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. Ans. 37d. 3. Reduce of a £ to the fraction of a penny. Ans. sec. of a bushel to the fraction of a pint. 9. Reduce of a day to the fraction of a second. 10 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. One One Q. How much is one-half of a £? One-third of a shilling? half of a penny? How much is one-half of a lb. Avoirdupois? fourth 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? lb. troy? 2. What is the value of of an acre? Ans. 9oz. 12pwt. 6. What is the value of % of a hogshead? Ans. 1gr. 7lb. Ans. 2R. 20P. Ans. S d. Ans. 52gal. 2qt. 7. What is the value of 18 of a hogshead? 8. What is the value of 9. What is the value of 504 Ans. gal. qt. of a guinea. Ans. 4s. 8d. of a lb. Troy? Ans. oz. pwt. 10. What is the value of of a tun of wine? CASE IV. Ans. 3hhd. 31gal. 2qt. § 101. To reduce a denominate number to a fraction of a given denomination RULE. Reduce the number to the lowest denomination mentioned in it: then if the reduction is to be made to a denomination still less, reduce as in Case II.; but if to a higher denomination reduce as in Case I. EXAMPLES. 1. Reduce 4s 7d to the fraction of a £. We first reduce the given number to the lowest denomination named in it, viz. pence. Then as the reduction OPERATION. 4s 7d=55d. 55 Then, 55 of of £50. is to be made to pounds, a higher denomination, we reduce by Case I. 2. What part of a bushel is 2pk. 3qt. We first reduce to quarts, this being the lowest denomination. We then reduce to bushels by Case I. OPERATION. 2pk. 3qt.=19qt. 19 of of 12 bu. 3. Reduce 2 feet 2 inches to the fraction of a yard. Ans. 13yd. 4. Reduce 3 gallons 2 quarts to the fraction of a hogs head. Ans. hhd. 5. Reduce 1qr. 77b. to the fraction of a hundred. 6. What part of a hogshead is 3qt. 1pt.? 7. What part of a mile is 6ft. 7in. ? 8. What part of a mile is 1 inch? 9. What part of a month of 30 days, is 1 hour 1 minute 1 second? 10. What part of 1 day is 3hr. 3mi. ? Ans. 1440 Ans. 183 11. What part is 3hr. 3m. of two days? Of 3? Of 4? Of 10? Of 25? ADDITION OF VULGAR FRACTIONS. § 102. Addition of integer numbers teaches how to express all the units of several numbers by a single number. Addition of fractions teaches how to express the value of several fractions by a single fraction. It is plain, that we cannot add fractions so long as they have different units: for, of a £ and of a shilling make neither £1 nor 1 shilling. Neither can we add parts of the same unit unless they are like parts; for of a £ and of a £ nor 2 of a £. of a £ make neither But of a £ and of a £ may be added: they make of a £. So, of a £ and 2 of a £ make of a £. Hence, before fractions can be added, two things are necessary. 1st. That the fractions be reduced to the same denomi nation. 2nd. That they be reduced to a common denominator. Q. What does addition of integer numbers teach? What does addition of fractions teach? What two things are necessary before fractions can be added? Can one-half of a £ be added to one-half of a shilling without reduction? Can one-half be added to one-fourth without reduction? CASE 1. § 103. When the fractions to be added are of the same denomination and have a common denominator. RULE. Add the numerators together, and place their sum over the common denominator: then reduce the fraction to its lowest terms, or to its equivalent mixed number. Q. When the fractions are of the same denomination and have a common denominator, how do you find their sum? What is the sum of one-third and two-thirds? Of three-fourths, one-fourth, and fourfourths? Of three-fifths, six-fifths, and two-fifths? Of three-sixths, seven-sixths, and nine-sixths? Of one-eighth, three-eighths and foureighths? 2. Add of a £, of a £, and of a £ together. Ans. 15 of a £.=£27, 3. What is the sum of 3+++13+16. Ans. 42=14. 4. What is the sum of ++++3⁄4. Ans. 2. 3 14 8 CASE II. 9 5 14 14 14 § 104. When the fractions are of the same denomi nation but have different denominators. RULE. Reduce compound fractions to simple ones, mixed numbers to improper fractions, and all the fractions to a common denominator. Then add them as in Case I. EXAMPLES. 1. Add,, and together. By reducing to a common denominator the new fractions are 38+30+32=142, which, by reducing to the lowest terms becomes 411. OPERATION. 6x3x5=90 1st numerator. 4×2×5=40 2nd numerator. 2×3×2=12 3rd numerator. 2x3x5=30 the denominator. Q. How do you add fractions which have different denominators? How do you reduce fractions of different denominators to equivalent fractions having a common denominator? 2. Add of a £, 3 of a £, and & of a £ together. Ans. £15-£175=£135. 3. Add together,}, 4, and 63. 126 Ans. 10348. |