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is called a Compound Proportion. The rulc by which the fourth term of a compound proportion is found, is called the Double Rule of Three, and may be understood from the precede ing analysis.
RULE. 202. Make that number, which is of the same kind as the required answer, the second term. Take any two of the remaining terms which are of the same kind, and place one for a first, and the other for a third term, as directed in the Single Rule of Three (198); then take any other two of the same kind, and place them in the same way, and so on till all are used. Multiply the product of the third terms by the second term, and divide the result by the product of the first terms: the quotient will be the required answer.
QUESTIONS FOR PRACTICE. 3. If 120 bushels of oats will 7. If the intereet of $45 for serve 14 horses 56 days, how 6 months be $1.80, what is the many days will 94 bushels | rate per annum? serve 6 horses ?
Ans. 8 per cent Ans. 1021. days.
8. If 8 men spend 48 dolls. 4. If $100 gain $6 in 12 | in 24 weeks, how much wil months, what will be the inter- | 40 men spend in 48 weeks, at est of $350 for 2 years and 7 the same rate ? Ans. $480. months ?
9. If the freight of 5 tierces 2y. 7mo.31mo.
of salt, each weighing 5} cwt. 100 : 6 ::: 350
80 miles, cost $28, what will 12: 31
be the freight of 75 sacks of Ans. $54.25.
salt, each weighing 21 cwt,
150 miles 5. If a sum of money at 6
Ans. $322.15911 per cento, simple interest, double in 200 months, what will
10. A man lent $350 to rebe the interest of $300 for 8 ceive interest, and when it months?
had continued 9 months, he $
received principal and interest 100-: 100 :: 300 together, $360.50; at what 200.:
rate per cent. did he lend his "Ans. $12.
Ans. 4 per cent. 6. If the transportation of 11. With how many pounds 20cwt. 37 miles, cost 16 dolls., sterling could I gain £5 per what will the transportation annum, if with £450 I gain of 12cwt. 50 miles cost ? in 16 months, £30 ? Ans. $12.972.
$20 B's share;} Ang
ANALYSIS. 203. 1. Two men, A and B, trade in company; A puts in $100, and B 8200, and they gain $30. What is each man's sharo of the gain?
Each man's gain must evidently have the same relation to the whole gain, that the money which he puts in, has to the whole amount put in. In other words, the whole amount put in, will be to the whole gain as each man's share of the amount put in, is to his share of the gain, i. e.
$300 : $30 : :
S $100 $10 A's share. 204. 2. A and B hired a pasture for 12 dollars ; A put in 3 cows for 8 weeks, and B put in 4 cows for 9 weeks; what part of the rent ought each to pay ?
Three cows 8 weeks are equal to 1 cow (3x8=) 24 weeks, and 4 cows 9 weeks are equal to 1 cow (4x9=) 36 weeks; their shares, then, of the pasturage are 24 weeks and 36 weeks, equal to 60. weeks' pasturage. Then, as the whole pasturage is to the whole rent, so is each man's share of the pasturage to his share of the rent; that is, 60 w. : $12 : : $3X8–24w. : $4.80 A's share. Ans.
4X9=36w. : $7.20 B's share. To prove the correctness of the work, we add together the shares, and find them to amount to (4.80+7.205) $12, the whole rent (5-1).
DEFINITIONS. 205. Money, or property employed in trade, is called capital, or stock,-gain to be divided, the dividend. Fellowship is a general rule, by which merchants, or others, trading in company with a joint stock, compute each person's particular share of the gain or loss.
RULE. 206. When the stocks are employed for equal times, say: As the whole stock : is to the whole gain or loss :: so is each man's share of the stock : to his share of the gain or loss (203). When the times are unequal, multiply each man's stock by the time of its continuance in trade; then say, As the sum of the products : is to the whole gain, or loss :: so is each man's product : to his share of the gain, or loss (204).
QUESTIONS FOR PRACTICE. 3. A and B made a joint 4. Three persons make a stock of $500, of which A put joint stock, of which each puts in $350, and B $150; they in an equal share ; A continues gain $75; what is each man's his stock in trade 4 months, B share of the gain?
his 6 months, and C his 10
Ans. months, and they gained $480 500 : 75 : :
what was each man's share ? 150 : 22.50 B's.
144 B's. And 75.00 pril
240 C's S
5. A, B and C companied; end of the 18 months they had À put in £480, B £680, C gained £263; what is each £840, and they gained £1010; man's share ? what is each man's share ?
9. Three men hire a pasture
among men, so that their shares shall for $100; A puts in 40 oxen be as 1, 2, 3, and 4.
for 20 days, B 30 oxen for 40
days, and C 50 oxen for 10 32
days; how much must each Ans. 48
$100 proof. sons; to the eldest he gave
10. Three farmers hired a $560, to the next $500 and to the other $450; but when his 5 cows for 4 months, B put
pasture for $60,50. A put in debts were paid, there were
in 8 for 5 months, and C put $950 left; what was each
in 9 for 6 months; how much son's share ?
must each pay of the rent ? $352.317+1st.
$11.25 A's. 1 314.569 +2d. Ans.
