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Q. How may the value of the product always be determined? By placing as many marks or accents at the right of the product as "here are inarks at the right of both multiplier and multiplicand counted together.

Q. What, then, would 70!!!! Cifths) multiplicd by S. (sizths) produce ? A. 50 mill, that is, 56 elevenths,

Q. What would 7" (seconds) mulciplied by 5'' (thirds) produce? A. 35'||||| that is, 35 fifths. Q. What rould 81 multiplied by 3'' produce ?

A. 24!!!!, (fourths.) Q. Froin the preceding, what appears to be the value of feet multiplied bv primes or inches, or what do fect multiplied by primes gide ?

.4. Primes. Q. What do primes mulliplied by primes give ? A. Seconds. Q. What do primes multiplied by seconds give ? A. Thirds. Q. What do secunds multiplied by seconds give ? A. Fourths. Q. Whut il seconds multiplied by thirds give ? A. Fifths. Q. What do thirds multiplicd by thirds give ?

A. Sixths.
Note. This might bo extended in the same manner to any indefinito length
The following table contains a few of these denominations.
Repeat the

TABLE.
12''" (fourths) make 1" (third.)
12" (thirds)

1" (seconds)
12" (seconds) 1' (inch or prime.)

12' (inches or primes) 1 foot. Q. How may duodecimals be added and subtracted? A. In the same manner as compound numbers ; 12 of a less denomiErtion always making 1 of a greater, as in the foregoing table.

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MULTIPLICATION OF DUODECIMALS. Q. What are duodecimals used for? A. For measuring any thing respecting which length and breadth, also depth, are considered.

1. How inany square feet in a board 10 ft. 8 in. long, and 1 it. 5 in. broad :

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

8 inches or primes = of a foot Length, 10 ft. 8 Breadth, 1 5

and 5' (primes) = 1 of 2 foot

then, X=*% of a foot, 4 5

that is, 40" (secous) = 3' (inches) 10 8

and 4" (seconds): wo now writé down 41 a the right of the incheier

, Ans., 15 11

reserving the 3' to be carried to the

inonics. In multiplying 10 Peeling 10 X 5=50

- 50 (inches), and the 3' we reserved

1218 maxes 53', == 4 feet and 5', which we place under feet and

the 3', we say,

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inches in their proper places. Then, multiplying 10 ft. B FT I st. makes 10 H 8', which we write under the 4 ft. 5'. W now proceed to add these two products together, which. bu carrying 12, after the manner of compound rules, make la ft 1' (inch) 4'' (seconds), the Answer.

It will be found most convenient in practice to begin by multiplying the mulLiplicand first by the feet, or highest denomination of the muliiplier, then by the inches, &c., thus : OPERATION.

and I ft. X 10 n. = 10 ft Then

X%=, ,(to Thi 10 4.8

down); 10 X 5' = 509 +31 (to ca-ry) 1 5

ft. and '5', which we write down underneath the

10 and 8. Then, the sum of these two producis, 10 8

adiled together as before, is 15 ft. 1'4" Ans. the 4 5 4"

same result as the other.

Note. Had we been required to myltiply !5 st. 15 11 411

1' 4" by fcet and inches again, we should havo

proceeded in the same manner, carrying " (thirds) one place farther towards tho right, and "'' (fourths) another place still, and

30 on.

From these examples we derive the following

RULE.
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 product one place further towards the right than the former, recollecting to carry by 12, as in com. fund rules.

More Exercises for the Slate. 2. How many feet in a board 2 ft. 6' wide, and 12 ft. 3' long? sins. 30 ft.? 07.

3. In a load of wood 8 ft. 4! long, 2 ft. 6' high, and 3 ft. 3 wide, how niany solid feet? 4. 67 ft. 8' 6".

Artiticers compute their work by different measures. Glazing and mas on's flat work are cumputod by the eqnare foot; painting, paving, plurter. ing, &c. by. the square yard; flooring, roofing, tiling, &c. by the square a 100 foot; brick work by the rod of 164 feet, whoso square is 2724 ; the con. tents of bales, cases, &c. by the ton of 40 cubic feet; and the tonna ge of ships by the ton 95 feet

4. What wil be the expense of plastering the walls of a room 8 ft. 6' high, and mch side 16 ft. 3' long, at $,50 per square

Ă. $30,6947 5. How many cubit fea in a block 4 ft. 3' wide, 4 ft. 6' long, and : ft. thick ? A. 57 ft. 419".

6. How much will a marble sinh cost, that is 7 ft. 4' long, and, 1 ft. 31 wide, at $1 per foot ? A. $9,16.

.7. How many square feet in a boaro. 17 ft. 7' long, 1 ft. Bo wide ? A. 24 ft, 10' 41".

Note.

yard ?

