Q. How may the value of the product always be determined? A. By placing as many marks or accents at the right of the product as there are marks at the right of both multiplier and multiplicand counted together. Q. What, then, would 7 (fifths) multiplied by 8 (sixths) produce? A. 56, that is, 56 elevenths. Q. What would 7" (seconds) multiplied by 511 (thirds) produce? A. 35''//', that is, 35 fifths. Q. What would 8" multiplied by 3 produce? A. 24, (fourths.) Q. What do primes multiplied by seconds give? A. Thirds. A. Fifths. Q. What do thirds multiplied by thirds give? A. Sixths. Note. This might be extended in the same manner to any indefinite length. The following table contains a few of these denominations. Repeat the TABLE. 12 (fourths) make 1'" (third.) 12" (thirds) 1" (second.) 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 denomination always making 1 of a greater, as in the foregoing table. 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 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. OPERATION. Length, 10 ft. 8' 5/ 4 5/ 4" 10 8/ Ans., 15 1/ 4" the 5', we say, 10 X 5=50 12 12 then, X=44 of a foot, that is, 40" (seconds): 3' (inches) and 4" (seconds): we now write down 4" at the right of he inches, reserving the 3' to be ca ried to the inches. In multiplying 10 feet by or 50' (inches), and the 3' we reserved makes 53,4 feet and 5', which we place under feet and W 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'. 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. 8/ 1 5! 10 8' 4 5/ 15 1/ 4" 1X88', and 1 ft. X 10 ft. = 10 ft Then, 58403', (to carry,) and 4", (to write down); 10X550+3' (to carry)=534 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. 14 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 Jo on. From these examples we derive the following RULE. How do you multiply in Duodecimals? 4. 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 pound rules. More Exercises for the Slate. 2. How many feet in a board 2 ft. 6' wide, and 12 ft. 3′ long? Ans. 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 ft. 8' 6". Note. Artificers compute their work by different measures. Glazing and mason's flat work are computed by the square foot; painting, paving, plastering, &c. by the square yard; flooring, roofing, tiling, &c. by the square o 100 feet; brick work by the rod of 16 feet, whose square is 272; 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 cubit feet in a block 4 ft. 3' wide, 4 ft. 6' long, and 3 ft. thick? A. 57 ft. 41 6". 6. How much will a marble slab cost, that is 7 ft. 4' long, and 1 ft. 3' wide, at $1 per foot? A. $9,168. 7. How many square feet in a board 17 ft. 7' long, 1 ft. 5′ wide? A. 24 ft. 10' 41". 8. How many cubic feet of wood in a load 6 ft. 7′ long, 3 ft. 5' 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 ft. 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. 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 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 be n spending $44? $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. 1 mo.? 9 mo.? 2mo.? 1 mo.? 15 days? 10 days? 6 days? 5 days? 1 day? 13. What is the amount of $1 for 6 mo.? 3 mo.? 2mo. ? 1 mo.? 15 days? 14. Suppose I owe a man $115, payable in 1 yr. 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 oqually 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 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 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 in 1 cow 4 months, and B 2 cows 3 months: what ought each to pay ? 23. 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 leave? 32. How much is 1x of? (1) How much is 1 x of Z ? 33. What is the quotient of divided by ? 34. How much does 35. How much does exceed ,75? exceed TT? 36. 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 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 high 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 inches 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 square 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 : Or, (163): (164) X360, read 16 +4=20X3= ? A. 20. Ꭿ. 2. =48. 4X3= 12. Then, 481260, the same as before 43. (9+3)+8= how many? 44. (9-3)-4= how many? 45. (155) X 4 = how many 46. (155) X 4 = how many 47. (93)÷3= how many ? 48. (9-3)3=how many? 49. (12-8)(3+1): A. 80, or 60+20. A. 40, or 60 20. A. 4, or 3+1. A. 2, or 3-1. how many? A. 1, or 3-2. A line, or vinculum, drawn over numbers, is sometimes usea 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£8s. 52. (2 s. 6 d.) x 2 = how much? 53. (7 s. 6 d.) X 4 = how much? 54. 3 x 3 x 2 × 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 n that time; in what time, then, will the sun travel over 15° (degrees) and why? A. 1 hour for 360°÷24 h. = 15° |