SUPPLEMENT TO THE SINGLE RULE OF THREE. QUESTIONS. 1. What is proportion? 2. How many numbers are required to form a ratio ? 3. How many to form a proportion? 4. What is the first term of a ratio called ? 5. the sec ond term ? 6. Which is taken for the numerator, and which for the denominator of the fraction expressing the ratio ? 7. How may it be known when 4 numbers are in proportion? 8. Having three terms in a proportion given, how may the fourth term be found ? 9. What is the operation, by which the fourth term is found, called ? 10. How does a ratio become inverted ? 11. What is the rule in proportion ? 12. In what denomination will the fourth term, or answer, be found ? 13. If the first and second terms contain different denominations, what is to be done ? 14 What is compound proportion, or double rule of three ? 15. Rule ? EXERCISES. 1, If I buy 76 yds. of cloth for $113-17, what does it cost per ell English ? Ans. $16861. 2. Bought 4 pieces of Holland, each containing 24 ells English, for $96; how much was that per yard ? Ans. $0'80. 3. A garrison had provision for 8 months, at the rate of 15 ounces to each person per day; how much must be allowed per day, in order that the provision may last 94 months ? Ans. 12 12 oz. 4. How much land, at $2'50 per acre, must be given in exchange for 360 acres, at $3-75 per acre Ans. 540 acres. 5. Borrowed 185 quarters of corn when the price was 19 s.; how much must I pay when the price is 17 s. 4 d. ? Ans. 2024 1 6. A person, owning of a coal mine, sells * of his share for 171 %; what is the whole mine worth? Ans. 380 £. 7. If of a gallon cost $ of a dollar, what costs of a tun? Ans. $140. 8. At 14 £. per cwt., what cost 3} lbs. ? Ans. 9. If 4f cwt. can be carried 36 miles for 35 shillings, how many pounds can be carried 20 miles for the same money ? Ans. 907 lbs. 10. If the sun appears to move from east to west 360 degrees in 24 hours, how much is that in each hour? in each minute in each second ? Ans. to last, 15" of a deg. 11. If a family of 9 persons spend $450 in 5 months, how much would be sufficient to maintain them 8 months if 5 persons more were added to the family? Ans. $1120. Note. Exercises 14th, 15th, 16th, 17th, 18th, 19th, and 20th,“ Supplement to Fractions," afford additional examples in single and double proportion, should more examples he thought necessary. 105 d. FELLOWSHIP. 1 98. 1. Two men own a ticket; the first owns t, and the second owns of it; the ticket draws a prize of 40 dollars; what is each man's share of the money ? 2. Two men purchase a ticket for 4 dollars, of which one pays 1 dollar, and the other 3 dollars; the ticket draws 40 dollars; what is each man's share of the money? 3. A and B bought a quantity of cotton ; A paid 100 dollars, and B 200 dol. lars; they sold it so as to gain 30 dollars; what were their respective shares of the gain? The process of ascertaining the respective gains or losses of individuals, engaged in joint trade, is called the Rule of Fellowship. The money, or value of the articles employed in trade, is called the Capital, or Stock; the gain or loss to be shared is called the Dividend, It is plain, that each man's gain or loss ought to have the same relation to the whole gain or loss, as his share of the stock does to the whole stock. Hence we have this RULE: As the whole stock : to each man's share of the stock : ; the whole gain or loss ; his share of the gain or loss. 4. Two persons have a joint stock in trade; A put in $250, and B $350; they gain $400; what is each man's share of the profit ? OPERATION. $350 600 ; 250 : : 400 : 166-6663 dollars, A's gain. Whole stock, $600 600 : 350 : : 400 : 233-333} dollars, B's gain. The pupil will perceive that the process may be contracted by cutting off an equal number of ciphers from the first and second, or first and third terms ; thus, 6 : 250 :: 4 : 166-6663, &c. It is obvious, the correctness of the work may be ascertained by finding whether the sums of the shares of the gains are equal to the whole gain; thus, $166-666; + $233-333} $400, whole gain. 5. A, B and C trade in company; A's capital was $175, B's $200, and C's $500; by misfortune they lose $250; what loss must each sustain ? $ 50 A's loss. Ans. $ 57'1429, B's loss. $142-857 77, C's loss, 6. Divide $600 among 3 persons, so that their shares may be to each other as 1, 2, 3, respectively. Ans. $100, $200, and $300. 