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 penny to the fraction of a £. 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 3oz avoirdupois to the denomination of tons. Ans. 143360T. 4. Reduce of a pint to the fraction of a hogshead. Ans. 1536hhd. 5. Reduce of a shilling to the fraction of a £. Ans. £30. 6. Reduce of a farthing to the fraction of a £. Ans. £2880 of a gallon to the fraction of a hogshead. Ans. hhd. 7. Reduce Ans. £180 9. Reduce 17 of a minute to the fraction of a day. Ans. 182880 187 da. 10. Reduce of a pound to the fraction of a cut. Ans. cut. 11. Reduce of an ounce, to the fraction of a ton. Ans. 71880T. 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. OPERATION. 1. Reduce of a £ to the fraction of a penny. is equal to 1 shilling is equal to 12 pence; of 20 of 12=24°d. hence of a £4 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. 3. Reduce Ans. 4481b. Ans. 37d. Ans. 480m. of a £ to the fraction of a penny. 4. Reduce Ans. 480P. 6. Reduce of a £ to the fraction of a farthing. 7. Reduce gallon. 8. Reduce of a hogshead to the fraction of a Ans. 5760 far. Ans. gal. Ans. 25pt. Ans. 259209 se of a bushel to the fraction of a pint. 9. Reduce of a day to the fraction of a second. 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. Multiply the numerator by that number which makes one of the next lower denomination, and divide the product by the denominator. II. If there be a remainder, multiply it by that number which makes one of 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 cwt.? 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 lb. troy? Ans. 9oz. 12pwt. EXAMPLES. 1. Add,,, and 3 together. It is evident, since all the parts are halves, that the true sum will be expressed by the number of halves that is by thirteen two's. OPERATION. 1+3+6+3 13. Hence, sum. = 4. Find the least common denominator (see § 96), and add the fraction,,, and . Ans. 118. 5. Find the least common denominator and add 6 3 , and. 12 5' Ans. 19 120 NOTE. 105. When there are mixed numbers, instead of reducing them to improper fractions we may add the whole numbers and the fractional parts separately, and then add their sums. 6. Add 19, 63, and 4 together. OPERATION. Whole numbers. 19+6+4=29. OPERATION. Fractional parts. +3+3=168=1105. Hence, 29+164-30645, the sum. 7. Add 31, 65, 895, and 65%. CASE III. 64 Ans. 84578. § 106. When the fractions are of different denominations. RULE. Reduce the fractions to the same denomination. Then reduce all the fractions to a common denominator, and then add them as in Case I EXAMPLES. 1. Add of a £ to 1⁄2 of a shilling. Then, of a £3 of 20-40 of a shilling: S= Or, the of a shilling might have been reduced to the fraction of a £ thus, Then, 5 of 20=150 of a £=24 of a £. 8 5 3+2+1=1 of a £: which being re100, gives 14s 2d. 2. Add of a yard to 5 of an inch. duced by 3. Add together. Ans. of a week, of a day, and 4. Add of a cwt., 85lb. and 3oz. together. Ans. 2qr. 171b. 13oz. |