RULE. Multiply the fraction of the higher denomination by the numbers in the descending scale successively, between the given and the required denomination. NOTE. Cancellation may be applied wherever practicable. EXAMPLES FOR PRACTICE. 1. Reduce of a lb. Troy to the fraction of a pennyweight. 4 Ans. pwt. 5 1 672 2. Reduce of a hhd. to the fraction of a pint. 3 3. Reduce of a mile to the fraction of a yard.. प Ans. ğ yd. 19 3 4. Reduce of a gallon to the fraction of a gill. 5. What part of a dram is 5' of of 3 of 6 of 34 pounds avoirdupois weight? 45 Ans. 192 dr. 4375 1680 Ans. § 1. 7 6. Reduce 10ʊ of a dollar to the fraction of a cent. RULE. I. Multiply the fraction by that number in the scale which will reduce it to the next lower denomination, and if the result be an improper fraction, reduce it to a whole or mixed number. II. Proceed with the fractional part, if any, as before, until reduced to the denominations required. III. The units of the several denominations, arranged in their order, will be the required result. EXAMPLES FOR PRACTICE. 1. Reduce of a yard to integers of lower denominations. Ans. 2 ft. 8 in. 2. Reduce of a month to lower denominations. 640 7 of a short ton to lower denominations. 4. What is the value of of a long ton? Ans. 11 cwt. 12 lb. 71 oz. 5. What is the value of 3 of 24 pounds apothecaries' weight? 6. What is the value of of an acre ? Ans. 2 R. 6 P. 4 sq. yd. 5 sq. ft. 1275 sq. in. 7. Reduce of a mile to integers of lower denominations. Ans. 6 gross 10 doz. 33. 9. What is the value in geographic miles of circle? 16 of a great Ans. 12150 mi. 10. What is the value of of 33 cords of wood? Ans. 2 Cd. 5 cd. ft. 93 cu. ft. 11. The distance from Buffalo to Cincinnati is 438 miles; having traveled of this distance, how far have I yet to travel? 12. What is the value of 13. What is the value of Ans. 262 mi. 6 fur. 16 rd. Ans. 33 gal. 3 qt. 1 pt. 173 gi. 14. A man having a hogshead of wine, sold of it; how much remained? CASE IV. 370. To reduce a denominate decimal to integers of lower denominations. 1. Reduce .125 of a barrel to integers of lower denominations. OPERATION. .125 31.5 3.9375 gal. 3.7500 qt. 1.50 pt. 4 2.0 gi. 3 gal. 3 qt. 1 pt. 2 gi., Ans. ANALYSIS. We first multiply the given decimal, .125 of a barrel, by 31.5 (31) to reduce it to gallons, and obtain 3.9375 gallons. Omitting the 3 gallons, we multiply the decimal, .9375 gal., by 4 to reduce it to quarts, and obtain 3.75 quarts. We next multiply the decimal part of this result by 2, to reduce it to pints, and obtain 1.5 pints. And the decimal part of this result we multiply by 4 to reduce it to gills, and obtain 2 gills. The integers of the several denominations, arranged in their order, form the answer. Hence, RULE. I. Multiply the given denominate decimal by that number in the descending scale which will reduce it to the next lower denomination, and point off the result as in multiplication of decimals. II. Proceed with the decimal part of the product in the same manner until reduced to the required denominations. The integers at the left will be the answer required. EXAMPLES FOR PRACTICE. 1. What is the value of .645 of a day? Ans. 15 h. 28 min. 48 sec. 2. What is the value of .765 of a pound Troy? 10. What is the value of .375625 of a barrel of pork? 11. What is the value of .1150390625 Cong.? Ans. 14 f3 5 f3 48 m. 12. What is the value of .61 of a tun of wine? Ans. 1 P. 27 gal. 2 qt. 1 pt. 3.04 gi. REDUCTION ASCENDING. CASE I. 371. To reduce a denominate number to a compound number of higher denominations. 1. Reduce 157540 minutes to weeks. OPERATION. 60) 157540 min. 24) 2625 h. + 40 min. 7) 109 da. 9 h. 15 wk. + 4 da. 15 wk. 4 da. 9 h. 40 min., Ans. ANALYSIS. Dividing the given number of minutes by 60, because there are as many hours as minutes, and we obtain 2645 h. plus a remainder of 40 min. We next divide the 2645 h. by 24, because there are as many days as hours, and we find that 2645 h. =109 da. plus a remainder of 9 h. Lastly we divide the 109 da. by 7, because there areas many weeks as days, and we find that 109 da. = 15 wk. plus a remainder of 4 da. The last quotient with the several remainders annexed in the order of the succeeding denominations, form the mainder of 24 yd., and this reduced to its equivalent compound number, (169) = 2 yd. 1 ft. 6 in. In forming our final result, the 6 in. of this number are added to the first remainder, 2 in., making the 8 in. as given in the answer. From these examples and analyses we deduce the following RULE. I. Divide the given concrete or denominate number by that number of the ascending scale which will reduce it to the next higher denomination. II. Divide the quotient by the next higher number in the scale; and so proceed to the highest denomination required. The last quotient, with the several remainders annexed in a reversed order, will be the answer. NOTE. The several corresponding cases in reduction descending and reduction ascending, being opposites, mutually prove each other. EXAMPLES FOR PRACTICE. 1. Reduce 1913551 drams to tons. 2. In 97920 gr. of medicine how many lb.? 3. Reduce 1000000 in. to mi. Ans. 17 lb. 4. How many acres in a field 120 rd. long and 56 rd. wide? 5. In a pile of wood 60 ft. long, 15 ft. wide, and 10 ft. high, how many cords? Ans. 70 Cd. 2 cd. ft. 8 cu. ft. 6. How many fathoms deep is a pond that measures 28 ft. 6 in. ? 7. In 30876 gi. how many hhd.? 8. How many bushels of corn in 27072 qt.? Ans. 4 fath. Ans. 846 bu. 9. At 2 cts. a gill, how much alcohol may be bought for $2.54? 10. In 1234567 far. how many £? 11. Reduce 2468 pence to half crowns. Ans. £1286 13 d. Ans. 475. 12. In $88.35 how many francs? 13. In 622080 cu. in. how many tons of round timber? 14. In 84621 m how many Cong.? 15. If 135 million Gillott steel pens are manufactured yearly, how many great-gross will they make? Ans. 78125. 16. Reduce 1020300" to S. Ans. 9 S. 13° 25' 17. In 411405 sec. how many da.? |