4. Reduce,, 21, and 25. 5. Reduce, 4, 11, and . 6. Reduce, 131, 4, and 2. 7. Reduce,,, and . 8. Reduce,, 11, and 71. 9. Reduce,, ¿, and 3§. 10. Reduce, §, 4, and 43. 11. Reduce, 14, 11, and 1. 12. Reduce, 12, 17, and 21. 13. Reduce, 30, 4, and 6. 14. Reduce, 2, 16, and 2. 15. Reduce, 7, 8, and 54. Reduce the following fractions to a common denominator: 16. Reduce, and to fractions having a common de nominator. 17. Reduce, §, and. 18. Reduce, 4, and 183. 5390 Ans. 12812, 096 8008 7007 14014 14014, 14014 14014 27. Reduce 2, 121, and 1728. Ans. 28506816 37088064 1784315529 T78431552 722813 178431552 ADDITION OF COMMON FRACTIONS. 227. Addition of fractions is the process of finding the valu 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. 3 Ex. 1. Add 12, 12, 12, and 11 together. Ans. 29 These fractions 13 24. all being twelfths, that is, having 12 Thus we for a common denominator, we add their numerators together, and write their sum, 26, over the common denominator, 12. 21, the sum required. obtain 22, which, being reduced, 2. What is the sum of 7, 12, 14, and 18? $ 12 16 20 3 4 OPERATION. Ans. 21. 20 X 4 X 3 830 X 1220 X 240 16 15 x 11 2012 × 13 The given fractions not expressing the same kind of fractional unit, we reduce them to their least common denominator, and thus make the fractional parts all of the same kind. The fractions now all expressing two-hundred-fortieths, we add their numerators, and write the result, 631, over the least common denominator, 240, and obtain g 2151, the answer required. = RULE. - Reduce the fractions, if necessary, to a common, or the least common denominator, and write the sum of the numerators over their common denominator. NOTE 1.- Mixed numbers must be reduced to improper fractions, and compound fractions to simple fractions, and each fraction to its lowest terms, before attempting to obtain the common denominator. NOTE 2. In adding mixed numbers, the fractional parts may be added separately, and their sum added to the amount of the whole numbers. EXAMPLES. 9 3. Add 17, 17, 17, 14, 14, and 1 together. 9 4. Add 2, 23, 1, 3, and together. 5. What is the sum of 47, 17, 37, and 77? 6. What is the sum of 144, 144, 24, and 107? Ans. 14. 7. What is the sum of 81, 491, 89, and ''T? Ans. 183 19. Add 63, 78, and 48 together. Ans. 40. 22. What is the sum of 37111, 61418, and 812? Ans. 106887. 23. Add of 18, and 11 of 4 of 63 together. Ans. 12881 24. Add & of 18, and of 1 of 7 together. 용 229. To add any two fractions, whose numerators are alike. product, which is 20; and the 9 being written as a numerator of a fraction, and the 20 as its denominator, the result,, is the answer required. The reason of the operation is, that the process reduces the fractions to a common denominator, and then adds their numerators. Hence, to add two fractions whose numerators are a unit, Write the sum of the given denominators over their product. 2. Add 2 to 3. Sum of the denominators X OPERATION. Ans. 1. 20 by one of the numerators, (4 +5) × 3 27 Product of the denominators, 4 X 5 By multiplying the sum of the denominators by one of the numer ators for a new numerator, and the denominators together for a new denominator, we reduce the fractions to a common denominator, and add their numerators, and thus obtain 717, the answer required. Hence, to add fractions whose numerators are alike, and greater than a unit, Write the product of the sum of the given denominators by one of the numerators over the product of the denominators. 6 6 8 8 8 3 T 10. Add to,&to, &to, & 8 to, to, to 8 to, to, to, to, to, to 3. to 3, &to, &to, &to, & tor to 6,4 to 63, to f, fr to f, to 19 8 14. Add to, to, to to 5 to 17, to. II 139 13 15,1 3 to 10 to 14, 1o to P, o to, o to 18. SUBTRACTION OF COMMON FRACTIONS: 230. SUBTRACTION of Fractions is the process of finding the difference between two fractions. - When the fractions express different fractional units, they require to be brought to those of the same kind before the subtraction can be performed. The fractions both being twelfths, having 12 for a common denominator, we subtract the less numerator from the greater, and write the difference, 6, over the common denominator, 12. Thus, we have as the difference required. of the same kind. We next find the difference of the new numerators, which we write over the common denominator, and obtain 4, the answer required. RULE. - Reduce the fractions, if necessary, to a common, or the least common denominator. Write the difference of the numerators over their common denominator. NOTE. If the minuend or subtrahend, or both, are compound fractions, they must be reduced to simple ones. 231. To subtract a proper fraction or a mixed number from a whole number. Ex. 1. From 7 take 3§. OPERATION. From 7 Take 3ğ Rem. 3 Ans. 38. below Since we have no fraction from which to subtract the, we must add 1, or its equal, g, to the minuend, and say from leaves. We write the the line, and carry 1 to the 3 in the subtrahend, and subtract as in subtraction of simple whole numbers. The result will be obtained, if we |