28. When tea is sold at $1.25 per lb. there is lost 25 per cent.; what would be the gain or loss per cent. if it should be sold at $1.40 per lb. ? Ans. 16 per cent. loss.

29. A exchanges with B 50 lbs. of indigo at $1.00 per lb. cash, and in barter $1.35; but he is willing to lose 12 per cent. to have ready money. What is the cash price of 1 yard of cloth delivered by B at $5.00 per yard to equal A's bartering price, and how many yards were delivered?

Ans. $4.20.3 cash price of 1 yard. 7 yards delivered by B.

SECTION LI.

DUODECIMALS AND CROSS MULTIPLICATION.

DUODECIMALS are so called, because they decrease by twelves from the place of feet towards the right.

Inches are called primes and are marked thus '; the next division after is called seconds, marked thus"; the next is thirds, marked thus "; and so on.

Duodecimals are commonly used by workmen and artificers in finding the contents of their work.

EXAMPLES.

1. Multiply 6 feet 8 inches by 4 feet 5 inches.

6 8' 4 5

26 8'

OPERATION.

29 4"

29 5' 4"

As feet are the integers or units, it is evident, that feet multiplied by feet will produce feet; and as inches are twelfths of a foot, the product of inches by feet will be twelfths of a foot. For the same reason, inches multiplied by inches will produce twelfths of an inch, or one hundred and fortyfourths of a foot. Hence we deduce the following

RULE.

12

Under the multiplicand write the same names or denominations of the multiplier; that is, feet under feet, inches under inches, &c. Multiply each term in the multiplicand, beginning at the lowest by the feet of the multiplier, and write each result under its respective term, observing to carry a unit for every from each denomination to its next superior. In the same manner the multiplicand by the inches of the multiplier, and write the result of each term one place further towards the right of those in the multiplicand. Proceed in the same manner with the

seconds and all the rest of the denominations, and the sum of all the lines will be the product required.

The denomination of the particular products will be as follows.
Feet multiplied by feet, give feet.

Feet multiplied by primes, give primes.
Feet multiplied by seconds, give seconds.
Primes multiplied by primes, give seconds.
Primes multiplied by seconds, give thirds.
Primes multiplied by thirds, give fourths.
Seconds multiplied by seconds, give fourths.
Seconds multiplied by thirds, give fifths.
Seconds multiplied by fourths, give sixths.
Thirds multiplied by thirds give sixths.
Thirds multiplied by fourths, give sevenths.
Thirds multiplied by fifths, give eighths, &c.

2. Multiply 4 feet 7' by 6 feet 4'.
3. Multiply 14 feet 9' by 4 feet 6'.
4. Multiply 4 feet 7' 8" by 9 feet 6'.
5. Multiply 10 feet 4' 5" by 7 feet 8′ 6′′.

Ans. 29ft. 0′ 4′′.

Ans. 66ft. 4' 6". Ans. 44ft. 0' 10".

Ans. 79ft. 11' 0" 6" 6'""'.

6. Multiply 39 feet 10' 7" by 18 feet 8' 4".

Ans. 745 ft. 6′ 10′′ 2′′ 4"".

7. How many square feet in a floor 48 feet 6' long 24 feet 3' broad? Ans. 1176ft. 1' 6". 8. What are the contents of a marble slab, whose length is 5 feet 7', and breadth 1 foot 10' ? Ans. 10ft. 2′ 10′′.

9. The length of a room being 20 feet, its breadth 14 feet 6 inches, and height 10 feet 4 inches; how many yards of painting are in it, deducting a surplus of 4 feet by 4 feet 4 inches, and 2 windows, each 6 feet by 3 feet 2 inches. Ans. 732 yards.

10. Required the solid contents of a wall 53 feet 6 inches long, 10 feet 3 inches high, and 2 feet thick ? Ans. 1096ft. 9'. 11. There is a house with four tiers of windows, and 4 windows in a tier; the height of the first is 6 feet 8 inches; of the second 5 feet 9 inches; of the third 4 feet 6 inches; of the fourth 3 feet 10 inches; and the breadth is 3 feet 5 inches; how many square feet do they contain in the whole ?

Ans. 288ft. 7in. 12. How many feet of boards would it require to make 15 boxes, each of which is 7 feet 9 inches long, 3 feet 4 inches wide, and 2 feet 10 inches high; and how many cubic yards would they contain ? Ans. 1717ft. lin. 4015 cubic yds.

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13. A mason has plastered 3 rooms, the ceiling of each is 20 feet by 16 feet 6 inches, and the walls of each 9 feet 6 inches high; there is to be 90 yards deducted for doors, windows, &c. How many yards must he be paid for? 17

Ans. 251 yards. 1ft. 6in.

14. How many feet in a board which is 17 feet 6 inches long and 1 foot 7 inches wide? Ans. 27 ft. 8' 6".

15. How many feet in a board 27 feet 9 inches long, 29 inches wide? Ans. 67 ft. O' 9".

16. How many feet of boards will it take to cover the side of a building 47 feet long, 17 feet 9 inches high? Ans. 834 ft. 3'.

NOTE.-A board to be merchantable should be 1 inch thick; therefore to reduce a plank to board measure, the superficial contents of the plank should be multiplied by its thickness.

