to a simple fraction. =4÷÷÷6=√, Ans., by Art. 143, Rule 2; or, Х 5 == Ans., by Art. 84 (a) and Art. 142 (a). 7 42' to a simple fraction. to a simple fraction. of % of Ans., by Art. 145 (c).. of 14 1 of 2 to its simplest form. Ans. 1. PROBLEM 9. 147. To reduce fractions that have not a common denominator to equivalent fractions that have a common denominator. Ex. 1. Reduce and to equivalent fractions having a common denominator. Ans. and . 213 ས་ པ་ 7 3 21 15 21 Multiplying both terms of each fraction by the denominator of the other fraction will not alter the value of either fraction (Art. 84, a), but it will necessarily make the denominators alike, for each new denominator is the product of the two given denominators. Similar reasoning applies, however many fractions are to be reduced. Hence, RULE 1. Multiply all the denominators together for a common denominator, and multiply each numerator into the continued product of all the denominators, except its own, for new numer ators. 147. Common denominator, how found by Rule 1? How the numerators? Explanation? 2. Reduce,, and to equivalent fractions having a com mon denominator. = OPERATION. X 7 X 9 252, common denominator, 3. Reduce,, and . 4. Reduce 3, 3, and 4. Ans. 4, 195, and 6. Ans. 131, 18, and 388. 5. Reduce,,, and . Ans. 168, 188, 188, and 18. 6. Reduce, 4, and §. 11. Reduce, 8, 7, and 13. 7. Reduce, †, and §. 12. Reduce, 4, 1, and 71. 8. Reduce 1,, and. 13. Reduce ‡, †,†, and 14. 1, 9. Reduce, %, and 15. 14. Reduce, 27, 4, and 3. 10. Reducer,, and. 15. Reduce 1, 17, 5, and f. (a) The foregoing rule will always give a common denominator, but not always the least common denominator; this, however, may always be effected by RULE 2. Reduce each fraction, if necessary, to its lowest terms (Art. 141). Find the least common multiple of the denominators (Art. 127) for a common denominator. Divide this multiple by each given denominator, and multiply the several quotients by the respective numerators for new numerators. NOTE 1. Each of these rules is founded on the principle that multiplying both terms of a fraction by the same number does not alter its value. 16. Reduce, §, and 72. OPERATION BY THE SECOND RULE. 2 X 2 X 3 X 2 24, least common multiple of denominators, 3 5 7 2) 8' 6' 12 24 X 5 14, 3d numerator; 9, 1st numerator, 20, 2d numerator, 147. Rule for finding the least common denominator? Rule for finding the numerators? Principle? 17. Reduce 1o,,, and . 18. Reduce, fo, do, and go. 19. Reduce, 1‰, 1, and 13. 32 30 Ans. 28, 18, 33, 38 NOTE 2. The first clause of Rule 2 is omitted by many authors, but its necessity is apparent from the following example: 20. Reduce,, and to equivalent fractions having the least common denominator. Disregarding the first clause of the rule, we find 72 to be the least common multiple of the denominators, and the fractions §, , and, reduce to 4, 4, and ; but, regarding the first clause, we have,, and 4, 3, and 4=12, 12, and 12, 1: which have a common denominator less than 72. 21. Reduce, †, 12, and ‡. 22. Reduce §, 1%, 11⁄2, and 1. 23. Reduce, 18, 1, and 18. 24. Reduce 18, 18, 1%, and 48. 15 Ans. 18, 18, 15, and 18. NOTE 3. In this and the following problems, each fraction should be in its simplest form before applying the rule. REMARK. The numerators, as well as the denominators, of fractions, may be made alike by reduction; thus, } and 1⁄2 are equal in value to 1 and 1; also ‡ and 14 and 1}; also 4, 1, and 44, 44, and 24; etc. The process is simple, but of little practical importance, and therefore seldom presented in Arithmetic. 147. May the numerators of fractions be made alike? How? PROBLEM 10. 148. To reduce a fraction of a higher denomination to a fraction of a lower denomination. Ex. 1. Reduce of a penny, to the fraction of a farthing. As 1 penny is equal to 4 farthings, so any fraction of a penny will be 4 times as great a fraction of a farthing; .. }d.=4 times qr.qr., Ans. 2. Reduce 4 of a shilling to the fraction of a farthing. As 1s. is equal to 12d., so 48. d. 4 times qr. = 12 times 4d. = ‡d., and #qr., Ans. Hence, RULE. Multiply the fraction by such numbers as are necessary to reduce the given to the required denomination. 3. Reduces. to the fraction of a farthing. Js. (=d. X 12) = 3d. (=3qr. × 4) = 28qr., Ans. ; or, 7 X 12 X 4 7 X 12 X 4 36 = 36 3 28 3 qr., Ans., as before. NOTE 1. The sign of multiplication, in these examples, is written only between the numbers which are given before the canceling is begun; thus, in Ex. 3, no sign is written between 36 and 3, for they are not to be multiplied together, but the 3 is obtained by canceling 12 in 36. So in Ex. 4, the 12 comes from canceling 20 in 240, and the 3 from canceling 4 in 12. 4. Reduce of a ton to the fraction of a dram. 7 × 20 × 4 × 25 × 16 × 16 240 44800 dr., Ans. 3 NOTE 2. In the first statement of Ex. 5, the 163, in the numerator, is equal to 33, and, in the second statement, the 33 is retained in the numerator as a factor in the dividend, and the 2 is put in the denominator as a factor in the divisor. 148. Rule for reducing a fraction from a higher to a lower denomination? Explanation? How is Ex. 5 solved? 6. Reduce of a pound, Troy Weight, to the fraction of a grain. Ans. 188. 7. Reduce of a pound, Apothecaries' Weight, to the fraction of a grain. Ans. 108. 8. Reduce 4200 of a day to the fraction of a second. of a gallon to the fraction of a gill. Ans. 144. 9. Reduce of a bushel to the fraction of a pint. Ans. 128. 10. Reduce 11. Reduce 12. Reduce 13. Reduce 14. Reduce 15. Reduce 16. Reduce 17. Reduce 18. Reduce of an acre to the fraction of a square yard. yd. of cloth to the fraction of an inch. circ. to the fraction of a second. of a ton to the fraction of an ounce. 19. Reduce 3024 of a day to the fraction of a second. £ to the fraction of a farthing. of a bushel to the fraction of a pint. 20. Reduce 21. Reduce PROBLEM 11. 149. To reduce a fraction of a lower denomination to a fraction of a higher denomination. Ex. 1. Reduce of a barleycorn to the fraction of an inch. In 15 barleycorns there is only of 15 inches, so in of a barleycorn there is only of of an inch of an inch, Ans. 2. Reduce of a gill to the fraction of a quart. As 1 gill is a like reason, of a pint, so = = gi. is of pt. pt. and, for qt.qt., Ans. Hence, RULE. Divide the given fraction by such numbers as are required to reduce the given to the required denomination. 149. Rule for reducing a fraction from a lower to a higher denomination? Explanation? |