306. To add or subtract denominate fractions, common or decimal, of different denominations. Ex. 2.-From .825 T. subtract .62 cwt. Proceed by either method shown under Example 1. Ex. 3.—Add .875 fb, .7 3, and .4 3. In adding decimals of different denominations, the second method is generally preferable. Reduce .875 b to ounces, and add in .73. Reserving 11, the integer, reduce .23 to drams, and add in .43. Collect the integers for the answer. Ans. 113 23. 12 10.500 .7 11.23 8 1.63 .4 2.03 RULE.-1. Reduce the given fractions to integers of lower denominations; then add or subtract, as required. 2. Or, reduce the fraction of the highest denomination to integers of lower denominations, taking care, as each is reached, to add or subtract, as may be required, any given fractional term belonging to that denomination. 15. Add cwt., qr., and lb. 16. Add bu., † pk., and 1 qt. Ans. 2 qr. 8 lb. 9 oz. 51 dr. 806. In how many ways may we add or subtract fractions of different denominations? Illustrate these two modes with the given examples. Recite the rule. 17. From oz. take 3 pwt. Ans. 7pwt. 15 gr. Ans. 8h. 59 min. 48 sec. Ans. 211. 4.5144 in. Ans. 15s. 10d. 0.74 far. Ans. 4 da. 21 h. 8 min. yd., 4ft., 297 in. Ans. S Same ans. Ans. 17 c. ft. 130 c. in. 240 A. 2 R. 30.4 sq. rd. Ans. 2 R. 42 P. 18. From .375 da. take .2 min. 19. From .22 ch. take .431. 20. Add £.75, .8s., .36d., .9 far. 21. Add & wk., } da., †h. 22. Add mi., fur., rd., 23. Add mi., fur., 123 rd., yd., ft., in. for both. 24. Add .6 cu. yd., .875 cu. ft., .4 cu. in. 25. Add .375 sq. mi., .54 A., .6 R. 26. From A. take of 3 roods. 27. From 1hhd. take & qt. 28. From .32 lb take .93. 29. From of 4 of a day take 30. Add lb. Troy, oz., and 31. A man had to plough 3 A, R., §P. was ploughed, how much had he to do? Ans. 2 A. Ans. 6 gal. 3 qt. pt. Ans. 2 oz. 7 dr. 1 sc. 11.2 gr. of 1 hours. pwt. Ans. 2 oz. 13 pwt. 3 gr. When A. R. † P. 1R. 10 P. Ans. 8373. Ans. 223.25 cu. in. Ans. 3763.235 cu. in. 32. How many cu. in. in 3 gal. 2 qt. 1 pt., Wine? 35. How many feet in 4 of a chain + fur.? 36. From a piece of cloth containing 20 yd. 2 qr. Ans. 247 ft. nails, 3 suits, each requiring 41 yd., were cut. One third of the remainder was gold for $10.684; what did it bring per yard? Ans. $4.50. 2.375 yd. $10.6875 +2.375 = $4.50. 37. What cost 4 bu. 3 pk. 6 qt. of potatoes, at 75c. a bushel? As the price is given by the bushel, reduce, by § 285, 4 bu. 3 pk. 6 qt. to bushels and the decimal of a bushel (4.9375 bu.), and multiply by the price. Ans. $3.70. 38. What cost 4 bu. 3 pk. 6 qt. of potatoes, at 18c. a pk.? As the price is given by the peck, reduce 4 bu. 3 pk. 6 qt. to pecks and the decimal of a peck, and multiply by the price. 4 bu. 3 pk. = 19 pk. 6 qt. .75 pk. 19.75 x .18 = $3.555. Ans. 39. What cost 57 A. 2 R. 20 P., at $20 an acre? Ans. $1152.50. Ans. $864.375. At $3.75 a rood? 40. What cost 7 gal. 3 qt. 1 pt. of wine, at $8 a gal.? Ans. $63. At $1.50 a quart? Ans. $47.25. Ans. $187.52. 41. What cost 5 T. 17 cwt. 20 lb. of hay, at $30.50 a ton? At $1.60 a hundred-weight? 42. What cost 3 lb. 6 oz. 1 dr. 2 sc. of quinine, at $2.75 per oz. ? 43. Find the cost of a gold ornament, weighing 4 oz. 18pwt. 20 gr., at £4 9s. per ounce. Ans. £21 19s. 9d. 2.8 far. 44. What is the cost of a block of marble, 9 ft. long, 4ft. 4 in. wide, and 3 ft. 6 in. thick, at $5 a cubic foot? Ans. $682.50. 45. What cost 3 bundles 8 quires of paper, at $6 a ream? 46. What cost a field 4 ch. 80 1. square, at $6.25 a rood? 47. What cost 41 A. 31 R. 1P. of land, at $4.25 a rood? 48. What cost T. } cwt. 25 lb., at $3 per cwt.? Ans. $11.35. 