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both the numerator and denominator of a fraction by the same number, does not alter its value. (Art. 191.)
63. Reduce 3, 3, 4, and & to a common denominator. 64. Reduce , Š, š, and to a common denominator. Reduce the following fractions to a common denominator: 65. Reduce $, ž, $, and j. 69. Reduce 1, 5, and 77. 66. Reduce 1, 4, 9, and . 70. Reduce 76, 700, and 67. Reduce $, , , and 72. 71. Reduce 1, l, and 15. 68. Reduce it, 4, 1, and š. 72. Reduce 1, 6, and 27.
73. Reduce }, }, and to the least common denominator.
Analysis.-We first find the least Operation. common multiple of all the given de- 2)3" 4" 8 nominators, which is 24. (Art. 176.) 2)3
2 " 4 The next step is to reduce the given 3 11 2 fractions to twenty-fourths without Now 2X2X3X2=24, the altering their value. This
evi- least common denominator, dently be done by multiplying both terms of each fraction by such a number as will make its denominator 24. (Art. 191.) Thus 3, the denominator of the first fraction, is contained in 24, 8 times; now, multiplying both terms of the fraction } by 8, it becomes a The denominator 4, is contained in 24, 6 times ; hence, multiplying the second fraction by 6, it becomes 2. The denominator 8, is contained in 24, 3 times ; and multiplying the third fraction by 3, it becomes . Therefore , #, and are the fractions required. Hence,
201. To reduce fractions to their least common denominator.
1. Find the least common multiple of all the denominators of the given fractions, and it will be the least common denominator. (Art. 176.)
II. Divide the least common denominator by the denominator of each given fraction, and multiply the quotient by the numerator ; the products will be the numerators of the fractions required.
QUEST.-201. How are fractions reduced to the least common denominator :
OBs. 1. This process, in effect, multiplies both the numerator and denominator of the given fractions by the same number, and consequently does not alter their value. (Art. 191.)
2. The rule supposes each of the given fractions to be reduced to its lowest terms; otherwise, the least common multiple of their denominators may not be the least common denominator to which the given fractions are capable of being reduced. Thus, the fractions , &, and in, when reduced to the least common denominator as they stand, become in, 107, and 12. But it is obvious that these fractions are not reduced to their least common denominator; for, they can be reduced to 1, , and ... Now, if the given fractions are reduced to the lowest terms, they become }, }, and f, and the least common multiple of their denominators, is also 4. (Art. 176.)
3. By a moment's reflection the student will often discover the least common denominator of the given fractions, without going through the ordinary pro- . cess of finding the least common multiple of their denominators. Take the fractions }, \, and 12; the least common denominator, it will be seen at a glance, is 4. Now if we multiply both terms of 1 by 2, it becomes f; and if we divide both terms of Ba by 3, or reduce it to its lowest terms, it becomes 1. Thus the given fractions are equal to , and I, and are reduced to the least common denominator.
74. Reduce, , and } to the least common denominator. Operation. Now 2X2 X3x2=24, the least com. denom. 2)4" 6" 8 Then 24:4=6, and 6X3=18, the 1st num. 2)2 311 4
24-6=4, and 4x5=20, the 2d
Ans. I, 4, and H.
75. Reduce and to the least common denominator. Reduce the following fractions to the least common denominator : 76. 15, 5, 7, and 1
84. 1, 5, 1%, and is. 77. , , and .
85. 12, 5, 12, and 72. 78. 4, 5, 7, and 12.
86. ÁT, 13, 14, and . 79. 4, , $, and 12.
87. 36, 2, 1, and 6. 80. 3, 4, \, and 1.
88. 48, 36, 49, and H. 81. }, 1, 13, and
89. 45, 67, , and 44. 82. 72, 19, o, and . 90. 41, 42, 43, and 83. 4, 6, H, and H.
91. 31, , 40, and
Quest.--Obs. Does this process alter the value of the given fractions? Why not 1 What does this rule suppose respecting the given fractions ?
ADDITION OF FRACTIONS,
Ex. 1. A beggar meeting four persons, obtained of a dollar from the first, from the second, 6 from the third, and from the fourth : how much did he receive from all ?
Solution.-Since the several donations are all in the same parts of a dollar, viz: sixths, it is plain they may be added together in the same manner as whole dollars, whole yards, &c. Thus, 1 sixth and 3 sixths are 4 sixths, and 4 are 8 sixths, and 5 are 13 sixths. Ans. 13, or 25 dollars.
Ex. 2. What is the sum of s and ?
OBs. A difficulty here presents itself to the learner; for, it is evident, that 2 thirds and 3 fourths neither make 5 thirds, nor 5 fourths. (Art. 51.) This difficulty may be removed by reducing the given fractions to a common denominator. (Art. 200.) Thus,
the new numerators. 3X3=9S
3X4=12, the common denominator. The fractions, when reduced, are in and ; now 8 twelfths + 9 twelfths=17 twelfths. Ans. 17, or 11.
