PARTITIVE PROPORTION. $ 228. PARTITIVE PROPORTION is proportion applied to dividing a given quantity into two or more parts, which shall be in a given ratio, one to another. The terms of the given ratio, or ratios, may be called the proportional terms. For example: to divide $300 between A, B, and C, in the proportion of 2, 3, and 5; that is, so that A's share shall be to B's as 2 to 3, and B's to C's, as 3 to 5. In this example, 2, 3, and 5, are the proportional terms, the consequent of the first ratio 2 to 3 being the antecedent of the second ratio 3 to 5. Note. This part of Arithmetic is commonly called PARTNERSHIP OF FELLOWSHIP. § 229. To divide a given quantity into Two or MORE PARTS which shall be in a given RATIO, one to another. 1. Add together all the given proportional terms. Then, 2. The sum of those terms will be to each term, separately, as the quantity to be divided is to each corresponding part of that quantity. EXAMPLE. To divide $300 between A, B, and C, in the proportion of 2, 3, and 5. The sum of the proportional terms is 2+3+5=10. Then, 10 2: $300: $300X2÷10=$60, A's share; 103 $300: $300×3÷10-$90, B's share; and 105 $300: $300×5÷÷10=$150, C's share. Analysis. Suppose the whole sum $300 to be divided into 2+3+5=10 equal parts. Then it is evident that A must have 2, B 3, and C 5, of those parts; that is, A's share is B's share is of $300-$300×2÷10=$60 ; 16 of $300=$300×5÷10=$150. of $300-$300×3÷10=$90; The expressions $300X2÷10, &c., for the several shares, found by the Analysis, are the same as those found by the Rule. Hence the Analysis demonstrates the Rule. EXERCISES. 1. Divide $240 between three persons in such a manner that their shares shall be as 5, 4, and 3, respectively. Ans. $100; $80; and $60. 2. A gentleman bequeathed to his son and daughter $5000, the son's share of it to be to the daughter's as 3 to 2. What was the share of each ? Ans. Son's $3000; daughter's $2000. 3. A merchant employed 3 clerks, at the annual salaries of $300, $400, and $500, respectively. At the end of the year, the merchant proving bankrupt, has but $650 to be divided proportionably among them. What will be the portion of each? The proportional terms are 300, 400, and 500; or without altering the ratios, 3, 4, and 5; since 2002, and 188=1. Ans. The 1st, $162.5; the 2d, $216.666'; the 3d, $270.833'. 4. An insolvent debtor owes to A $250, to B $100, and to C $300. He is able to pay $420; what would each creditor receive of the $420? Ans. A $161.538'; B $64.615'; C $193.846'. 5. It is required to divide the number 180 into three parts which shall be to one another as 1, 3, and 4. Analysis. Reducing the proportional terms to a common denominator, we have them, 18%, and 11⁄2 ; and these are to one another as the numerators 6, 8, and 9; (§ 217). Hence 6, 8, and 9 may be taken for the proportional terms. Ans. 46; 62; and 70§. 6. A father proposed to divide $100 between his two sons in the ratio of to, provided either of them could ascertain the portion offered to him. What would each portion be? Ans. The 1st, $40; the 2d, $60. 7. The sum of $500 is to be divided among A, B, and C, in the proportion of 3, 14, and 2, respectively. What will be the share of each? Ans. A's $75.471'; B's $141.509'; C's $283. 8. Two persons form a partnership in trade, with a capital of $3000, of which the first contributed $1800, and the second the remainder. They gain $900: what is each one's share? Ans. The first, $540; the second, $360. 9. A bankrupt is indebted to A $425.50; to B $200; to C $100; and to D $85.75. He is able to pay $500. If this sum be divided among his creditors proportionably to their respective claims, what will be the share of each? Ans. A's $262.249'; B's $123.266'; C's $61.633'; D's $52.85'. When each given Ratio has a separate Antecedent and Consequent. § 230. In Partitive Proportion, when each given ratio has a separate antecedent and consequent,-take, for the proportional terms, the given 1st and 2d terms, the fourth proportional to the 3d, 4th, and 2d terms, the fourth proportional to the 5th and 6th terms, and last fourth proportional,—and so on to the last term. EXAMPLE. 10. Divide $1700 between A, B, C, and D, so that A's share may be to B's as 1 to 2, B's to C's as to 1, and C's to D's as 3 to 4. The given first and second terms are 1 and 2 ; finding a fourth proportional to the 3d, 4th, and 2d terms, Rule XLII, we have 1:2:2×1÷3=6; and finding a fourth proportional to the 5th and 6th terms, and last fourth proportional 6, we have 3 4:6:6X4-3=8. Now since B's share is to C's as to 1, or as 2 to 6; and since C's share is to D's as 3 to 4, or as 6 to 8; the four shares will be to one another as the numbers 1, 2, 6, and 8, which we accordingly take for the proportional terms. Then 17: 1 :: $1700 $100, A's share. In like manner B's is $200, C's $600, and D's $800. 11. Divide $70 between A, B, and C, in such a manner that A's share shall be to B's as 2 to 3, and B's to C's as 4 to 5. Ans. A's share $16, B's $24, and C's $30. 