MULTIPLICATION OF DUODECIMALS.

Q. What are duodecimals used for?

A. For measuring anything respecting which length and breadth, also depth are considered.

1. How many square feet 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.

10 X 550

plying 10 feet by the 5', we say 12 12 or 50 (inches), and he 3' we reserved makes 53', 4 feet and 5', which we place ander feet and inches in their proper places. Then, multiply. 'ng 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. 1' (inch) 4" (seconds), the Answer.

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

OPERATION. 10 ft. 8'

5'

1

10

8'

4

15

5' 4"

1' 4"

1 x 8'8', and 1 ft. × 10 ft. 10 ft. Then 5' x 8' =40" 3' (to carry), and 4" (to write down); 10 X 5'50'+3' (to carry) =53′ = 4 ft. and 5', which we write down underneath the 10 and 8'. Then, the sum of these two products, added together as before, is 15 ft. 1' 4" Ans., the same result as the other.

Note.-Had we been required to multiply 15 ft. 1' 4" by feet and inches again, we should have proceeded in the same manner, carrying "" (thirds) one place farther towards the right, and """ (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 product one place farther towards the right than the former, recollecting to carry by 12, as in compound rules.

More Exercises for the Slate.

2. How many feet in a board 2 ft. 6' wide, and 12 ft. 3' long? A. 30 t. 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 .8' 6".

Note. Artificers compute their work by different measures. Glazing and mason's that work are computed by the square foot; painting, paving, plas tering, &c., by the square yard; flooring, roofing, tiling, &c., by the square of 100 feet; brick work by the rod of 16 feet, whose square is 270; the contents of bales, cases, &c., by the ton of 40 cubic feet; and the 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? A. $30,694+.

5. How many cubic feet in a block 4 ft. 3′ wide, 4 ft. 6' long, and 3 ft. thick? A. 57 ft. 4' 6".

6. How much will a marble slab cost, that is 7 ft. 4' long, and 1 ft. 3' wide, at $1 per foot. $9,16}.

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

8. How many cubic feet of wood in a load 6 ft. 7' long, 3 ft. 5' high, and 3 fi. 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. $788,661. 11. How many solid feet are there in a stick of timber 70 ft. ong, 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. 6d. per yard? At 3 s. per yard? At 4 s. 6 d.? At 6 s? At

9 s 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 oz., 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. cost?

7. If a man spend $24 per day, how many days would he in spending $4? $61? $12 ? $20?

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

2 days?

4 days?

20 days?

29 days? mo. ? 9

10. What part of 1 month is 1 day? 5 days? 6 days? 7 days? 10 days? 11. What is the interest of $1 for 12 mo. ? 10 mo. ? 6 mo. ? 3 mo. ? 1 mo. ? 15 days?

2 yrs. ?

12. What is the interest of $6 for 1 yr. 2 mo. ? 1 yr. 1 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 2 yrs. 6 mo., without interest, and I wish to pay him now, how much ought I to pay him?

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

16. William has of an orange, and Thomas ; what part of an orange do both own?

17. Harry had of an orange, which he wished to divide equally between his two little sisters; can you tell 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 gallon of cider from a barrel containing 30 gallons? How many times of a gallon? How many times of a gallon? How many times of a gallon?

20. A man, failing in trade, is able to pay his creditors only $,33 on the dollar; how 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 $,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 gave $3, and the other $7: what part of the whole did each par › 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 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 each one's share of the gain?

27 Two men, A and B, hired a pasture for $12; A put 1. 1 cow 4 months, and B 2 cows 3 months; what ought eac 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 of a dollar, Thomas, 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 of an orange, gave to Thomas, who gave of his part to his little brother, and kept the remainder himself; what part did he keep? How much is of ? How much does of of from 32. How much is 1X of ?-1. of ??

leave?

How much is 1×9

33. What is the quotient of divided by ?

34. How much does

35. How much does

exceed ,75?

exceed?

10

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

37. How many square feet in a board 12 inches wide, and 48 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 3 feet wide, 2 feet high, and of the usual length ? How many feet in a load 6 feet hign and 3 feet wide? 2 feet high and 6 feet wide? 4 feet high and 2 feet wide ?

40. How many solid feet in a block 12 inches long, 12 nches thick, and 12 inches wide? 12 inches long, 12 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 quare and 3 square rods?

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

(16+4) X 360, read 16+4=20 × 3 = 60.

Or, (16 × 3) = 48. 4 X 3 = 12. Then, 481260, the same as before.

43. (9+3)+8: how many?

44. (9-3)-4 = how many?

45. (15+5) X4

A. 20.

A. 2.

how many

46. (15-5) X 4 = how many?

47. (93) 3 how many?

48. (9-3)-3= how many?

A. 4, or 3+1.

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.

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 15° (degrees)? and why? A. 1 hour for 360° — 24 h. — 15o. 58. In what time will he travel over 1° (degree) of motion? A of 1 hour, or 15 of 60 min., = 4 min. 59. In what time will he travel over 1' (minute of motion)? of 1 min., or of 60 sec.,

A.

=

4 seconds.

Q. By the foregoing we see that every degree of motion makes a differ ence in time of 1 minutes, and every minute of motion a difference of