7. Two merchants, A and B, loaded a ship with 500 hhds. of rúm ; A loaded 350 hhds., and B the rest; in a storm, the seamen were obliged to throw overboard 100 hhds.; how much must each sustain of the loss ? Ans. A 70, and B 30 hhds. 8. A and B companied ; A put in $45, and took out 3 of the gain ; how much did B put in ? Ans. $30. Note. They took out in the same proportion as they put in; if 3 fifths of the stock is $15, how much is 2 fifths of it? 9. A and B companied, and trade with a joint capital of $400; A receives for his share of the gain f as much as B; what was the stock of each ? Ans. { $2666667; B's stock. 10. A bankrupt is indebted to A $780, to B $460, and to C $760; his estate is worth only $600; how must it be divided ? Note. The question evidently involves the principles of fellowship, and may be wrought by it. Ans. A $234, B $138, and C $228. 11. A and B venture equal stocks in trade, and clear $164: by agreement, A was to have 5 per cent. of the profits, because he managed the concerns; B was to have but 2 per cent.; what was each one's gain, and how much did A receive for his trouble ? Ans. A's gain was $117'1424, and B's $16-8577, and A received $70-285 for his trouble. 12. A cotton factory, valued at $12000, is divided into 100 shares; if the profits amount to 15 per cent. yearly, what will be the profit accruing to 1 share ? to 2 shares ? to 5 shares ? to 25 shares ? Ans. to the last, $450. 13. In the above-mentioned factory, repairs are to be made which will cost $340 ; what will be the lax on each share necessary to raise the sum ? on 2 shares ? on 3 shares ? on 10 shares ? Ans. to the last, $34. 14. If a town raise a tax of $1850, and the whole town be valued at $37000, what will that be on $1 ? What will be the tax of a man whose property is valued at $1780 ? Ans. $05 on a dollar, and $89 on $1780. 1199. In assessing taxes, it is necessary to have an inventory of the property, both real and personal, of the whole town, and also of the whole number of polls; and, as the polls are rated at so much each, we must first take out from the whole tax what the polls amount to, and the remainder is to be assessed on the property. Wemay then find the tax upon 1 dollar, and $30 make a table containing the taxes on 1, 2, 3, &c., to 10 dollars; then on 20, 30, &c., to 100 dollars; and then on 100, 200, &c., to 1000 dollars. Then, knowing the inventory of any individual, it is easy to find the tax upon his property. 15. A certain town, valued at $64530, raises a tax of $2259'90 ; there are 540 polls, which are taxed $'60 each; what is the tax on a dollar, and what will be A's tax, whose real estate is valued at $1340, his personal property at $874, and who pays for 2 polls ? 540 x 60 = $324, amount of the poll taxes, and $2259-90 $324 1935'90, to be assessed on property. $64530 : $1935-90 : : $1 : '03; or, 1935'90 '03 tax on $1. 64530 TABLE. dolls. dolls. Tax on 1 is '03 Tax on 10 is '30 Tax on 100 is 3 200 is 64 300 is 9 4 is '12 40 is 1620 400 is 12 500 is 15 600 is 189 700 is 21 • 800 is 24 900 is 27 1000 is 300 Now, to find A's tax, his real estate being $1340, I find, by the table, that The tax on $1000 is The tax on 300 9 The tax on 40 1620 Tax on his real estate $40'20 In like manner I find the tax on his personal property to be 26'22 2 polls at '60 each, are 1'20 Amount, $67-62 16. What will B's tax amount to, whose inventory is 874 dollars real, and 210 dollars personal property, and who pays for 3 polls ? Ans. $31'32. 17. What will be the tax of a man, paying for 1 poll, whose property is valued at $3482 ? at $768? at $940 ? at $4657 ? Ans, to the last, $140'31. 18. Two men paid 10 dollars for the use of a pasture 1 month; A kept in 24 cows, and B 16 cows; how much should each pay ? 19. Two men hired a pasture for $10; A put in 8 cows 3 months, and B put in 4 cows 4 months ; how much should each pay? T 100. The pasturage of 8 cows for 3 months is the same as of 24 cows for 1 month, and the pasturage of 4 cows for 4 months is the same as of 16 cows for 1 month. The shares of A and B, therefore, are 24 to 16, as in the former question. Hence, when time is regarded in fellowship,--Multiply each one's stock by the time he continues it in trade, and use the product for his share. This is called Double Fellowship. Ans. A $6, and B $4. 