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Take the difference between each price, and the mean price, and set them down alternately, and they will be the quantities required.

EXAMPLES.

1. How much wheat at 1 dol. rye at 66 cents, and barley at 50 cents per bushel, must be mixed to make the mixture worth 75 cents per bushel?

9+25=34

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barley 50

25 =25

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This is proved by Alligation Medial, thus, 34 bushels at 1 dol. is $34,00

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2. How much canary wine, at dol. 1,50 per gallon, Malaga at dol. 1,20, and white wine at I dol. must be mixed together, that the mixture may be worth dol. 1,30 per gallon?

2 gallons of Canary..

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3. How much sugar at 9 dols. per cwt. at 7 dols. per cwt. and at 10 dols. per cwt. must be mixed together, that the mixture may be worth 9 dols. per cwt?

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NOTE. Other cases in Alligation might be treated of, but as they are of no real advantage, they are omitted.

DUODECIMALS.

Duodecimals are fractions of a foot, or of an inch, or of any part of an inch, having 12 for their denominators, useful in measuring mechanics' work, bales, boxes, &c.

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Duodecimals are written thus: 5 7 2 5 8 &c. and are read thus: 5 feet, 7 inches, 2 seconds, 5 thirds, 8 fourths, &c.

NOTE. Some call the inches primes, and then the above

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would stand thus: 5 25 8 &c.

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MULTIPLICATION OF DUODECIMALS.
By some called Cross Multiplication.

Feet

NOTE 1. Feet multiplied by feet produce feet. multiplied by inches, or inches by feet, produce inches. Feet multiplied by seconds, or seconds by feet, produce seconds. Inches multiplied by inches, produce seconds. Inches multiplied by seconds, or seconds by inches, produce thirds. Seconds multiplied by seconds, produce fourths, &c.

NOTE 2. The multiplicand must be multiplied by each denomination in the multiplier, and the several products added together, to obtain the true product.

NOTE 3. Sometimes it is easier to take parts, as in practice, with the less denominations instead of multiplying,

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In this example I begin by multiplying by 3, saying 3 times 3 are 9 seconds, which I set down under the seconds; then say, 3 times 7 are 21 minutes, which is 9 more than 12; I set down the 9 under the minutes, and carry one for the 12, to the next denomination, saying 3 times 6 are. 18, and 1 that I carry will make it 19 feet; this being the highest denomination, I set down all over 10, viz. 9, and carry one for the ten to the next, saying, 3 times 7 are 21, and 1 I carry makes 22; this being the left hand, or last figure to multiply, I set the whole down, viz. 22. I now come to the figure 6 in the multiplier, standing in the place of inches, and say 6 times 3 are 18, this is 6 over 12, and as inches and seconds multiplied together, produce thirds, I set down 6 in the place of thirds, and carry one to the next, &c.

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The several products added together will be the true product.

NOTE. Sometimes it will be easier to take parts with inches, &c. instead of multiplying by them.

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5. A floor is 18 feet, 7 inches long, and 1.6 feet, 9 inches wide, what is the area of the floor?

Ans. 311 F. 3' 3". 6. What is the area of a floor, which is 26 feet, 3 inches long, and 9 feet, 9 inches wide?

Ans. 255 F. 11' 3".

7. A roof of a building measures 52 feet, 7 inches, by 56 feet, 3 inches, how many squares of 100 superficial feet each does this roof contain ?*

*

Ans. 29 squares, 57 F. 9' 9".

Flooring and roofing are done by the square of 100 superficial feet. Some partitions and ceilings are done by the square yard of 9 superficial feet each: also, plastering, paving, and painting, are done by the square yard.

8. A mason has paved a part of a street, which measures in length 236 feet, 8 inches, and in breadth 37 feet 8 inches; how many square yards does it contain ?

Ans. 990 yds. 4 F. 5'.

9. A carpenter has ceiled the sides of a store, the walls of which are 9 feet, 10 inches high, and 150 feet, 6 inches about, and to be paid by the square yards; how many yards must he be paid for? Ans. 164 yds. 3 F. 11".

10. A mason has plastered 3 rooms, the ceiling of each is 20 feet by 16 feet, and the walls of each, 9 feet high, and 73 feet about; there is to be 90 yards deducted for doors, windows, chimneys, &c. from the whole; how many yards must he be paid for?

Ans. 251 yds. 1 F. 6'.

11. A painter has painted the side of a building 40 feet long, 27 feet high, (no deduction) how much will the painting come to, at 40 cents per yard?

Ans. $48,88,88.

DUODECIMALS APPLIED TO CUBIC MEASURE.

Stone walls of cellars, &c. are laid by the perch of 243 cubic feet each, the number of perches are found by first finding the number of solid (or cubic) feet; then multiply the cubic feet by 4, to reduce them to quarters, and divide the product by 99 (the number of quarters in 244) the quotient will be perches; the remainder so many ninety-ninths of a perch.

The cubic feet in a wall, or any other thing, are found by multiplying the length, breadth, and height together.

EXAMPLES.

1. In a cellar wall which is 9 feet high, 24 feet thick, and the sides and ends together, were 131 feet, 5 inches, in length; how many perches did it contain?

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