CHAPTER XIII. REDUCTION OF COMPOUND NUMBERS. Art. 267. Operations in denominate numbers are reduction, addition, subtraction, multiplication, or division. REDUCTION OF DENOMINATE INTEGERS. Art. 268. Reduction of denominate numbers is the process of changing them from one denomination to another. This does not alter their value. It is of two kinds, namely, descending, and ascending. Descending Reduction is changing a number to a lower denomination. It is performed by multiplication. Ascending Reduction is changing a number to a higher denomination. It is performed by division. CASE I. Art. 269. To reduce an integer to a lower denomina tion. Ex. 1. How many rods are 25 miles? WRITTEN PROCESS. 25 miles. 8 fur. in 1 mi. 200 fur. in 25 mi. 40 rd. in 1 fur. 8000 rd. in 25 mi. Ans. 8000. EXPLANATION. Since 1 mile is 8 fur., 25 mi. are 25 times 8 furlongs, or 200 fur. Since 1 fur is 40 rd., 200 fur. are 200 times 40 rd., or 8000 rd. NOTE.-The following simple process obtains the result:-Since 1 mile is 320 rods, 25 miles are 25 times 320 rods, or 8000 rods. Ex. 2. Reduce 17 mi. 5 fur. 19 rd. 8 ft. 4 in. to inches. Ans. 1120582 inches. EXPLANATION. Since 1 mi. is 8 fur., 17 mi. are 17 times 8 fur., or 136 fur., which, with its 5 fur. given, makes 141 fur. Since I fur. is 40 rd., 141 fur. are 141 times 40 rd., or 5640 rd., which, with the 19 rd. given, makes 5659 rd. Since 1 rd. is 16 ft., 5659 rd. are 5659 times 16 ft., or 93373 ft., which, with the 8 ft. given, makes 933814 feet. Since 1 foot is 12 in., 93381 ft. are 93381 times 12 in., or 1120578 in., which, with its 4 in. given, makes 1120582 inches. Rule.-Multiply the number in the highest denomination by that number of the next lower denomination which makes one of the denomination multiplied. If the number to be reduced is compound, add to this product that part of the number which is of the same denomination as the product. Change this result in like manner to the next lower denomination, and so on till the required denomination is reached. EXAMPLES FOR PRACTICE. 3. Reduce 1 mi. 2 fur. 4 yd. 2 ft. to inches. 4. In 1 lea. how many ft.? Ans. 79368 in. 5. What will it cost to inclose a farm 11 mi. square at 11. After selling 45 A. from a farm containing 63 A., how many square rods remain ? Ans. 2880. 12. How many yards of carpet 1 yd. wide will be required to cover the floor of a room 11 yd. long and 8 yd. wide? 13. In 13 A. 6 sq. ch. 10 P. 105 sq. 1. how many square links? 14. In 43 cu. ft. how many cu. in.? Ans. 74304. 15. How many cubic feet in a cellar 15 ft. long, 12 ft. wide, and 5.5 feet deep? Ans. 990. 16. How many cubic yards in a room 12 yd. long, 9.5 yd. wide, and 43 yd. high? Ans. 532. 17. How many cu. ft. of wood in a pile 96 ft. long, 6 ft. wide, and 5 ft. high? Ans. 2880. 18. How many cubic feet in a wall 123.75 ft. long, 11⁄2 ft. thick, and 4 ft. high? Ans. 742.5. 19. How many feet in a 1 inch board 18 ft. long, and averaging 18 inches wide? Ans. 27. 20. In a plank 15 feet long, 18 inches wide, and 2 inches thick, how many board feet? Ans. 45. 21. How many board feet in a log 16 feet long, and averaging 15 inches square? Ans. 300. Ans. 188 gi. 22. Reduce 5 gal. 3 qt. 1 pt. to gills. 24. Reduce 3 lb. 2 oz. 5 pwt. 7 gr. to grains. 25. Reduce 2 lb. 23 13 29 to grains. 26. Reduce 3 cwt. 4 lb. 10 oz. to drams. 27. Reduce 5° 25′ 18′′ to seconds. Ans. 165 pt. Ans. 18367 gr. Ans. 12580 gr. Ans. 5592 far. 28. Reduce 2 da. 7 hr. 20 min. to seconds. 