tions of time, and repeat the table. Name the months of the year, and tell how many days are in each. 9. In Counting, what units or denominations are used? Repeat the table. In buying and selling paper? Repeat the table. 10. In integers and in decimals, how do the orders of units increase and decrease? In compound numbers, is there any regular rate or scale of increase and decrease? How do the different orders of units in integers, decimals, and compound numbers increase? Where are the highest orders, or denominations, placed? The lowest? SECTION XXI. REDUCTION OF DENOMINATE NUMBERS. 410. Reduction of Denominate Numbers is the process of changing them from one denomination to another without changing their value. 411. Reduction of denominate numbers is of two kinds : Reduction Descending and Reduction Ascending. 412. Reduction Descending is the process of changing a denominate number to an equivalent number expressed in lower denominations. 413. Reduction Ascending is the process of changing a denominate number to an equivalent number expressed in higher denominations. CASE I. Reduction Descending. 1. How many inches in 1 ft.? In 2 ft.? In 3 ft.? 2. Change 2 ft. 6 in. to inches. 414. Analysis. Since 1 foot is 12 inches, 2 feet are 2 times 12 inches, or 24 inches; 24 inches and 6 inches are 30 inches. 3. In 2 gal. are how many quarts? How many pints? 4. Express 3 bu. 2 pk. as pecks; as pints. 5 bu. 3 pk. 5. What is the value of 3 qt. 1 pt. of milk at 8 cents a pint? Of 4 gal. 2 qt.? Of 6 gal. 2 qt. 1 pt.? 6. At 50 cents a foot, what will 1 rd. of wire fence cost? At $.75 a foot, how much will 2 rd. 1 yd. of fence cost? 7. Change 1⁄2 lb. avoirdupois to ounces. 4 lb. 8 lb. 8. At 10 cents a quart, what will bu. of chestnuts cost? 415. Principle. A denominate number is changed to lower denominations by multiplication. Written Exercises. 416. Example. — Change 2 mi. 50 rd. 3 yd. to yards. EXPLANATION.-Since 1 mile is 320 rods, 2 miles are 2 times 320 rods, or 640 rods; 640 rods and 50 rods are 690 rods. 345 Since 1 rod is 5 yards, 690 rods are 690 times 5 yards, or 3450 yards plus 345 yards, or 3795 yards; 3795 yards and 3 yards are 3798 yards, Hence, 2 mi. 50 rd. 3 yd. are 3798 yds. 417. Rule for Reduction Descending. 1. Multiply the number of the highest denomination given by the number of units of the next lower denomination which equals one of that higher; and to the product add the given number of the lower denomination, if any. II. In the same manner reduce this result to the next lower denomination; and so proceed until the given number is changed to the denomination required. Problems. Change, or reduce 1. 8 rods to inches; 1 mi. 36 rd. 2 ft. to feet. 2. 2 mi. 7 yd. 14 in. to inches; 5 sq. yd. to square inches. ft. 100 sq. in. to sq. in. cu. ft. to cu. in. 3. 2 A. to sq. 5. 100 gal. to pints'; 41⁄2 bbl. to quarts; 3.5 hhd. to gills. 6. 20 gals. 3 pt. 23 gi. to gills; 1 lb. Troy to grains. 7. 1 T. to pounds; 10 T. 5 cwt. 75 lb. 123 oz. to ounces. 8. $10 to mills; $1.07 to cents; $10.005 to mills. 9. £1 to farthings; £50 10 s. 81 d. to pence; to far. 10. 1° to seconds; 75° 20′ 35′′ to seconds. 11. 2 da. to seconds; 10 da. 20 hr. 10.75 min. to seconds. 12. 365 da. 5 hr. 48 min. 49.7 seconds to seconds. 13. of a gallon to gills. SOLUTION. gal. = 4 qt. 5 qt. 24 12 qt. 2 pt. 34 3 pt. 2 pt.2 X 4 gi. EXPLANATION.-Since 1 gallon is 4 quarts, of a gallon are of 4 quarts, or 2 of a quart. Since 1 quart is 2 pints, 12 of a quart are 2 of 2 pints, or 24 of a pint. Since 1 pint is 4 gills, of a pint are of 4 gills, or 25 of a gill, which equals 19 gills. Change, or reduce 14. g of a mi. to yd.; g rd. to ft.; yd. to in.; 42 ft. to in. EXPLANATION.