8. Reduce 451 to its equivalent improper fraction. Ans. 1618 10. Reduce 149119 to an improper fraction. 100 · Ans. 20529. 11. Reduce 12410 to a whole or mixed number. 137 Ans. 653 12. In 67856 of pounds of sugar, how many pounds? 13. In 23885 of hhd. of wine, how many hhd. ? 360 Ans. 61125hhd. 14. In 354 bushels of wheat, how many bushels? 15. Reduce ofofofofto a simple fraction. Ans. 1 16. In of of of 21 dollars, how many dollars? 17. Reduce of fraction. 18. I bought 9 Ans. 5. of of % of of 40 to a simple Ans. 5. of of a ship: what part did I buy? 8 19. Sold of of 265 yards of cloth: how many did I sell ? Ans. 131 yards. 20. In of 153 of 57 of 100 hogsheads of sugar, how many hogsheads? Ans. 575813. 21. Reduce, 7, 8, and 54 to a common denominator. 22. What is the least common denominator of 3, 7, 1. 31? Ans. 23. What is the common denominator of 3, 5, and 12? 54 60 888 Ans. 12 72 72 24. Reduce,, and § of to a common denomi nator. Ans. 25. What is the common denominator of of of 2, Ans. 30, 1503, 16. of of 84, and of 9? 26. What is the common denominator of of 2 of of 4, of 5 of 1 of 6 of 1 of 8, and 3 of 24 of of 3 of 3 of 1? Ans. 432 416 567 432 432' 432' REDUCTION OF DENOMINATE FRACTIONS. 138. We have seen (ART. 61) that a denominate number 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 shilling. 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. 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? shillings in a £? How many in £2? In 3? In 4? pence in 1s.? In 2? In 3? In 2s. 8d.? In 3s. 6d.? How many fee in 3 yards 2ft.? How many inches? How many How many . 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 of 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. OPERATION. of of £ 1. Reduce of a penny to the fraction of a £. The given fraction is of a penny. But one penny is equal to of a shilling: hence of a penny is equal to one shilling is equal to of of a shilling. But of a pound: hence of a penny is equal to of1⁄2 of of a £=£10. The reason of the rule is therefore evident. 2. Reduce of a barleycorn to the denomination of yards. Since 3 barleycorns make an inch, we first 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 Ans. 1280007. tons. 3 4. Reduce of a pint to the fraction of a hogshead. Ans. hhd. 5. Reduce of a farthing to the fraction of a £. Ans. £280 139. 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? 6. Reduce of a gallon to the fraction of a hogshead. Ans. hhd. 7. Reduce of a shilling to the fraction of a £. 1 Ans. £180 8. Reduce of a minute to the fraction of a day. Ans. 12da. 9. Reduce of a pound to the fraction of a cut. Ans. cwt. 10. Reduce of an ounce to the fraction of a ton. Ans. 64000T CASE II. 140. To reduce a denominate fraction from a higher to a lower denomination. I. 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 is equal to of 20 shillings. But 1 shilling is equal to 12 OPERATION. 4 of 20 of 12=20od. pence: hence of a £= of 20 of 12-240d. Hence, the reason of the rule is manifest. 2. Reduce cwt. to the fraction of a pound. Ans. 400 lb. 3. Reduce of a £ to the fraction of a penny. Ans. 37d. 140. What do you first do in reducing a denominate fraction to a lower denomination? What next? What next? 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. 576 far. of a hogshead to the fraction of a Ans. gal. Ans. 256pt. of a bushel to the fraction of a pint. 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 still 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. 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 cwt.? One-half of a quarter? 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? |