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 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 RE

QUIRED 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 he

use?

Taking the differences as above, we find the proportions to be 2 of brandy to 5 of rum; consequently, 1 gallon or 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? what? if 10 lbs., what? if 20 lbs., what?

if 6 lbs.,

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, 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 $1'50 per bushel, rye at $1'16, corn at $75, and barley at $50, so as to sell the mixture at $ '84 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? What would be the quantity of the compound? Ans. to last, 7 ounces. 12. If he would have the whole compound consist of 15 -oz., how much must he use of each kind? · if of 30 ɔz., how much of each kind? if of 373 oz., how much? Ans. to the last, 5 oz. of the 3 first, and 22 oz. of the last.

Hence, when the quantity of the compound is given, we may say, As the sum of the PROPORTIONAL quantities, founa by the ABOVE RULE, is to the quantity REQUIRED, so is each PROPORTIONAL quantity, found by the rule, 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?

Then, 2 + 2 2: 20 lbs. at 8 cts.

We find the proportions to be, 2, 2, and 6. +6 = 10, and

10 100 ::

2: 20 lbs. at 10 cts. Ans.

6

: 60 lbs. at 14 cts.

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. 171 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.

QUESTIONS.

medial? 3.

the

1. What is alligation? 2. 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 given, 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 ?

DUODECIMALS.

T103. Duodecimals are fractions of a foot. The word s derived from the Latin word duodecim, which signifies twelve. A foot, instead of being divided decimally into ten equal 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 into twelve equal parts, called fourths, "); and so on to any extent.

In this way of dividing a foot, it is obvious, that

inch, or prime, is

1" second is

1" third is

of

of 12 of 12

of a foot.

fourth ish of th of th of 12, = 20136 of a foot. 1''''' fifth isofofofof 12,248532 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

Note. The marks, ', "', '"', &c., which distinguish the different parts, are called the indices of the parts or denomi

nations.

MULTIPLICATION OF DUODECIMALS.

Duodecimals are chiefly used in measuring surfaces and

solids.

1. How many square feet in a board 16 feet 7 inches long, and 1 foot 3 inches wide?

Note. Length × breadth = superficial contents, (¶ 25.)

OPERATION.
ft.

Length, 16
Breadth, 1 3'

4 1' 9"
16 771

Ans. 20

8' 9"

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 l' to be carried forward to its proper place. To multiply 16 feet by 3

is to take of 1649, that is, 48'; and the 1' which we reserved makes 49', 4 feet 1'; we therefore set down the 1', and carry forward the 4 feet to its proper place. Then, multiplying the multiplicand by the 1 foot in the multiplier, and adding the two products together, we obtain the Answer, 20 feet, 8', and 9".

The only difficulty that can arise in the multiplication of duodecimals is, in finding of what denomination is the product of any two denominations. This may be ascertained as above, and in all cases it will be found to hold true, that the product of any two denominations will always be of the denomination denoted by the sum of their INDICES. Thus, in the above example, the sum of the indices of 7' 3' is " sequently, the product is 21"; and thus primes multiplied by primes will produce seconds; primes multiplied by seconds produce thirds; fourths multiplied by fifths produce ninths, &c. It is generally most convenient, in practice, to multiply the multiplicand first by the feet of the multiplier, then by the inches, &c., thus:

; con

16 ft. X 1 ft. = 16 ft., and 7 × 1 ft. 7. Then, 16 ft. × 3′ = 48′ = 4 ft., and 7 × 3′ 21′′ = 1′ 9′′. The two products, added together, give for the Answer, 20 ft. 8' 9", as before.

2. How many solid feet in a block 15 ft. 8' long, 1 ft. 5' wide, and 1 ft. 4' thick?