EXAMPLES. 2. Reduce 167 to a mixed number. Ans. 1135 3. Reduce 1631 to a mixed number. Ans. 14116 4. Reduce 137 to a mixed number. 5. Reduce iti to a mixed number. Ans. 3131 6. Change 1900 to a mixed number. Ans. 111) 7. Change 4123 to a mixed number Ans. 9121 8. Change 123 to a whole number. Ans. 125. 9. Change 37 to a whole number. 222. To reduce a whole. or mixed number to an improper fraction. Ex. 1. Reduce 19 to a fraction whose denominator shall be 7. Since there are 7 sevenths in 1 19 x 7 = 133. whole one, 19 whole ones = 133 133 sevenths = Ans. sevenths OPERATION. 133, 123. RULE. — Multiply the whole number by the given denominator, and to the product add the numerator of the fractional part, if any; and write the result over the denominator. Note. A whole number may be expressed in its simplest fractional form, by taking it for a numerator with 1 for a denominator. Thus, 4 may be written , and read 4 ones. EXAMPLES. Ans. 3. Reduce 15 to fourths. Ans. 1044 6. Ans. 1 8. Change 5 to a fraction whose denominator shall be 17. Ans. i. 2. Reduce 98%; to an improper fraction. Ans. 254. 10. Reduce 116% 1 to an improper fraction. Ans. 1492. 11. 7187equal how many ninety-sevenths? Ans. 696.91. 12. Reduce 100198 to an improper fraction. Ans. 24832 13. Reduce 7 to an improper fraction. 14. Reduce 19 to a fraction whose denominator shall be 13. Ans. 213 15. 1161 yards equal how many fourths of a yard? Ans. 465 fourths. 223. To reduce a compound fraction to a simple fraction. Ex. 1. Reduce of } to a simple fraction. Ans. 3. By multiplying the denominator of ; by 4, the denominator of \, it is evident, we i X = 1, Ans. obtain of l = 3, since the parts into which the number is divided are 4 times as many, and consequently only as large as before ; and since of B = ja, of will be 3 times 3 RULE. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator. Note 1. — All whole and mixed numbers in the compound fraction must be reduced to improper fractions, before multiplying. NOTE 2. - When there are factors common to both numerator and denomi. nator, they may be cancelled in the operation. OPERATION. EXAMPLES. 2. Reduce of 75 of 4 of 3 to a simple fraction. Ans. It OPERATION, Ans. 1725 38 1 29 Ans. 7 15 29 38 11 3. What is of off of 11 ? = 4. What is of 1} of of i's ? 5. Reduce # of g of j of 1} to a simple fraction. Ans. 465 6. What is the value of i of 4 of 1 of 21 ? Ans. 339 = 27 . T756 7. What is the value of 1 of 15% of 570 of 100 ? Ans. 575811 8. What is of 4 of 1? 9. What is the value of 1 of of ft of $71? Ans. $ 1.75. 10. What is the value of of it of 13 of 3} gallons ? Ans.gal. 11. What part of a ship is ļof of ? 12. What is the value of 4 of zo of 18 of 14 of $ 34 ? Ans. $ 6.75. A COMMON DENOMINATOR. 224. Fractions have a common denominator when all their denominators are alike. 225. A common denominator of two or more fractions is a common multiple of their denominators; and their least common denominator is the least common multiple of their denominators. 226. To reduce fractions to a common denominator. Ex. 1. Reduce št, and it to other fractions of equal value, having a common denominator. FIRST OPERATION, 66 640 1536 7 x 12 x16=13 4 4 new numerator. } 1 1344 5 X 8 X 16 640 ਧ Ans. 11 X 8 X 12= 10 5 6 56 16 = 1838 8 X 12 X16=153 6 common denominator. We first multiply the numerator of } by the denominators 12 and 16, and obtain 1344 for a new numerator. We next multiply the numerator of is by the denominators 8 and 16, and obtain 640 for a new numerator; and then we multiply the numerator of it by the denominators 8 and 12, and obtain 1056 for a new numerator. Finally, we multiply all the denominators together for a common denominator, and write it under the several numerators, as in the operation. By this process, since the numerator and denominator of each fraction are multiplied by the same numbers, their relation to each other is not changed, and the value of the fraction remains the same. (Art. 217.) SECOND OPERATION. 148 least common denominator. 