8. How many cubic feet of wond in a load 6 ft. 7' long, 3 ft. 5' high, and 3 ft. 8' wide ? A. 82 ft.5' 8" 4ir.

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 wero there in the whole liouse ? A. 700 sq. ft.

10. What will the paving of a court yard, which is 70 ft. lung, and 56 ft. 4' wido, come to, at $,20 per square ?

A. $788,663 U. How many solid feet are there in a stick of timber 70 ft song, 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. 6 d. per yard ? At 3 s. per yard ? At 4 s. 6 d.? At 6 s.? At 9s ? At 10 s. 6 d. ?

3. If a man consume 1 lb. 9 oz. of bread in a week, how much would he consume in 1 month ?,

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

7. If a man spend $24 per day, how many days would he be in spending $15? $64? $12? $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.? 10 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. i mo.? 9 mo.? 2 mo.? 1 mo.? 15 days ? 10 days? 6 days? 5. days? 1 day?

13. What is the amount of $1 for 6 mo.? 3 mo.? 2 mo.? 1. mo.? 15 days ?

14. Suppose I owe a man $115, payable in 1 yr. 6 mo., with. out interest, and I wish to pay him now, how much ought I to

15. What is the discount of $115, for 2 yrs. O mo? 16. William has $ of an Grange, and Thomas t ; what part of an orange do both own?

pay him?

17. Harry had l of an orange, which he wished to divide qually between his two little sisters ; can you tell me what part of an orange cach one would receive ?

18. Which is the most, ,5 of 20, or ,23 of 40 ?

19. How many times can you draws of a gallon or cider from a barrel containing 30 gallons ? How many times of o gallon? How many times of a gallon? How many times ] of a gallon ?

20. A man, failing in trade, is able to pay his crcditors ou ly $,33f on the dollar; how inuch will he pay on $3? On $4 On $12? On $13? On $300 ?

21. A man, failing in trade, was able to pay only $,163 on the dollar ; how much would he pay on a debt of $4? $6? $10: $9 ? $20 ? $100 ? $600 ?

22. Two men bought a barrel of flour for $10; one gavc $3 and the other $7: what part of the whole did each pay? What part of the flour must each have ?

23. If 30 bushels of oats cost $10, what is that a bushel ? What will 5 bushels cost ? What will 20 bushels ?

24. If 3 men mow a field in 8 days, how many men will mow the same in 2 days? In 1 day? In 4 days?

25. Two men, A and B, hired a pasture; A paid $3, and B $5; what fractional part of the whole did each pay? The prosts from the pastu re 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 each one's share of the gain?

27. Two men, A and B, hired a pasture for $12; A put in 1 cow 4 months, and B?cows 3 months: what ought cach to pay?

28. A merchant, hyving 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 $(f a dollar, Thomas Ty, and Harry $; how many cents have the r 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 of an orange, gave š to Thomas, who gave of his part to his little brother, and kept the remainder himself; what part did he keep? How much is } off? How much does off off from leave?

32. How much is 1 x off? (1) Ilow much is 1 x 4 of 7?
33. What is the quotient of divided by i?
34. How much does & exceed ,75 ?
35 How much does g exceed I'I?

How many

36. How many strokes does a regular clock strike in 2 hours? 3 h.4 h.? 5h.? 6 h.? 7 h.? 8 h.? 9 h.? 10 h.? 11 h.. !2 h.? 21 h. ?

37. How many square feet in a board 12 inches wide, and 43 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 ?

3.). How many cord feet of wood are contained in a load 3 feet wide, 2 feet high, and of the usual length? foot in a load-6 feet high and 3 feet wide ? 2 feet high and 6 fect wide ? 4 feet high and 24 feet wide ?

46 Ho v many solid seet in a block 19 inches long, 12 inches thick, and 12 inches wide ? 12 inches long, 12 inches wide, and 6 inches thiek ?

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 fect square ? 10 miles square and 10 square miles ? 3 rods squars. and 3 square tods ?

A parenthesis, enclosing severai numbers, signifies that these numbers are to be taken together, or as ono whole number ; but, when performing subtraction and addition with these and other numbers, they may be taken either as beforo, or one by one, thus :

(16+4)X3= 60, read 16+4= 20 X3 = 60. Or (16 X 3) 4X3= 12. Then, 18 + 12 = 60, the same as befors 13. (9+3)+8= how many ?

A. 20. 44. (9 — 3) — 4= how many ?

A. 2. 45. (15+5) X4=how many ? A. 80, or 60 + 20: 46. (15—5) X4=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 linc, or vinculum, drawn over numbers, is sometimes usea instead of a parenthesis.

+8 x 12= how many ? 9. 144, or 48 + 96. 51. 2£48. 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 x 3 x 2 x 10

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

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

37. 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 15° (degrees)' and why? A. 1 hour for 3600 • 24 h. 150

=48.

50.

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