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MULTIPLICATION OF DUODECIMALS. Q. What are duodecimals used for ?

A. For measuring any thing respecting whichi length and breadth, also depth, are considered.

1. How many square feat in a board 10 ft. 8 in. long, and 1 ft. 5 in. broad?

We have seen how such an example may be performed by common decimals : we will now perform it by duodecimals.

OPERATION. Length, 10 ft. 8

8 inches or primes=1 of a Breadth, 1

foot, and 5' (primes)= of a 4 5' 4"

foot; then, exp=of

a foot, that is, 40" (seconds) 10 8

3' (inches) and 411 (seconds) :

we now write down 411 at the Ans. 15 1' 4"

right of the inches, reserving the 3' to be carried to the inches.

10 X 5 = 30 'n multiplying 10 feet by the 5', we say

Tor 50 vinches), and the 3' we reserved makes 53',=4 feet and 5', which we place under feet, and inches in their proper places. Then, multiplying 10 ft. 8' by 1 ft. makes 10 ft. 8', which we write under the 4 ft. 5'. We now proceed to add these two products together, which, by carrying 12, after the manner of compound rules, make 15 ft. I (inch) 4" (seconds), the Answer.

It will be found most convenient in practice to begin by multiplying the mulaplicand first by the feet, or bighest denomination of the multiplier, then by the inches, &c., thus :

1 X 8=&, and 1 ft. X 10 ft. = 10 ft. Then 5' x OPERATION

8= 401 = 3', (to carry,) and 4", (to write down); 10 ft. 8 10 x 51 = 501 + 3. (to carry) = 53 = 4 ft. and 5,

which we wfite down underneath the 10 and 8. 1 5

Then, the sum of these two products, added together as before, is 15 ft. 11 4'Ans., the same result as the

other. 10 8

Note.-Had we beon required to multiply 15 fe 4 5 4" 1' 4" by feet and inches again, we should have pro

ceeded in the same manner, carrying ! (thirds) one 15 1' 4"

place farther towards the right, and hnl (fourths)

another place still, and so on.
From these examples we derive the following

RULE.
Q. How do you multiply in duodecimals ?

A. Begin with the highest denomination of the multiplier and the lowest denomination of the multiplicand, placing the first figure in each prod uct one place farther towards the right than the sormer, recollecting to carry by 12, as in com pound rules.

More Exercises for the Slate. 2. How many feet in a board 2 ft. 6' wide, and 12 ft. 3' long

A. 30 ft. 7! 6". 3. in a load of wood 8 ft. 4! long, 2 ft. 6' high, and 3 ft. 3 wide, how many solid feet? A. 67 11. 81 611.

Note.- Artificers compute their work by different measures. Glazing and mason's Mat work are computed by the square foot ; painting, paving, plaster ing, &c. by the squarc yard; flooring, roofing, tiling, &c., by the square of 100 feet; brick work by the rod of 164 feet, whose square is 2721; the contents of bales, cases, &c., by the ton of 40 cubic feet; and he tonnage of ships by the ton of 95 feet.

4. What will be the expense of plastering the walls of a room 8 ft. 6' high, and each side 16 ft. 3' long, at $,50 per square yard? $30,694+.

5. How many cubic feet in a block 4 ft. 3' wide, 4 ft. G long, and 3 ft. thick ? A. 57 ft. 4' 61:.

6. How much will a marble slab cost, that is 7 ft. 4' long, and 1 ft. 3' wide, at $1 per foot ? A. $9,163.

7. How many square feet in a board 17 ft. 71 long, 1 ft. 5 wide ? 1. 24 ft. 10' 11".

8. How many cubic feet of wood in a load 6 ft. 71 long, 3 ft 51 high, and 3 ft. 8' wide ? A. 82 ft. 5' 8" 4!!!.

9. A man built a house consisting of 3 stories ; in the upper story there were 10 windows, each containing 12 panes of glass, each pane 14 long, 12 wide; the first and second stories contained 14 windows, each 15 panes, and each pane 16 long 12 wide : how many square feet of glass were there in the whole house ? A. 700 sq. ft.

10. What will the paving of a court-yard, which is 70 ft long, and 56 ft. 4! wide, come to, at $,20 per square ?

A. $788,663 11. How many solid feet are there in a stick of timber 70 long, 15' thick, and 18' wide ? A. 131 ft. 3'.

Questions on the foregoing. 1. How many pence are there in 1 s. 6 d.? How many cents ?

2. What will 4 yards of cloth cost, in cents, at 1 s. Od per yard ? At 3 s. per yard ? At 4 s. 6 d.? At 6 s.? Ad Os. At 10 s. 6 d.?

3. If a man consume 1 lb. 9 oz. of bread in a 10k, 2017 much would he consume in 1 month ?

4. At 4 cents for 1 oz., what would 1 lb. cost ? 5. At 4 cents for 2 02., what would 1 lb. cost ? 6. At 4 cents for 8 oz., what would 2 lbs. cost?

7. If a man spend $24 per day, how many days would he be in spending $41? $61 ? $121? $20?

8. How many marbles, at 4 cents apiece, must be given for 24 apples, at 2 cents apiece?

9. How many yards of cloth, at $4 per yard, must be given for 6 bbls. of cider, at $2 per bbl. ? For 8 bbls. ? For 12 bbls. ? For 18 bbls. ?

10. What part of 1 month is 1 day? 2 days? 4 days? 5 days? 6 days? 7 days? 10 days ? 20 days! 29 days?

11. What is the interest of $1 for 12 mo.? 1) mo.? 9 mo.? 6 mo.? 3 mo.? 1 mo.? 15 days?

12. What is the interest of $6 for 1 yr. 2 mo.? 2 yrs. ? 1 yr. 1 mo.? 9 mo.? 2 mo.? i mo.? 15 days ? 10 dnya ? 6 days? 5 days? 1 day?

