seems unnecessary to consider these rules in detail. A few questions, falling under some of the more important, are subjoined. 1st Example. A silk mercer bought 100 yards of riband at 3 yards for 1 sh., and 100 yards at 2 yards for I sh., and sold the whole at the rate of 5 yards for 2 sh. Did he gain or lose? How much in all? And how much per cent.? 3 yds. 100 yds.::1 sh.: sh.=1£. 13 sh. 4 d. price of 100 yds. at 3 yds. for 1 sh. ; 100 100 2 yds. 100 yds.::1 sh.: sh. 2£. 10 sh. 0 d. price of 100 yds. at 2 yds. for 1 sh.; ... 1£. 13 sh. 4 d. +2 £. 10 sh.=4 £. 3 sh. 4 d.=price paid for the whole. .. 4 £. 3 sh. 4d-4£.-3 sh. 4 d. the loss upon the whole. 1 25 6 Again, since 4 £. 3 sh. 4d.=4 £.= £.; 3 sh. 4 d.= 6 100 1 ; and that £. : £. 25:1::100: <=4£.; it appears that the loss is 25 1 6 6 4 per cent. This question is one of a class which, in treatises of commercial arithmetic, is considered under the rule of Profit and Loss. 2d Example. How many lbs. of tea at 5 sh. 9 d. per lb. must be given in exchange for 426 yds. of cloth at 13 sh. 4 d. per yard? The first step towards a solution is, to find the price of 426 yds. at 13 sh. 4 d., or £. per yard. 142 2 The second step is, to find how many lbs. of tea at 5 sh. 9 d. per lb. can be bought for 284 £. Therefore 987 lb. of tea at 5 sh. 9 d. per lb., are equivalent to 426 yards of cloth at 13 sh. 4 d. per yard. This is a simple case of the exchange of commodities by means of their money values. Such exchanges are described by the term Barter. Any sum of money expressed in terms of the coins or circulating medium of one country may be reduced to an equivalent sum expressed in terms of the coins or circulating medium of another country (if the money of the two countries has a given relation) by a process like that of Example 2, or by the method of Article 316. The process for the reduction of any sum of the money of one country to an equivalent sum of the money of another country is termed Exchange. 3d Example. How many Spanish piastres at 3 sh. 5 d. per piastre are equivalent to 1000 £. 16 sh. 10 d. sterling? The calculation becomes considerably more complex when the money of one country is to be changed into that of another; the relative value of the money of the two countries being given indirectly in terms of the money of a third country, or of a third and fourth country, &c. 4th Example. Suppose that 1 £. or 20 sh. are equal to 25 francs, 15 francs are equal to 8 German florins, 50 florins are equal to 9 Hamburgh ducats, 19 ducats are equal to 40 Russian roubles, it is required to find how many Russian roubles are equivalent to 100£. sterling? |