20.00 B's. Ans. 283.112 -30.)
29.25 C's. 8. Two merchants entered into partnership for 18 months.
$60.50 proof A at first put in £100, and at the end of 8 months put in
11. D and E companied ; £50 more; B at first put in :D put in $125, and took out £275, and at the end of four is of the gain ; what did E inonths took out £70; at the I put in?
ANALYSIS. 207. 1. If I mix 6 quarts of currants, which are worth 8 cents a quart, with 2 quarts worth 12 cents a quart, what will a quart of the mixture bo worth ?' (60)
Six quarts at 8 cents are worth (8X6=) 48 cents, and 2 quarts at 12 canls are worth (12x2=) 24 cents; then 48+-2472 cents, the world of the whole mixture, and 7%;8(6+2, the whole mixture) =9 cents, the worth of 1 quart of the mixture. When the prices and quantities of the simples are given, and it is required to find the price of a given quantity of the mixture, as in the preceding example, it is called
RULE. 208. Multiply cach quantity by its price, and divide the sum of the products by the sum of the quantities, the quotient will be the rate of the compound required.
QUESTIONS FOR PRACTICE. 2. If I mix 8 bushels of lons of wine at 4s. 10d. a galwheat at $1.20 per bushel, 12 lon, with 12 gallons at 5s.6d., bushels of rye at 60 cents, and and 8 at 6g. 3 d. a gallon; 10 bushels of corn at 50 cents, what is a gallon of the mixture together; what is a bushel of worth?
Ans. 5s. 7d. the mixture worth?
4. If 5lb. of tea at 6s. per lb., 1.20 60 50 8
8lb. at 5s., and 41b. at 48. 6d., 8 12 .10 12
be mixed together, what is a
10 9.60 7.20 5.00
pound of the mixture worth?
Ans. 5s. 2 d. 7.20
30 sum of 5.00 (the quantities. 5. A goldsmith melted to
gether 10 oz. of gold 20 carats 21.80 sum of prod.
fine, 8 oz. 22 carats fine, and Then 30 )21.80( $0.724 per 1 lb. 8 oz. 21 carats fine ; what bushel, Ans.
is the fineness of the mixture? 3. A merchant mixed 6 gal- Ans. 2015 carats fine.
ALLIGATION ALTERNATE. 209. When the prices of the simples, and also the price, or rate of the mixture, are given, the method of finding the proportion, or quantities of the several simples, is called Alligation Alternate.
1. A person has lea worth 40 cents a pound, which he wishes to mix with la worth 60 cents a pound, in such manner that the mixture shall be worth 50 cents a pound; in what proportion must it be mixed ? Ans. Equal quantities of each for the price of one kind exceeds the mean just as much as the price of the other falls short of it, the difference between the given rate and the mean being 5 in each case.
2. In what proportion must I mix currants worth 9 cents a pound, with churrants worth 12 cents a pound, in order that the mixture may be worth 10 cents a pound ? Here a pound at 9 cents falls one cent short of the mean, and a pound at 12 cents exceeds the mean 2 cents; hence, 2 lb. at 9 cents will fall short of the mean by the same quantity that ond' lb. at 12 cents exceeds it ; we must therefore take twice as mar of the 9 cent cuirants as we do of Haşe worth 12 cents, in order that ibe mixture may be worlle 10 cents.
From the foregoing examples it appears, that the less the prico of any simple differs from that of the mixture, the quantity required of that simple to form the mixture will be proportionately greater, and the greater the dirference the less the quantiiy; and that the differences between the values of the simples and the given value of a mixture of those simples, mutually exchanged, express the relative quantities of those simples necessary to make a mixture of the given value. Exchanging these differences in the above examples, we have in the first, 5 lb. at 40 cents, with 5 lb. at 60 cts., or equal quantities of each; and in the second, we have 2 lb. at 9 cts. with 1 lb. at 12.
RULE. 210. Reduce the rates of all the simples to the same de nomination, and write them in a column with the rate of the required compound at the left hand. Connect each rate which is less than the rate of the compound, with one that is greater, and each that is greater with one that is less. Write the difference between each rate and that of the compound against the number with which it is connected. Then if only one difference stand against any rate, it will express the relative quantity to be taken of that rate ; but if there be more than one, their sum will express the relative quantity to be taken of that rate in making up the compound.
QUESTIONS FOR PRACTICE. 3. A farmer wishes to mix rye worth 4s., corn worth 387.. barley worth 2s. 6d., and oats worth 2s., so that the mixture may be worth 23. 10d. per bushel ; what proportion must he take of each sort? 2s. 24d.
d. bu. 29. 6d. =30d.
4 = 4 48
10+ 4=14 4. A merchant would mix 5. How must barley at 40 wines at 14s., 15., 19s. and cents, rye at 60 cents, and 228. a gallon, so that the mix-wheat at 80 cents a bushel, be ture may be worth 188. a gal- mixed together, that the com lon; 'how much must he take pound may be worth 624 cente, af each sort?
a bushel ? 4 gal. at 14s.
17) bush. barley.
17) bush. rye,
25 bush. wheat. gal. at 228.
4 corn, -10 rye.