nominator, and the products will be the numerators of the required fractions; under which write the least common denominator. EXAMPLES. 1. Reduce, and to their least common denomi nator. OPERATION. 2)7--8--6 7--4--3 and 3×4×7×2=168 the least common denominator. 2. Reduce, and to their least common denomi nator. 45 Ans. 36 40 45, 48 and 9 3. Reduce 145, 63 and 51, to their least common denominator. Ans. 4. Reduce 3 24 and to their least common denomi nator. 6 1209 409 5. Reduce 67 nator. 320 60 Ans. 72 360 360 360° , to their least common denomi Ans. 67 18 300 120 120 120 6. Reduce 1, 31, 41 and 8 to a common denominator. Ans. 82 605 450 800 100 100 100 100 7. Reduce 31, 412, 81, 1417%, to their least common denominator. 10. Reduce,,, 3, 1, and to equivalent fractions having the least common denominator. Q. How do you reduce fractions to their least common denominator? Does this reduction affect the values of the fractions? REDUCTION OF DENOMINATE FRACTIONS. § 97. We have seen § 45, that a denominate number is one in which the kind of unit is denominated or expressed. For the same 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. Q. 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 one-half of a foot, what is the unit? When are fractions of the same denomination? When of different denominations? Are one-third of a £ and onefourth 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 Is? In 2? In 3? In 2s 8d? In 3s 6d? In 5s 8d? How many feet in 3 yards 2ft.? How many inches? CASE I. 98. To reduce a denominate fraction from a lower to a higher denomination. RULE. 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 the compound fraction to a simple one by Case V.: EXAMPLES. OPERATION. 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 a penny is equal to shilling is equal to is equal to of } of 11⁄2 of 1=£7}0 2 of of a shilling. of a pound: hence But one of a penny of 12 of 20 of a £=£720. 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 OPERATION. ofofof=84 yards. 12 inches make a foot, we place 1 over 12, and as 3 feet make a yard, we next place 1 over 3. Q. 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. Reduce oz avoirdupois to the denomination of tons. Ans. 143360T. 4. Reduce of a pint to the fraction of a hogshead. of a shilling to the fraction of a £. Ans. hhd. Ans. £300. Ans. £2880 5. Reduce 7. Reduce of a gallon to the fraction of a hogshead. Ans. hhd. 8. Reduce of a shilling to the fraction of a £. Ans. £ 9. Reduce 17 of a minute to the fraction of a day. 87 Ans. 187 182880da. 10. Reduce of a pound to the fraction of a cut. Ans. 11. Reduce of an ounce to the fraction of a ton. cwt CASE II. 99. To reduce a denominate fraction from a higher to a lower denomination. RULE. 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. Then reduce the compound fraction 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 pence; OPERATION. of 20 of 12-24°d. hence of a £ of 20 of 12=24°d. Hence the reason of the rule is manifest. Q. What do you first do in reducing a denominate fraction to a lower denomination? What next? What next? 2. Reduce cwt. to the fraction of a pound. Ans. 4481b. 3. Reduce of a £ to the fraction of a penny. Ans. 32d. 4. Reduce of a day to the fraction of a minute. 5. Reduce Ans. 480m. of an acre to the fraction of a pole. Ans. P. 6. Reduce of a £ to the fraction of a farthing. Ans. 5760 far 7. Reduce gallon. of a hogshead to the fraction of a Ans. gal 10. Reduce of a tun to the fraction of a gill. CASE III. Ans. 40320 gill. § 100. To find the value of a fraction in integers of a less denomination. RULE. 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. Q. How much is one-half of a £? One-third of a shilling? Onehalf of a penny? How much is one-half of a lb. Avoirdupois? Onefourth of a ton? One-fourth of a cut. ? One-half of a quarter? Onefourth of a quarter? One-seventh of a quarter? One-fourteenth of a quarter? One-twenty-eighth of a quarter? How do you find the value of a fraction in terms of integers of a less denomination? 2. What is the value of 3. What is the value of 4. What is the value of 5. What is the value of lb. troy? of a cwt.? of an acre? of a £? Ans. 9oz. 12pwt. 6. What is the value of 1⁄2 of a hogshead? Ans. 1qr. 7lb. Ans. 2R. 20P. Ans. S d. Ans. 52gal. 2qt. |