20. A and B enter into partnership; A puts in $100 6 months, and then puts in $50 more; B puts in $200 4 months, and then takes out $80; at the close of the year they find that they have gained $95; what is the profit of each? Ans. $43-711, A's share; $516288, B's share. 21. A, with a capital of $500, began trade Jan. 1, 1826, and, meeting with success, took in B as a partner, with a capital of $600, on the first of March following; four months after, they admit C as a partner, who brought $800 stock; at the close of the year, they find the gain to be $700 ; how must it be divided among the partners ? Ans. $250, A's share; $250, B's share ; $200, C's share. QUESTIONS. 1. What is fellowship? 2. What is the rule for operating? 3. When time is regarded in fellowship, what is it called ? 4. What is the method of operating in double fellowship ? 5 How are taxes assessed ? 6. How is fellowship proved ? . at 70 : 210 -1016 cts. Ans. Ans. 1211. ALLIGATION. 101. Alligation is the method of mixing two or more simples, of different qualities, so that the composition may be of a mean or middle quality. When the quantities and prices of the simples are given, to find the mean price of the mixture compounded of them, the process is called Alligation Medial. 1. A farmer mixed together 4 bushels of wheat, worth 150 cents per bushel, 3 bushels of rye, worth 70 cents per bushel, and 2 bushels of corn, worth 50 cents per bushel; what is a bushel of the inixture worth? It is plain, that the cost of the whole, divided by the number of bushels, will give the price of one bushel. 4 bushels, at 150 cents, cost 600 cents. ojo 9 bushels cost 910 cents. 2. A grocer mixed 5 lbs. of sugar, worth 10 cents per lb., 8 lbs. worth 12 cents, 20 lbs. worth 14 cents; what is a pound of the mixture worth? 3. A goldsrnith melted together 3 ounces of gold 20 carats fine, and 5 ounces 22 carats fine; what is the fineness of the mixture ? Ans. 211. 4. A grocer puts 6 gallons of water into a cask containing 40 gallons of rum, worth 42 cents per gallon; what is a gallon of the mixture worth? Ans. 361 cents. 5. On a certain day the mercury was observed to stand in the thermometer as follows : 5 hours of the day it stood at 64 degrees ; 4 hours, at 70 degrees ; 2 hours, at 75 degrees, and 3 hours, at 73 degrees; what was the mean tem. perature for that day? It is plain this question does not differ, in the mode of its operation, from the former. Ans. 69_3 degrees. 17 1 102. When the mean price or rate, and the prices or rates of the several simples are given, to find the proportionate quantities of each simple, the process is called Alligation Alternate: alligation alternate is, there. fore, the reverse of alligation medial, and may be proved by it. 1. A man has oats worth 40 cents per bushel, which he wishes to mix with corn worth 50 cents per bushel, so that the mixture may be worth 42 cents per bushel ; what proportions, or quantities of each, must he take ? Had the price of the mixture required etceeded the price of the oats by just as much as it fell short of the price of the corn, it is plain, he must have taken equal quantities of oats and corn; had the price of the mixture exceeded the price of the oats by only as much as it fell short of the price of the corn, the compound would have required 2 times as much oats as corn; and in all cases, the less the difference between the price of the mixture and that of one of the simples, the greater must be the quantity of that simple in proportion to the other; that is, the quantities of the simples must be inversely as the differences of their prices from the price of the mixture; therefore, if these differences be mutually exchanged, they will, directly, express the relative quantities of each simple necessary to form the compound required. In the above example, the price of the mixture is 42 cents, and the price of the onts is 40 cents; consequently, the difference of their prices is 2 cents: the price of the corn is 50 cents, which differs from the price of the mixture by 8 