202. From these illustrations we deduce the following general
RULE FOR ADDITION OF FRACTIONS.
Reduce the fractions to a common denominator ; add their numerators, and place the sum over the common denominator.
Obs. 1. Compound fractions must, of course, be reduced to simple ones, before attempting to reduce them to a common denominator. (Art. 198.)
2. Mixed numbers may be reduced to improper fractions, and then be added according to the rule; or, we may add the whole numbers and fractional parts separately, and then unite their sums.
3. In many instances the operation may be shortened by reducing the given fractions to ine least common denominator. (Art. 201.)
QUEST.-202. How are fractions added ? fractions? llow are mixed numbers added ? ened?
Obs. What must be done with compound
3. What is the sum of 3, 4, and 5 ? Ans. H=2. 4. What is the sum of 1, 3, 4, and ? 5. What is the sum of 3, , 4, and ? 6. What is the sum of 3, 4, 15, and }? 7. What is the sum of , , , and 13 ? 8. What is the sum of ,, ft, and 1? 9. What is the sum of , to, 4, and ? 10. What is the sum of b, 13, 7, and 12 ? 11. What is the sum of 1, }, }, s, and f? 12. What is the sum of í of }, i of 5, and $ ? 13. What is the sum of } of g, ğ of , and 17? 14. What is the sum of of z of 7 of 1, and ? ? 15. What is the sum of 5, 4 of 3, of }, and ? 16. What is the sum of 41, 81, 27, 63, and s? 17. What is the sum of of 6, of 2, 3), and 54 ? 18. What is the sum of 4, 6, 4, 5, and 18 ? 19. What is the sum of 211, 351, 3, and šof ? 20. What is the sum of $ of , 21, 61, 13, and ? 21. What is the sum of } and it? Note. It is obvious, if two fractions, each of whose numerators is 1, are reduced to a common denominator, the new numerators will be the same as the given denominators. (Art. 200.) Thus, if ş and ta are reduced to a common denominator, the new numerators will be 12 and 8, the same as the given denominators. Now, the sum of the new numerators, placed over the product of the denominators, will be the answer; (Art. 202;) that is
127-8 * 20
12X8 96 244, the answer required. Hence,
203. To find the sum of any two fractions whose numerators
Add the denominators together, place this sum over their product, and the result will be the answer required.
Obs. 1. The reason of this rule may be seen from the fact that the operation is the same as reducing the given fractions to a common denominator, then adding their numerators.
2. When the numerators of two fractions are the same, their sum may be found
QUEST.--203. How is the sum of any two fractions found whose numerator3 are 1 ? Obs. How find the sum of two fractions whose numerators are the same ?
by multiplying the sum of the two denominators by the common numerator, and placing the result over the product of the given denominators. Thus, the
(4+5)X3_9X3 27 sum of and is equal to
4X5 4X5 20 22. What is the sum of zł and 315? Of Lu and its ? 23. What is the sum of to and 7? Of it and to? 24. What is the sum of öts and 512? Of , and 9 ਵੈs ? ? 25. What is the sum of 5 and P? Of 15 and 7? 26. What is the sum of 15 and 15? Of 15 and 45? 27. What is the sum of and 39 ? Of 2and ? 28. What is the sum of 5 and j? Note. The object in this example is to unite the 5 with the į in a single expression; that is, to incorporate the whole number with the fraction.
Solution.-5=*. (Art. 197. Obs. 2.) Now +j=Ans. 204. Hence, to add a whole number and a fraction together.
Reduce the whole number to a fraction of the same denominator as that of the given fraction; then add their numerators together. (Art. 202.)
Note. The process of incorporating a whole number with a fraction, is the same as that of reducing a mixed number to an improper fraction. (Art. 197.)
29. What is the sum of 45 and ? 30. What is the sum of 320 and 75? 31. What is the sum of 452 and Blo? 32. What is the sum of 63516 +427+*+16257 ? 33. What is the sum of 1959++60049 +563031 +-1604 ? 34. What is the sum of 6713% +48313 +842147+4325} ? 35. What is the sum of 59011 +100% +400547 +30201 ? 36. What is the sum of 23914 +-64433 +165079+45001? 37. What is the sum of 65634 +1000} +18303+8396 ? 38. What is the sum of 356H+464+1653+6005 +321ỉ ? 39. What is the sum of 414 +-105%+-3001+241 +4721? 40. What is the sum of 86727+163645}+18003+6625130 ? 41. What is the sum of 260034+193523 +92831+686934 ? 42. What is the sum of 192566 +456005 +å of g of ? 43. What is the sum of of 28+67+453 +4 of 300 ?
QUEST.--204. How add a whole number and a fraction ?