12. Three persons in a joint speculation gain $1000 ; which is to be divided so that the first share shall be to the second as 3 to 2, and the second to the third, as 5 to 6. Required the shares. Ans. $405.405', $270.270', and $324.324'. 13. A, B, and C, in partnership, lose $800. A's portion of the capital employed was of B's, and B's was of C's: what amount of the loss should be assigned to each? Observe that A's capital was to B's as 3 to 4, and B's to C's as 2 to 3; and that their respective losses should be in like proportion. Ans. To A $184.615', to B $246.153', and to C $369.230'. 14. A farmer divided 500 acres of land between his three sons, giving to the first 1 times as much as to the second, and to the second 14 times as much as to the third. How much did he give to each ? The first share was to the second as 1 to 1, and the second to the third as 1 to 1. Ans. To the 1st, 227, to the 2d, 1517, to the 3d, 12137 A. Partitive Ratios dependent on Time. $231. When different periods of time are involved in Partitive Proportion, the proportional terms to be used will be found by multiplying each term reckoned with time, by its time. The different periods of time, before multiplying, must all be in the same denomination. EXAMPLE. 15. Two persons, A and B, trade together; A ventures $200 for 7 months, and B $300 for 9 months. They gain $100; how must it be divided between them? $200 for 7 mo. is equivalent to (200X7) $1400 for 1 mo. and $300 for 9 mo. is equivalent to (300×9) $2700 for 1 mo. Having the time the same, that is, 1 month, in both cases, we take 1400 and 2700, or 14 and 27, for the proportional terms. without regard to time. Ans. A must have $34.146', B $65,853'. 16. Three persons rent a pasture for $20. A put in 20 sheep for 4 months, B 36 sheep for 3 months, and C 45 sheep for 2 months: how much of the rent should accordingly be paid by each ? Ans. A must pay $5.755', B $7.769', C $6,474'. 17. E, F, and G, in partnership, have made $400. What will be the share of each, supposing E's stock in the business to have been $500 for 10 months, F's $900 for 1 yr. 3 m., and G's $600 for 2 years? Ans. E's $60.790', F's $164.133', G's $175.075. 18. A and B invested capital in a joint speculation as follows: A put in at first $1000, and 6 months after $500 more; B advanced at first $2000, and 4 months after withdrew $600. At the end of 12 months the profits amounted to $800: what was each one's share of the same? A employed $1000 for 6 m.; $1000×6=$6000 for 1 m.; and $1500 for 6 m.; $1500×6=$9000 for 1 m. Then A's capital was equivalent to $15000 for 1 month. In like manner find the equivalent for B's capital. Ans. A's $350.877', B's $449.122'. At 19. A, B, C, and D engaged in partnership for 2 years. the outset A advanced $2000, B $3000, C and D each $4000. Six months afterwards A added $500 to his stock in the business, B $300, and C and D each withdrew $1000. At the end of the 2 years, the profits were found to be $800; to what amount of profit was each one entitled? Ans. A $157.024', B $213.223', C and D each $214.876'. MEDIAL PROPORTION. § 232. MEDIAL PROPORTION is Proportion applied to finding in what ratio to one another two or more quantities, at different rates of value, must be taken, to form a compound of a given medial or mean rate of value. For example, to find in what ratio to each other, rye at 37 cents per bushel, and oats at 25 cents per bushel, must be mixed together, that the mixture may be worth 30 cents per bushel. Note. This part of Arithmetic is commonly called ALLI GATION. RULE XLVI. $ 233. To find the ratio of Two or MORE QUANTITIES at different rates of value, for a COMPOUND of a given mean rate of value. 1. For two different rates-take the quantities inversely as the differences between their respective rates and the mean rate. 2. For three or more different rates-find the ratio for one rate which is less, and another which is greater, than the mean rate as above; then for one of these two rates and another, or for two others, in like manner; and so on, until all the different rates are included, and add together all the proportional terms found for the same rate. EXAMPLE. To find in what ratio to each other, rye at 37 cents, and oats at 25 cents a bushel, must be mixed together, that the mixture may be worth 30 cents a bushel. The differences between the rates of the two ingredients and the mean rate 30 cents, are for the rye 37-30-7, and for the oats 30-25=5. Then the quantity of RYE will be to that of OATS inversely as 7 to 5; and by converting the inverse into a direct proportion (§ 222), we find the quantity of RYE to that of OATS as 5 to 7; that is, the mixture must be in the ratio of 5 bushels of rye to 7 bushels of oats. In other words, since 5+7=12, f of the mixture must be RYE, and must be oats, whatever be the quantity of the mixture. |