cents. Therefore, by exchanging these differences, we have 8 bushels of oats to 2 bushels of corn, for the proportion required. Ans. 8 bushels of oats to 2 bushels of corn, or in that proportion. The correctness of this result may now be ascertained by the last rule ; thus, the cost of 8 bushels of oats, at 40 cents, is 320 cents; and 2 bushels of corn, at 50 cents, is 100 cents; then, 320 + 100 = 420, and 420, divided by the number of bushels, (8 + 2) = 10, gives 42 cents for the price of the mixture Or, 12 2. A merchant has several kinds of tea; some at 8 shillings, some at 9 shil. lings, some at 11 shillings, and some at 12 shillings per pound; what propor. tions of each must he mix, thai e may sell the coinpound at 10 shillings per pound? Here we have 4 simples; but it is plain that what has just been proved of two will apply to any number of pairs, if in each pair the price of one simple is greater, and that of the other less, than the price of the mixture required. Hence we have this RULE. The mean rate and the several prices being reduced to the same denomination,-connect with a continued line each price that is less than the mean rate with one or more that is greater, and cach price greater than the mean rate with one or more that is less. Write the difference between the mean rate, or price, and the price of each simple opposite the price with which it is connected; (thus the difference of the two prices in each pair will be mutually exchanged ;) then the sum of the differences, standing against any price, will express the relative quantity to be taken of that price. By attentively considering the rule, the pupil will perceive that there may be as many different ways of mixing the simples, and consequently as many different answers, as there are different ways of linking the several prices, We will now apply the rule to solve the last question : OPERATIONS, 8 -2 + 9s. 9 1 10 . Ans. Ans. lls. 11 +2=3 12s. -2 Here we set down the prices of the simples, one directly under another, in order, from least to greatest, as this is most convenient, and write the mean rate, (10 s.,) at the left hand. In the first way of linking, we find that we may take in the proportion of 2 pounds of the teas at 8 and 12 s. to 1 pound at 9 and 11 s. In the second way, we find for the answer, 3 pounds at 8 and 11 s. to 1 pound-at 9 and 12 s. 3. What proportions of sugar, at 8 cențs, 10 cents, and 14 cents per pound, will compose a mixture worth 12 cents per pound ? Ans. In the proportion of 2 lbs. at 8 and 10 cts. to 6 lbs. at 14 cts. Note. As these quantities only express the proportions of each kind, it is piain, that a compound of the same mean price will be formed by taking 3 times, 4 times, one half, or any proportion, of each quantity. Hence, When the quantity of one simple is given, after finding the proportional quantities, by the above rule, we may say,--As the PROPORTIONAL quantity : is to the given quantity :: so is each of the other PROPORTIONAL quantities : to the REQUIRED quantities of each. 4. If a man wishes to mix i gallon of brandy, worth 16 s. with rum at 9 s. per gallon, so that the mixture may be worth ii s. per gallon, how much rum must he use ? Taking the differences as above, we find the proportions to be 2 of brandy to 5 of rum ; consequently, I gallon of brandy will require 2 gallons of rum. Ans. 24 gallons. 5. A grocer has sugars worth 7 cents, 9 cents, and 12 cents per pound, which he would mix so as to form a compound worth 10 cents per pound; what must be the proportions of each kind ? Ans. 2 lbs. of the first and second to 4 lbs. of the third kind. 6. If he use 1 lb. of the first kind, how much must he take of the others ? if 4 lbs., what? if 6 lbs., what? if 10 lbs., what ? if 20 lbs., what? Ans, to the last, 20 lbs. of the second, and 40 of the third. 7. A merchant has spices at 16 d. 20 d. and 32 d. per pound; he would mix 5 pounds of the first sort with the others, so as to form a compound worth 24 d. per pound; how much of each sort must he use? Ans. 5 lbs. of the second, and 74 lbs. of the third. 8. How many gallons of water, of no value, inust be mixed with 60 gallons of rum, worth 80 cents per gallon, to reduce its value to 70 cents per gallon ? Ans. 84 gallons. |