-Since 1 ton is 20 hundred-weight, .007 of a ton are .007 times 20 cwt., or .140 of a hundred-weight. Since 1 hundred-weight is 100 pounds, .140 of a cwt. are .140 times 100 pounds, or 14 pounds. Hence, .007 of a ton equals 14 lb. 20. .009 yr. to hr.; .018 da. to sec.; .025 leap yr. to hr. 21. .5 of a ream to sheets; .0075° to "; .15 grt. gr. to duz. 22. $.005 to mills; £.075 to far.; 4.5 mi. to rd. and to ft. 23. .0625 hhd. to qt.; .008 gal. to gills; .024 bbl. to qt. 24. .001 A. to sq. ft. ; .02 A. to sq. ch.; .05 sq. rd. to sq. in. 25. How many feet long is the telegraph-wire between Philadelphia and New York, the distance being 96 miles? 26. If a boy buys a bushel of pea-nuts for $2.50, and sells them at $.05 a pint, what does he clear? At 6 cents? How many shirts will 72 27. On one shirt are 6 buttons. gross of buttons trim? 28. A stationer bought 5 and sold it at 1 cent a sheet. 60 dozen? reams of paper at $3.50 a ream, What did he gain? a 29. How much younger is man who is 50 years old, than one who is 3 score and 10 years old? 30. If your pulse beats 75 times in a minute, how often does it beat in a day? In a week? A common year? 31. How many jugs, each containing 2 qt., can be filled from a barrel of vinegar? From a hogshead? 32. From 4 oz. 3 dr. 2 scr. of calomel, how many pills, each weighing 5 grains, can be made? Each 4 gr.? 33. How many papers, each weighing .25 of a pound, can be filled from 5 cwt. 5 lb. of garden-seed? 34. If 1 pk. of clover-seed is sown on an acre of land, how many acres will 4.5 bushels seed? 35. If one pound sterling is worth $4.866 in United States money, how much are £10 worth? 36. How much are 10 gal. 2 qt. of maple-syrup worth at $.20 a quart? At $.311? At $.373? At $.50 a gallon? CASE II. Reduction Ascending. 1. How many feet in 36 in.? How many yards? 2. Change 48 far. to shillings; 200 ct. to dollars. 3. In 31 quarts, how many pecks? 418. Analysis. Since 8 quarts are 1 peck, 31 quarts are as many pecks as the number of times 8 quarts are contained in 31 quarts, or 3 pecks and 7 quarts remaining. Hence, etc. 4. How many gallons are 22 pints? 59 pints? 37 pints? 5. Change 51 gi. to pints. To quarts. To qt. pt. gi. 6. At $2.56 for 32 qt. of berries, what is the cost of 2 bu.? 7. What part of a pound Troy are 6 oz.? 8 oz.? 10 oz.? 8. At 24 cents a yard, what will 2 ft. of ribbon cost? What will 18 in. cost? 12 in.? 9 in.? 8 in.? 6 in.? 419. Principle. A denominate number is changed to higher denominations by division. 2 Written Exercises. 420. Example.-Change 608 ft. to higher denominations. SOLUTION. 3 ft.) 608 ft. yd.) 202 yd. + 2 ft. 2 11 ) 404 half-yd. 8 36 rd. + yd. 4 yd. 608 ft. 36 rd. 4 yd. 2 ft. are 1 yard, 608 ft. are as many yards as the number of times 3 ft. are contained in 608 ft., or 202 times with 2 remaining, or 202 yd. 2 ft. Since 5 yd. are 1 rod, 202 yd. are as many rods as the number of times 5 yd. are contained in 202 yd. 5 yd. changed to half-yards are 11 half-yards, and 202 yd. are 404 half-yards. 11 half-yards are contained in 404 half-yards 36 times with 8 half-yards, or 4 yards, remaining. Hence, 608 feet are 36 rd. 4 yd. 2 ft., the higher denominations required. 421. Rule for Reduction Ascending. I. Divide the given number by the number of units of its own denomination which equals one of the next higher, and write the remainder, if any. II. In the same manner change this result to the next higher |