EXAMPLES IN REDUCTION OF FRACTIONS, Ans.. Ans. 11. Ans. 115 Ans. 3024 Ans. 152 456 100 100 1. Reduce 57 to its lowest terms. Ans. 2. Reduce 1429 to its lowest terms. 2858 1296 617 4319 Ans. 1618 10. Reduce 14913to an improper fraction. Ans. 20529. 11. Reduce 141029 to a whole or mixed number. Ans. 6538 12. In 67856 of pounds of sugar, how many pounds? 13. In 22 685 of hhd. of wine, how many hhd. ? Ans. 6132hhd. 14. In 33bushels of wheat, how many bushels ? 15. Reduce 1 of of } of į of } to a simple fraction. 16. In i of of 1 of 21 dollars, how many dollars ? 137 360 35 Ans. sa: Ans. 54 8 17. Reduce of of of of of 1 to a simple 흥 fraction. Ans. 5. 18. I bought ii of į of a ship: what part did I buy ? 19. Sold z of j; of 265 yards of cloth : how many did I sell ? Ans. 131 yards. 20. In 1 of 157 of 57 of 100 hogsheads of sugar, how many hogsheads? Ans. 575811 21. Reduce , 7, 8, and 54 to a common denominator. Ans. 14, 441. 504. 324. 22. What is the least common denominator of i, s k. 3 ? Ans. 23. What is the common denominator of số and 12, 92, 724 63, 63, 63. 5 60 Ans. 54 12} ? 888 24. Reduce , , and of 11 to a common denomi. nator. Ans. 25. What is the common denominator of į of of 2, of of 84, and of 9į? 26. What is the common denominator of of 2 of } of 3 of of 4, } of 5 of of 6 of of 8, and of 21 of 31 of 1? Ans. O 153 1 68 30. Ans. 133, 432, 43 2* 2 REDUCTION OF DENOMINATE FRACTIONS. a 138. We have seen (Art. 61) that a denominate num. ber is one in which the kind of unit is denominated or expressed. For a like reason, a denominate fraction is one which expresses the kind of unit that has been divided. Such unit is called the unit of the fraction. Thus, of a £ is a denominate fraction. It expresses that one £ is the unit which has been divided. The fraction of a shilling is also a denominate fraction, in which the unit that has been divided is one shil. ling. These two fractions are of different denominations, the unit of the first being one pound, and that of the second, one shilling. Fractions, therefore, are of the same denomination when they express parts of the same unit, and of different denominations when they express parts of different units. REDUCTION of denominate fractions consists in changing their denominations without altering their values. a 138. What is a denominate number? What is a denominate fraction? What is the unit called ? In two-thirds of a pound, what is the unit? In three-eighths of a shilling, what is the unit? In onehalf of a foot, what is the unit?' When are fractions of the same denomination? When of different denominations ? Are one-third of a £ and one-fourth of a £ of the same or different denominations ? One-fourth of a £ and one-sixth of a shilling? One-fifth of a shilling and one-half of a penny? What is reduction? How many shillings in a £? How many in £2? In 3? In 4? How many pence in 1s. ? In 2? In 3? In 2s. 8d.? In 3s. 6d.? In 58. 8d.? How many feet in 3 yards 2ft.? How many inches ? CASE I. 139. To reduce a denominate fraction from a lower to a higher denomination. I. Consider how many units of the given denomination make one unit of the next higher, and place 1 over that number forming a second fraction. II. Then consider how many units the second denomination make one unit of the denomination next higher, and place 1 over that number forming a third fraction; and so on, to the denomination to which you would reduce. III. Connect all the fractions together, forming a compound fraction ; then reduce it to a simple one by Case V. EXAMPLES. a OPERATION. OPERATION. 1. Reduce } of a penny to the fraction of a £. The given fraction is of a penny. But one penny is of 12 of žb=£ão: equal to t'a of a shilling : hence of a penny is equal to } of La of a shilling. But one shilling is equal to of a pound: hence of a penny is equal to 1 of 1 of 2 of a £=£720. The reason 20. of the rule is therefore evident. 2. Reduce of a barleycorn to the denomination of yards. Since 3 barleycorns make an inch, we first of } of Iz of }=a yards. place 1 over 3: then as 12 inches make a foot, we place 1 over 12, and as 3 feet make a yard, we next place 1 over 3. 3. Reduce oz. avoirdupois to the denomination of tons. Ans. 12:00T. 4. Reduce of a pint to the fraction of a hogshead. Ans. hhd. 5. Reduce j of a farthing to the fraction of a £. Ans. £850 139. How do you reduce a denominate fraction from a lower to a higher denomination? What is the first step? What the second? What the third ? 3 64 a 6. Reduce of a gallon to the fraction of a hogshead. Ans. & hhd. 7. Reduce ã of a shilling to the fraction of a £. Ans. £16 8. Reduce 187 of a minute to the fraction of a day. Ans. 1828787da. 9. Reduce g of a pound to the fraction of a cwt. Ans. cwt. 10. Reduce of an ounce to the fraction of a ton. Ans. wboot. 00 CASE II. 140. To reduce a denominate fraction from a higher to a lower denomination. 1. Consider how many units of the next lower denomination make one unit of the given denomination, and place 1 under that number forming a second fraction. II. Then consider how many units of the denomination still lower make one unit of the second denomination, and place 1 under that number forming a third fraction, and so on, to the denomination to which you would reduce. III. Connect all the fractions together, forming a compound fraction, and then reduce it to a simple one by Case V. EXAMPLES. 1. Reduce of a £ to the fraction of a penny. In this example of a pound OPERATION. is equal to 1 of 20 shillings. 4 of 4 of 1=249d. But 1 shilling is equal to 12 pence: hence 7 of a £=} of 21 of Y=240d. Hence, the reason of the rule is manifest. 2. Reduce cut. to the fraction of a pound. Ans. 44016. 3. Reduce 15 of a £ to the fraction of a penny. Ans. Ya. 140. What do you first do in reducing a denominate fraction to a lower denomination? What next? What next? a 4. Reduce } of a day to the fraction of a minute. Ans. 480m. 5. Reduce of an acre to the fraction of a pole. Ans. 480P. 6. Reduce of a £ to the fraction of a farthing: Ans. 57. far. 7. Reduce så of a hogshead to the fraction of a gallon. Ans. gal. 8. Reduce io of a bushel to the fraction of a pint. Ans. 48 pt. a a CASE III. 141. To find the value of a fraction in integers of a less denomination. I. Reduce the numerator to the next lower denomination, and then divide the result by the denominator. II. If there be a remainder, reduce it to the denomination stiil less, and divide again by the denominator. Proceed in the same way to the lowest denomination. The several quotients, being connected together, will form the equivalent denominate number. a EXAMPLES. a 40 1. What is the value of of a £? OPERATION. 2 We first bring the pounds to 20 shillings. This gives the fraction of shillings, which is 3)40 equal to 13 shillings and 1 over. 138. ...1 Rem. Reducing this to pence gives the 12 fraction of pence, which is 3)12 equal to 4 pence. 4d. Ans. 138. 4d. 141. How much is one-half of a £? One-third of a shilling? One-half of a penny? How much is one-half of a lb. Avoirdupois ? One-fourth of a ton? One-fourth of a cut.? One-half of a quar. ter? One-fourth of a quarter? One-seventh of a quarter? Onefourteenth of a quarter ? One twenty-eighth of a quarter? How do you find the value of a fraction in integers of a less denomination ? a |