8 12/16 8 6 X 7= 42, new numerator. } A 言 3 12 4 x 5=20, i's=Ans. 16 3 X 11= 33," id=43) 16 X 3 = 48, least common multiple, and least common denominator. Having first obtained the least common multiple of all the denominators of the given fractions, we assume this to be their least common denominator. We then take such a part of this number, 48, as is expressed by each of the fractions separately for their respective new numerators. Thus, to get a new numerator for }, we take of 48, the least common denominator, by dividing it by 8, and multiplying the quotient 6 by 7. We proceed in like manner with each of the fractions, and write the numerators thus obtained over the least common denominator. In this process the value of each fraction remains unchanged, as both terms are multiplied by the same number. (Art. 217.) The method used in the second operation, it will be perceived, expresses the fractions of the result in lower terms than that used in the first. On this account it is often to be preferred to the other. RULE. — Find the least common multiple of the denominators for the LEAST COMMON denominator. Divide the least common denominator by the denominator of each of the given fractions, and multiply the quotients by their respective numerators, for the new numerators. Or, Multiply each numerator by all the denominators except its own, for the new numerators; and all the denominators together for A COMMON denominator. NOTE 1. Compound fractions must be reduced to simple ones, whole and mixed numbers to improper fractions, before finding a common denominator, and all to their lowest terms, before finding the least common denominator. NOTE 2. Fractions may sometimes be reduced to a common denominator most readily by multiplying both terms of one or more of them by such a number as will make all the denominators alike. Thus } and may be brought to a common denominator simply by multiplying both terms of the } by 2, and changing in that way its form to ą. Note 3. — Fractions may often be reduced to lower terms, without destroying their common denominator, by dividing all their numerators and denominators by a common divisor. EXAMPLES. Reduce the following fractions to their least common denominator : 2. Reduce , Ś, }, and is Ans. 54, 42, 43, 4. 3. Reduce , 18, and Ans. 1573974, 1338, 2725 96 798 84 20, 20, 240, 240. 4. Reduce t, 1t, and 5. Reduce ti, 4, 11, and . Ans. 3, 34, 13, 41. 6. Reduce }}, , , and Ans. 219, 18045, , . 7. Reduce }, }, }, and . Ans. 38, 40, 45, 48. 8. Reduce 4, 4, it, and 11: Ans. $$4, 1, 2, . 9. Reduce , }, ¢, and 36. 10. Reduce , g, 4, and 43. Ans. 768, 1968, 168, 16g 11. Reduce 4, 1, 11, and Z. Ans. , 13, 33, 41. 12. Reduce ş, la, 1%, and LT: Ans. 24, 147, 238, 24, 13. Reduce 18, 30, 41, and 3. Ans. $78, 148, 266, %. 14. Reduce 5, 75, 7o, and 20 Ans. 210. 140. 105. 15. Reduce , 7, 8, and 57. Reduce the following fractions to a common denominator : 16. Reduce ş, , and to fractions having a common denominator. Ans. 1925, 120, 12 / . 17. Reduce 4, 5, and Ans. 360. 560 , . 18. Reduce 41, 4, and is. Ans. 343115701, OUT. 19. Reduce in, 143, and 71. 20. Reduce 11, , and is. Ans. 14, 1935, 23125 21. Reduce ș, ft, and 1115. Ans. 2948, 195, 2006 22. Reduce }, , 4, and 8. 23. Reduce $, TT, and sof 75. Ans. 112, 11313%. 24. Reduce 1, 1, ], and 17. 25. Reduce 11, of 6, and 211. Ans. 118, 129, 576, 2580 26. Reduce 4, 11, 13, 4, and to Ans. , , , . 27. Reduce 26, 27, and 1738. Ans. 2850 68 16 96 105 630, 630, 630 Ans. Á 1, 42, 43, 42 28 24 3 3 6 37088064 7 228 13 1784 315 523 1784 315 52, 1784315 52• ADDITION OF COMMON FRACTIONS. 227. Addition of fractions is the process of finding the value of two or more fractions in one sum. NOTE. — Only units of the same kind, whether integral or fractional, can be collected into one sum; if, therefore, the fractions to be added do not express the same fractional unit, they require to be brought to the same, by being reduced to a common or the least common denominator. 228. To add together two or more fractions. |