nominator 8 is contained in 24, 3 times; and multiplying the third fraction by 3, it becomes 1. Now, 1, and are respectively equal to the given fractions,, and §, and are the answer required. Hence, 126. To reduce fractions to their least common denomi nator. I. Find the least common multiple of all the denominators of the given fractions, and it will be the least common denominator. II. Divide the least common denominator by the denominator of each of the given fractions, and multiply the numerator by the quotient; the products will be the numerators required. OBS. 1. If the example contains compound fractions, whole or mixed numbers, they must first be reduced to simple fractions, then all must be reduced to their lowest terms; otherwise the least common multiple of their denominators, may not be the least common denominator. 2. It is evident this process does not alter the value of the given fractions: for the numerator and denominator of each fraction are multiplied by the same number, consequently their value remains the same. (Art. 116.) 14. Reduce,, to the least common denominator. 2×3×2=12, the least common denominator. Now 12+3=4, and 4 × 2= 8, the 1st numerator. Operation. 2)3" 4" 6 3)3 " 2 3 "1 27. 28, 3 of 3 and 45. ૐ 29. 63 849 13 of 4, and 63. 31.92 11 of 17, and 734. QUEST.-126. How reduce fractions to the least common denominator! Obs. Does this process alter the value of the given fractions? Why not? ADDITION OF FRACTIONS. 126.a. If two or more fractions have a common denominator, the parts of a unit expressed by their numerators, are of the same value or denomination. (Art. 107. Obs. 3.) Hence, When fractions have a common denominator, their numerators are added like whole numbers, and the result placed over the common denominator, will be the sum of the given fractions. Note. This and the following articles refer to fractions which arise from Simple Numbers. For the method of adding and subtracting Fractional Compound Numbers, see Arts. 168, 169. Ex. 1. A man gave of a dollar to one of his children, to another, to another, and to another: how much did he give to all? Suggestion.-Write the frac tions one after another with the sign+between them, and add Operation. +++=', or 13 Ans. their numerators. Thus, 1 eighth and 2 eighths are 3 eighths, and 3 are 6 eighths, and 5 are 11 eighths. He therefore gave , or 12 dollars to all. 2. A man bought a barrel of flour at one time, 3 of a barrel at another, and of a barrel at another: how much did he buy in all? Operation. Suggestion. Since these fractions have not a common denominator, it is evident their numerators cannot be added as in the last example; for 1 half, 2 thirds, and 3 fourths will make neither 6 halves, nor 6 thirds, nor 6 fourths. (Art. 22.) We therefore reduce them to a common denominator, then add their numerators as above. When reduced, the fractions become 1, 18, and 1. Now placing the sum of the numerators 46, over the common denominator, the result is 4, which reduced to a mixed number becomes 12, or 11. 12 16 66 1x3x4=12, 1st numerator. QULST.-123.a. How do you add fractions which have a common denominator? Explain the reason? 127. From these illustrations and principles, we deduce the following general RULE FOR ADDITION OF FRACTIONS. Reduce the fractions to a common denominator; add their umerators, and place the sum over the common denominator. OBS. 1. Compound fractions must be reduced to simple ones, whole and mixeł bumbers to improper fractions, and all of them to a common denominator, then add them as above. (Art. 125. Obs. 2, 3.) 2. In adding mixed numbers, it is generally more convenient to add the whole numbers and fractional parts separately, and then unite their sums. 3. The operation may frequently be shortened by reducing the given fractions to their least common denominator, and adding their numerators. (Art. 126.) 3. What is the sum of of, 21, 14 and 7? Suggestion. of 3=3, 21=1, and 7=7. Now reducing, 2, and 7 to a common denominator as in the margin, and adding their numerators, the result is 230, which is equal to 914, or 9. 24 Operation. 8 24 54 24 168 Ans. 91, or 97. 26. A grocer sold 473 pounds of sugar to one customer, 833 pounds to another, and 685 pounds to another: how much did he sell to all? 27. If you travel 85,5 miles in one day, 78, in another, and 12517 in another, how far will you travel in all? 28. If a man buys 3 pieces of cloth, containing 1277 yards, 168 yards, and 2563 yards, how much will he then have? QUEST.-127. What is the rule for addition of fractions? Obs. What must be done with compound fractions, whole and mixed numbers? How else can mixed numbers be added? How may the operation be shortened? SUBTRACTION OF FRACTIONS. 128. When two fractions have a common denominator, the less numerator may be subtracted from the greater, as in whole numbers, and the result placed over the common denominator, will be the difference between the fractions. (Art. 126.a.) Ex. 1. If I buy 31⁄2 of an acre of land, and afterwards sell }? i an acre, how much shall I have left? Suggestion.-We write the less fraction after the greater with the sign between them, then taking 19 fifty-sevenths from 35 fifty-sevenths, the answer is 1 of an acre. Operation. }{-}}=}; Ans. 2. From of a yard of cloth, take ? of a yard. Operation. Suggestion. Since these fractions have not a common denominator, it is plain that one numerator cannot be taken from the other; for 3 fourths taken from 5 sixths, will leave neither 2 sixths, nor 2 fourths. We must, therefore, reduce them to a common denominator, then subtract as above. When reduced, the fractions become 20, and 18; and 18 twenty-fourths from 20 twenty-fourths leave, or yard, which is the answer. 5×4=201 numerators. 20 129. From these illustrations and principles, we deduce the following general RULE FOR SUBTRACTION OF FRACTIONS. Reduce the fractions to a common denominator; subtract the less numerator from the greater, and place the remainder over the common denominator. Oвs. 1. Compound fractions must be reduced to simple ones, whole and mixed numbers to improper fractions, and all of them to a common denominator, as in addition. 2. In subtracting mixed numbers, it is sometimes more convenient to take the fractional part of the less from the fractional part of the greater, then the integral part of the less from that of the greater. QUEST.-129. What is the rule for subtraction of fractions? Obs. What must be done with compound fractions, whole and mixed numbers? How else are mixed numbers subtracted? 3. In subtracting a proper fraction from a whole number, we may borrow a unit and take the fraction from this, then diminish the whole number by 1. (Art. 38.) 3. From 93 subtract 51. Suggestion.-Reducing the mixed numbers to improper fractions, then to a common denominator, they become 53, and 33. Subtracting the less from the greater, we have 25, or 41. Or, reducing the fractional parts to a common denominator, then subtracting the numerator of the less from that of the greater, the result is 41, the same as before. (Art. 129. Obs. 2.) Second operation. 20. From 6 take 3. 21. From 65 take 253. 22. From of take of . 23. From of take 1 of 3. 24. From 3 of 10 take 25. From of 24 take § of 27. Ans. 7, or 13. of 6. 26. A man bought a wagon for 855 dollars, and a sleigh for 692 dollars: how much more did he pay for one than the other? 23 27. A man having 2467 acres of land, sold of 195 acres: how many acres did he have left? 28. If from a piece of cloth containing 12518 yards, you cut 87 yards, how many yards will be left? 29. From 5637 pounds, take 19 of 2601 pounds. 30. From 1673 bushels, take of 1 of 356 bushels. 31. From of 3 of 256 miles, take of 33 miles. 82. From of of 385 33. From rods, take 4 of 673 rods. of 1 of 573 tons, take of 216§ tons. QUEST.-In what other manner is a fraction subtracted from a whole number } |