29. Reduce £5 16 s. 6 d. to farthings. 30. How many steps of 2 ft. 8 in. each, will a man take, in walking from Richmond to Petersburg, the distance being 18 miles? 31. B. bought 12 bu. of chestnuts at $4.25 a bushel, and retailed them at $.08 a pint; how much did he gain on the whole? 32. How many feet high is a horse that measures 17 hands? 33. Reduce 4 O. 5 f3. 4 fz. 24 m to minims. 34. Bought 2 T. 45 lb. of cheese for $400, and retailed it at 16 cts per lb.; what did I gain? 35. What cost 3 cwt. 24 lb. of beef at $.06 36. Reduce 2 lb. 5 3. 4 3. to scruples. 37. Reduce 8 M. M. to M. 38. Reduce 3.625 K.M. to M. 39. Reduce 6.05 M.2 to d.m.2 40. Reduce 4.00138 M.2 to m.m." Ans. $247.20. per pound? Ans. 8000 M. Ans. 605 d.m." Ans. 4324 d.m.3 Ans. 732.1 L. Ans. 15000 m.g. CASE II. 46. Reduce 2.165 K.G. to G. Art. 270. To reduce an integer to a higher denomina tion. Ex. 1. Reduce 1120582 inches to miles. Ans. 17 mi. 5 fur. 19 rd. 8 ft. 4 in. WRITTEN PROCESS. 12) 1120582 in. 169338 1 ft. 10 in. 2 33) 186762 40) 5659 rd. 15 ft.-7 ft. EXPLANATION. In 1120582 in. there are as many ft. as 12 in. are contained times in 1120582 in., namely, 93381 ft. and 10 in. remain. In 93381 ft. there are as many rd. as 16 ft. are contained times in 93381 ft.; or as many rd. as 33 half ft. are contained times in 186762 half ft., namely, 5659 rd., and 15 half ft. remain. In 5659 40 rd. are contained times in rd. there are as many fur. as 5659, namely, 141 fur., and 19 rd. remain. In 141 fur. there are as many miles as 8 fur. are contained times in 141 fur., namely, 17 mi., and 5 fur. remain. Hence 1120582 in. equal 17 mi. 5 fur. 19 rd. 7 ft. 10 in. But ft. is 6 in., which, added to 10 in., makes 16 in., or 1 ft. 4 in. The 1 ft. added to 7 ft. makes 8 ft. 17 mi. 5 fur. 17 mi. tur. 19 rd. 71⁄2 ft. 10 in. = 17 mi. 5 fur. 19 rd. 8 ft. 4 in. Rule.—Divide the given number by that number of its denomination which makes one of the next higher. Change this quotient in like manner to the next higher denomination, and so continue till the required denomination is reached. The answer is composed of the last quotient and all the remainders. NOTE 1.-Each remainder should be marked as being of the same denomination as that dividend from which it came. NOTE 2.-Descending and ascending reduction may each be used as a proof for the correctness of the other. EXAMPLES FOR PRACTICE. 2. Reduce 151338 in. to miles. Ans. 2 mi. 3 fur. 4 rd. 5 ft. 6 in. 3. In 63360 ft. how many lea.? Ans. 3. Ans. 6 fur. 21 rd. 4 yd. 2 ft. 6 in. Ans. 41 ch. Ans. 10 yd. 2 qr. 5. Reduce 8118 in. to ch. Ans. 7. Ans. 2 A. 3 R. 17 P. 9. Reduce 457 P. to A. 10. How many yards of carpet 1 yd. wide will be required to cover the floor of a room 33 ft. long and 24 feet wide? Ans. 88. 11. How many cubic yards in 124560 cu. in.? 12. What will it cost to dig a cellar 15 ft. long, 12 ft. wide, and 6 ft. deep, at $4.50 per cu. yd.? Ans. $180. 13. What will be the cost of flooring a hall 84 ft. long and 36 feet wide, at $12 per square? Ans. $362.88. 14. What is the cost of a pile of bark 32 ft. long, 12 ft. wide, and 8 ft. high, at $10.24 per cord? Ans. $245.76. 15. What will it cost to build a wall 74.25 ft. long, 1.75 Ans. $188.16. thick, and 6.4 ft. high, at $5.60 a perch? 16. A field is 60 rd. long; how wide must it be to contain 18 acres? Ans. 48 rd. 17. A room is 25 ft. long, 15 ft. wide, and contains 4875 cu. ft.; how high is it? Ans. 13 ft. |