49. 18 Cd. 8 cu. ft. of wood, at $8 a cord? Ans. $144.50. 50. 7 T. 14 cwt. 3 qr. 10 lb., at $75 a ton? Ans. $580.687 +. 51. 3lb. 63 15 gr. of calomel, at $1.50 an ounce ? Practice. 307. Practice is a short method of operating with compound numbers, by means of aliquot parts. It was applied to Federal Money on p. 123, and may be extended to compound numbers generally. The aliquot parts most frequently used are as follows: 807. What is Practice? Give the aliquot parts of £1. Of a shilling. Of a hundred-weight. Of a year. Of a month. = Ex. 52.-What cost 960 Grammars, at 1s. 8d. each? At £1 each, 960 Grammars would cost £960. of £1; therefore, at 1s. 8d., they will cost of £960, or £80. But 1s. 8d. 12) £960 Ans. £80 Ex. 53. If it costs $17.50 to insure a house 1 year, what will it cost to insure it for 3 yr. 1 mo. 15 da.? 1 mo. 15 da. = $17.50 3 $52.50 Ans. $54.6875 2.1875 For 3 years take 3 times the cost for 1 yr. For 1 mo. 15 da., which is of 1 yr., take of the cost for 1 yr. Find the whole by adding these two parts. Ex. 54.-How much seed will be needed for 10 A. 1 R. 30 P., allowing 1 bu. 2 pk. 4 qt. to an acre? For 10 A. take 10 times the quantity required for 1 A. For 1 R., which is of 1 A., take the quantity required for 1 A. 30 P. not being an aliquot part of 1 rood, take first for 20 P., which is of 1 R.; then for 10 P., which is 20 P. Find the whole by adding these parts. 1 R. 1 2 bu. pk. qt. 4 10 16 1 0 0 1 5 55. What cost 14 dozen Readers, at 3s. 4d. apiece? Ans. $226. Ans. £18. Ans. $33.20+. Ans. $184.84+. 60. What will it cost to travel 1200 miles, at 14d. a mile? 61. At $25 a month, what will be a man's wages for 1 year 7 months 12 days? 62. What will be the yield of 16 A. 25 P. of land, at the rate of 24 bu. 3 pk. 1 qt. per acre? Ans. 400 bu. 1 pk. 3.90625 qt. 63. What cost a plate of glass, measuring 7 ft. by 5 ft. 6 in., at 4s. 6d. a square foot? Ans. £8 13s. 3d. 64. What cost 10 panes of glass, each 4 ft. by 2 ft. 9 in., at 1s. 3d. a square foot? Ans. £6 17s. 6d. 65. Find the rent of a farm of 250 A. 2 R. 15 P., at £1 12s. 4d. Ans. £405 2s. 6d. 1.5 far. an acre. CHAPTER XVIII. DUODECIMALS. 308. Duodecimals are a system of compound numbers, sometimes used as measures of length, surface, and solidity. The foot, whether linear, square, or cubic, is the unit; and the other denominations arise from successive divisions by 12. Hence the term duodecimals, duodecim being the Latin for twelve. 1 of any denomination in this system makes 12 of the next lower; and, conversely, 12 of any denomination make 1 of the next higher. The marks used to distinguish the denominations ľ """""") are called In'dices (singular, Index). 309. When used in connection with one dimension simply, as length or breadth, the prime, being of a foot, is equivalent to 1 inch. When applied to surfaces, the prime, being of a foot, equals 12 square inches. The second, being of of a foot, equals 1 sq. in. When applied to solid contents, the prime, being of a foot, equals 144 cubic inches; the second = 12 cubic inches; the third = 1 cubic inch. 308. What are Duodecimals? What is the unit? How do the other denominations arise? Whence is the term duodecimals derived? Recite the Table. What are the marks used to distinguish the denominations called?-309. What is the prime equivalent to, when used in connection with one dimension simply? When applied to surfaces? When applied to solid contents ?-810. How are duodecimals added, subtracted, multiplied, and divided? |