13. What is the amount of $1 for 6 mo.? 3 mo? 2 me? 1 mo.?

15 days? 14. Suppose I owe a man $115, payable in 2 ms. 6 mong without interest, and I wish to pay him now, low MULES ought I to pay him?

15. What is the discount of $115, for 2 yrs. 6 20.?

16. William has of an orange, and 'Thom: S. ; what part of an orange do both own?

17. Harry had of an orange, which he wishe I to divide equally between his two little sisters; can you tel me what part of an orange each one would receive ?

18. Which is the most, ,5 of 20, or ,25 of 40:

19. How many times can you draw of a galle"? of cider from a barrel containing 30 gallons ? Liow many:imes of a gallon? How many times } of a gallon? How is any tims & of a gallon?

20. A man, failing in trade, is able to pay his creditors ostane ly $,33} on the dollar; aow much will he pay on $3? On $4 ? On $12. On $13? On $300 ?

21. A man, failing in trade, was able to pay only $,16'on the dollar; how much would he pay on a debt of $4? $67 $10 ? $9? $20 ? 100? $600 ?

22. Two men boughi a barrel of flour for $10 ; one gave $3, and the other $7: whit part of the whole did each pay? What part of the flour mustique a have ?

23. ITS) bushels of oats cost $10, what is that a bushel ? "Vhat will 5 bushels cost? What will 20 bushels ?

24. Há men mow a field in 8 days, how many men will wow the came in 2 days? In 1 day? In 4 days?

25. Twɔ men, A and B, hired a pasture ; A paid $3, and B $5; what fractional part of the whole did each pay? The profits from the pasture were $16; what was each man's share of the gain?

26. Three men, A, B, and C, are engaged in trade; A puts in $4, B $5, and C $6; they gained $60: what is eacb one's share of the gain ?

27. Two men, A and B, hired a pasture for $12; A put in I cow 4 n.onths, and B 2 cows 3 months; what ought each to pay ?

28. A merchant, having purchased a piece of broadcloth for $2 per yard, wishes to make 20 per cent. on it; what price must he ask for it?

29. William has ļ of a dollar, Thomas it, and Harry ; how many cents have they in all ?

30. A merchant sold calico at $,22 per yard, and thereby gained 10 per cent. ; what did it cost him per yard ?

31. Harry, having i of an orange, gave š to Thomas, who graves of his part to his little brother, and kept the remainder hirıself; what part did he keep? How much is } of ? How much does 4 of} of from leave?

32. How much is 1x of 2 ?-1. How much is 1 x

33. What is the quotient of divided by ì ?
34. How much does t exceed ,75?
35. How much does $ exceed 1?

30. How many strokes does a regular clock strike in 2 hours ? 3 h.? 4 h.? 5 h.? 6 h. ? 7 h.? 8 h.? 9 h.? 10 h.? 11 h.? 12 h.? 24 h.?

37. How many square feet in a board 12 inches wide, and 28 inches long? 36 in. long? 72 in. long?

38. What part of an acre of land is there in your father's garden, allowing it to be 4 rods long and 2 rods wide ? 4 rods wide ?

39. How many cord feet of wood are contained in a load 3feet wide, 2 feet high, and of the usual length? How Inny feet in a load 6 feet high and 3 feet wide? 2 feet highan ; feet wide ? 4 feet high and 24 feet wide ?

40. How many solid feet in a block 12 inches long, 12 inches thick, and 12 inches wide ? 12 inches long !2 inches wide, and 6 inches thick?

41. How long will it take to count $1000, at the rate of 50 a minute ?

42. What is the difference between 4 square feet and 4 feet square? 10 miles square and 10 square miles ? 3 rods square and 3 square rods?

A parenthesis, enclosing zeveral numbers, signifies that these numbers are to be taken together, or as one whole number; but, when performing subtrastion and aldition with these and other uumbers, they may be taken either as before, or oue by one, thus :

(16 + 4) X 3= 60, read 16 +4= 20 X 3= 60. Or, (16 x 3) = 48. 4X3= 12. Then, 48 + 12 = 60, the same as before. 43. (9+3)+8= how many ? A. 20. 44. (9-3) — 4= how

many
?

1. 2. 45. (15+5) X 4= how many? A. 80, or 60+20. 46. (15—5; X 4= how

many

? A. 40, or 60 — 20. 47. (9+3)=3=how many? A. 4, or 3 +1. 48. (9—3):-3= how many ? A. 2, or 3 – 1. 49. 12—8)= (3+1)= how many? A. 1, or 3 -2.

A line, or vinculum, drawn over numbers, is sometimes used instead of a parenthesis.

50. 4+8 x 12 = how many ? A. 144, or 48 +96.
51, 2£.4 s. X 2= how much? A. 4£. 8 s.
52. (2 s. 6 d.) X 2= how much ?
53. (7 s. 6 d.) x 4= how much?
54. 3 X3 X2 X 10= how many ?

55. How many minutes of motion make 2 degrees of motion? How many seconds of motion make 3 minutes of motion?

56. How many degrees is the circumference of the earth ?

57. The earth, you know, turns round once in every 24 hours, or, in common language, the sun moves round the earth in that time; in what time, then, will the sun travel over 150 (degrees)? and why? A. 1 hour for 360° ;- 24 h 58. In what time will he travel over 1° (degree) of motion ?

A. Its of 1 hour, or T's of 60 min., = 4 min. 09. In what time will be travel over 1' (minute) of motion?

A. Is of 1 min., or 1 of 60 sec., = 4 seconda Q. By the foregoing we see that every degree of motimn makes a difference in time of 4 minutes, and every minute of motion a differ

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