17. How many feet, board measure, are in a plank 18 feet 9 inches long, 1 foot 6 inches wide, and 3 inches thick ?

Ans. 84 ft. 4' 6". 18. How many feet, board measure, are in a plank 20 feet long, 1 foot 6 inches wide, and 2 inches thick? Ans. 75 ft. 19. How many feet in a plank 40 feet 6 inches long, 30 inches wide, and 24 inches thick? Ans. 278 ft. 5' 8".

NOTE. A pile of wood that is 8 feet long, 4 feet high, and 4 feet wide, contains 128 cubic feet, or a cerd; and every cord contains 8 cord-feet; and as 8 is 1-16 of 128, every cord-foot contains 16 cubic feet; therefore, dividing the cubic feet in a pile of wood by 16, the quotient is the cord-feet; and if cordfeet be divided by 8, the quotient is cords.

20. How many cords of wood in a pile 18 feet long, 6 feet nigh, and 4 feet wide? Ans. S cords. 21. How many cords in a pile 10 feet long, 5 feet high, 7 feet wide? Ans. 2 cords, 94 cubic feet. 22. How many cords in a pile 35 feet long, 4 feet wide, 4 feet high? Ans. 43 cords. 23. How many cords in a pile that measures 8 feet on each side? Ans. 4 cords.

24. How many cords in a pile that is 10 feet on each side? Ans. 718 cords.

NOTE.When wood is 'corded' in a pile 4 feet wide, by multiplying its length by its height, and dividing the product by 4, the quotient is the cordfeet; and if a load of wood be 8 feet long, and its height be multiplied by its width, and the product divided by 2, the quotient is the cord-feet.

25. How many cords of wood in a pile 4 feet wide, 70 feet 6 inches long, and 5 feet 3 inches high? Ans. 11 cords.

26. How many cords in a pile of wood, 97 feet 9 inches long, 4 feet wide, and 3 feet 6 inches high? Ans. 10 cords.

27. Required the number of cords of wood in a pile 100 feet long, 4 feet wide, and 6 feet 11 inches high? Ans. 215

3. Agreed with a man for 10 cords of wood, at $5.00 a cord; s to be cut 4 feet long, but by mistake it was cut only 46

inches long. How much in justice should be deducted from the stipulated price?

Ans $2.08.

29. If a load of wood be 8 feet long, 3 feet 8 inches wide, and 5 feet high; how much does it contain ?

Ans. 9 cord feet.

feet 10 inches wide,

30. If a load of wood be 8 feet long, 3 and 6 feet 6 inches high; how much does it contain ?

Ans. 12 cord-feet.

31. If a load of wood be 8 feet long, 3 feet 6 inches wide; how high should it be to contain 1 cord? Ans. 4 ft. 6' 10".

32. If a load of wood be 12 feet long and 3 feet 9 inches wide; how high should it be to contain 2 cords. Ans. 5 ft. 8' 3" }.

CROSS MULTIPLICATION.

CROSS MULTIPLICATION differs from duodecimals, in not having the inferior denominations confined to twelves; for any number, whether its inferior denominations decrease from the integer in the same ratio or not, may be multiplied crosswise; and for the better understanding it, the learner must observe the following problem.

33. Multiply 3£. 6s. 8d. by 2£. 5s. 7d.

Thus £.×£.£.s. And, if pence be multiplied by pence, the product will be two hundred and fortieths of a penny, &c. Therefore we see the propriety of the following

RULE.

That if we multiply any denomination by an integer, the value of an unit in the product will be equal to an unit in the multiplicand; but, if we multiply by any number of an inferi

or denomination, the value of an unit in the product will be so much inferior to the value of an unit in the multiplicand, as an unit of the multiplier is less than an integer.

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34. Multiply 1£. 19s. 114d. by 1£. 19s. 114d. Ans. 3£. 19s. 11d. 0qr. 35. Multiply 3 miles, 4 furlongs and 12 rods, by 2 miles, 6 furlongs, 8 rods, miles being the integer or unit.

Ans. 9 m. 6 fur. 21 rd. 4 ft. 11 in. 1br.

86. What is the difference in time between London and Boston, the latter place being 71° 4′ west of the former, the sun passing 1o in 4' of a solar day? Ans. 4h. 44′ 16′′. 37. If a mill be multiplied by a mill, what will be the product, a dollar being the unit?

Ans.1.000.000

of a dollar. 38. A, B and C bought a drove of sheep in company; A paid 14£. 5s., B 13£. 10s., and C 11£. 5s. They agreed to dispose of them at the market; that each man should take 18s. as pay for his time, &c.; and that the remainder should be divided in proportion to their several stocks. At the close of the sale, they found themselves possessed of 46£. 5s. What was each man's gain, exclusive of the pay for his time, &c.?

Ans.

1£. 13s. 3d. A's gain. 1£. 11s. 6d. B's gain. 1£. 6s. 3d. C's gain.

SECTION LII.

INVOLUTION.

INVOLUTION is the raising of powers from any given number,

as a root.

A power is a quantity produced by multiplying any given number, called a roɔt, a certain number of times continually by itself, thus,

2 2 is the root, or 1st power of 2X2 4 is the 2d power, or square of 2X2X2= 8 is the 3d power, or cube of

2=21.

2-22.

2=23.

2X2×2×2=16 is the 4th power, or biquadrate of 2=24.

The number denoting the power, is called the index or exponent of the power. Thus, the fourth power of 3 is 81, or 34; the ond power of 7 is 49, or 72.

raise a number to any power required.—