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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 pro portions, or quantities of each, must he take?
Had the price of the mixture required exceeded 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 com pound required. In the above example, the price of the mixture is 42 cents, and the price of the oats 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.
2. A merchant has several kinds of tea; some at 8 shillings, some at 9 shillings, some at 11 shillings, and some at 12 shillings per pound; what proportions of each must he mix, that he may sell the compound 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
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 each 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 :
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 cents, 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 cents to 6 lbs. at 14 cents.
Note. As these quantities only express the proportions of each kind, it is plain, 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 1 gallon of brandy worth 16 s. with rum at 9 s. per gallon, so that the mixture may be worth 11 s. per gallon, how much rum must be
Taking the differences as above, we find the proportions to be 2 of brandy to 5 of rum; consequently, 1 gallon of brandy will require 24 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 7 lbs. of the third. 8. How many gallons of water, of no value, must be mixed with 60 gallons of rum, worth 80 cents per gallon, to reduce its value to 70 cents per gallon? Ans. 84 gallons.
9. A man would mix 4 bushels of wheat, at $150 per bushel, rye at $1'16, corn at $75, and barley at $50, so as to sell the mixture at $ 34 per bushel; how much of each may he use?
10. A goldsmith would mix gold 17 carats fine with some 19, 21, and 24 carats fine, so that the compound may be 22 carats fine; what proportions of each must he use? Ans. 2 of the 3 first sorts to 9 of the last. 11. If he use 1 oz. of the first kind, how much must he use of the others? Wha would be the quantity of the compound? Ans. to last, 72 ounces. 12. If he would have the whole compound consist of 15 oz., how much must he use of each kind? if of 30 oz., how much of each kind? -if of 374 oz., how much? Ans. to the last, 5 oz. of the 3 first, and 224 oz. of the last.
Hence, when the quantity of the compound is given, we may say, As the sum of the PROPORTIONAL quantities, found by the ABOVE RULE, is to the quantity REQUIRED, so is each PROPORTIONAL quantity, found by the rube, to the REQUIRED quantity of EACH.
13. A man would mix 100 pounds of sugar, some at 8 cents, some at 10 cents, and some at 14 cents per pound, so that the compound may be worth 12 cents per pound; how much of each kind must he use?
We find the proportions to be, 2, 2, and 6. Then, 2 +2 +6 = 10, and
2: 20 lbs. at 18cts. (Ans.
10: 100 ::
6: 60 lbs. at 14 ets.
14. How many gallons of water, of no value, must be mixed with brandy at $1'20 per gallon, so as to fill a vessel of 75 gallons, which may be worth 92 cents per gallon? Ans. 174 gallons of water to 574 gallons of brandy.
15. A grocer has currants at 4 d., 6 d., 9d. and 11 d. per lb.; and he would make a mixture of 240 bls., so that the
mixture may be sold at 8 d. per lb.; how many pounds of
each sort may he take?
Ans. 72, 24, 48, and 96 lbs., or 48, 48, 72, 72, &c. Note. This question may have five different answers.
1. Whet is alligation? 2. medial? 3. rule for operating? 4. What is alligation alternate? 5. When the price of the mixture, and the price of the several simples, are given, how do you find the proportional quantities of each simple? 6. When the quantity of one simple is giver, how do you find the others? 7. When the quantity of the whole compound is given, how do you find the quantity of each simple?
T 103. Duodecimals are fractions of a foot. The word is derived from the Latin word duodecim, which signifies twelve. A foot, instead of being divided decimally into ten squal parts, is divided duodecimally into twelve equal parts,
called inches, or primes, marked thus, ('). Again, each of these parts is conceived to be divided into twelve other equal parts, called seconds, ("). In like manner, each second is conceived to be divided into twelve equal parts, called thirds, (""); each third irto twelve equal parts, called fourths, (); and so on to any extent.
In this way of dividing a foot, it is obvious, that
1' inch, or prime, is
of a foot.
1" second is
of a foot.
1" third is
of th of 12,
1728 of a foot. 1 fourth ish of th of th of 20136 of a foot. 1 fifth is of of th of =237 of a foot, &c. Duodecimals are added and subtracted in the same manner as compound numbers, 12 of a less denomination making 1 of a greater, as in the following
OPERATION. ft. Length, 16 7 Breadth, 1 3'
4 1' 9/ 16 7/
Ans. 20 8' 9"
Note. The marks, ', ", "", "", &c., which distinguish the different parts, are called the indices of the parts or denominations.
1' inch or prime,
MULTIPLICATION OF DUODECIMALS.
Duodecimals are chiefly used in measuring surfaces and
1. How many square feet in a board 16 feet 7 inches long, and 1 foot 3 inches wide?
Note. Length x breadth
superficial contents, ( 25.) 7 inches, or primes, of a foot, and 3 inches of a foot; consequently, the product of 7' x 3' of a foot, that is, 21′′ =1' and 9"; wherefore, we set down the 9", and reserve the 1' to be carried forward to its proper place. To multiply 16 feet by 3'