per gallon, and by so doing I lose 5 per cent. on the cost. How many gallons leaked out? Ans. 15 gallons. 32. Bought a hogshead of molasses for a certain sum; but 15 gallons having leaked out, I sell the remainder for $2.21,63 per gallon, and thereby lose 5 per cent. on the cost. was the cost?

What

Åns. $112.00. 33. Bought a hogshead of molasses for $112.00; but 15 gallons having leaked out, I sell the remainder at $2.21,63 per gallon. What per cent. is my loss? Ans. 5 per cent. 34. If I sell cloth at $5.60 per yard, and thereby lose 7 per cent., should I gain or lose, and how much per cent., by selling it at $6.25 per yard? Ans. 38 per cent. gain.

35. Sold a watch which cost me $30 for $ 35, on a credit of 8 months. What did I gain by the bargain? Ans. $3.65,31.

36. 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. 37. A exchanges with B 50lbs. of indigo at $1.00 per lb. cash, and in barter $1.35; but he is willing to lose 12 per cent. to have one third 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 reduced 12 per cent., and how many yards were delivered ? Ans. $4.20 cash price of 1 yard; 7 yards delivered by B.

SECTION LIX.

DUODECIMALS.

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.

OPERATION.

6 8'

4 5 26 8'

29 4"

29 5' 4"

RULE.

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.

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 12 from each denomination to its next superior. In the same manner multiply the multiplicand by the inches of the multiplier, and write the result of each term one place further towards the right than the corresponding terms in the preceding product. Proceed in the same manner with the seconds and all the rest of the denominations, and the sum of the several products 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 4ft. 7' by 6ft. 4'.
3. Multiply 14ft. 9' by 4ft. 6'.
4. Multiply 4ft. 7' 8" by 9ft. 6'.
5. Multiply 10ft. 4' 5" by 7ft. 8' 6".

Ans. 29ft. 0' 4". Ans. 66ft. 4' 6". Ans. 44ft. 0' 10".

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

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

Ans. 745ft. 6′ 10′′ 2′′′ 4′′".

7. How many square feet in a floor 48 feet 6 inches long, 24 feet 3 inches broad? Ans. 1176ft. 1' 6".

8. What are the contents of a marble slab, whose length is 5 feet 7 inches, and breadth 1 foot 10 inches?

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. 73

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; the second, 5 feet 9 inches; the third, 4 feet 6 inches; 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. 283ft. 7in.

12. How many square feet of paper would it require to line 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 the boxes contain? Ans. 1717ft. lin. 4031 cubic yds.

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

Ans. 251yd. 1ft. 6in 14. How many feet in a board which is 17 feet 6 inches long, and 1 foot 7 inches wide? Ans. 27ft. 8' 6". 15. How many feet in a board 27 feet 9 inches long, 29 inches wide? Ans. 67ft. 0' 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. 834ft. 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. 84ft. 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. 75ft. 19. How many feet in a plank 40 feet 6 inches long, 30 inches wide, and 23 inches thick ? Ans. 278ft. 5' 3".

NOTE. A pile of wood that is 8 feet long, 4 feet high, and 4 feet wide, contains 128 cubic feet, or a cord, and every cord contains 8 cord-feet; and as 8 is 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 cord-feet be divided by 8, the quotient is cords.

20. How many cords of wood in a pile 18 feet long, 6 feet high, and 4 feet wide? Ans. 33 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. 48 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. 71 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 cord-feet; 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. NOTE. Small fractions are rejected.

26. How many cords in a pile of wood 97 feet 9 inches long, 4 feet wide, and 3 feet 6 inches high? Ans. 1017 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. 2188.

28. Agreed with a man for 10 cords of wood, at $5.00 a cord; it was 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. 30. If a load of wood be 8 feet long, 3 feet 10 inches wide, 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. 4ft. 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. 5ft. 8' 3". 33. D. H. Sanborn's parlour is 17ft. 9in. long, 14ft. 8in. wide, and 8ft. 9in. high. There are two doors 3ft. 4in. wide, and 7ft. high, and four windows 5ft. 3in. high, and 3ft. 4in. wide; the mop-boards are 9in. high. B. Gordon, a first-rate mason, will charge 10 cents per square yard for plastering the

room.

The paper for the room is 20 inches wide, and costs 6 cents per yard. E. Eaton will " paper" the room for 4 cents per square yard. Each window has 12 lights of 10in. by 14in. glass, the price of which is 12 cents per square foot. The painter's bill for setting the glass is 8 cents per light, and for painting the floor, mop-boards, and doors is 25 cents per square yard. What is the amount of Mr. Sanborn's bill? Ans. $33.728

SECTION LX.

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 root, a certain number of times continually by itself; thus,

2
2 x 2 =

2 × 2 × 2 = 2×2×2×2

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

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, 81, is expressed by 34, and 4 is the index or exponent; and the second power of 7, 49, is expressed by 72.

=

To raise a number to any power required.

RULE. Multiply the given number continually by itself, till the number of multiplications be one less than the index of the power to be found, and the last product will be the power required.

EXAMPLES.

1. What is the 5th power of 4?
4 x 4 x4
2. What is the 3d power of 8?
3. What is the 10th power of 7 ?
4. What is the 6th power of 5?
5. What is the 3d power of?
6. What is the 5th power of?
7. What is the 4th power of 24 ?

4 x4 = 1024 Ans. Ans. 512.

Ans. 282475249.

Ans. 15625.
Ans.

Ans. 13.
Ans. 50.