ADDITION OF COMMON FRACTIONS. 227i Addition of fractions is the process of finding the valuo 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. 228i To add together two or more fractions. Ex. 1. Add -fe, TV, fa and together. Ans. f § = 2\. orrtunoB. These fractions A + A + A + ti - f5 - ¥ - 2*. all being tad/Jfa, that is, having 12 for a common denominator, we add their numerators together, and write their sum, 26, over the common denominator, 12. Thus we obtain which, being reduced, = 2\, the sum required. 2. What is the sum of J, -B, and? Ans. 2£££. OPERATION. new numerators. $ 12 1 6 j 2 0 2 4 0 = least common denominator. ~3 4" 8 30 x 7~= 210 12 20X 5 = 100 20x4x3 = 240 16 15x 11 = 165 20|l 2 x 13=156 Sum of numerators, 6 3 1 21 1 A Least com. denom., 2 4 0 ^"i^ ns" 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 f|£ = 2Mi, tne 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. 3. Add f'T, T6T, T9T, ff, tf, and |f together. Ans. 3-J-S. 4. Add A, ih ih and together. Ans. 2$%. 5. What is the sum of ¥4T, ^-f, §f, and |f? 6. What is the sum of TW, Tti, T%, and Mi? Ans- 1H 7. What is the sum of Uh I§t, Mi, and ^Vr? Ans. 2ffl. 8. Add £, and f together. 9. Add T\, and £ together. Ans. 2§§. . 10. Add T\, -j-i, and \ together. Ans. 2^§g. 11. Add -jsj-, 3^, -j>36B, and £ together. Ans. 1. 12. Add |i iI, H and T% together. Ans. 3^. 13. Add I, ^, £, and } together. Ans. ljfo. 14. Add |, f, and 5 J together. Ans. 6f&£. 15. Add T*T, and 9^ together. 16. Add £, and 4£ together. Ans. 6 J. 17. Add |, 7£, and 8f together. Ans. 17^. 18. Add I, 3£, and 5f together. Ans. 9£iJ. 19. Add 6f, 7f, and 4§ together. Ans. 18Ij^. 2Q What is the sum of 17f, 14£, and 13?? 21. What is the sum of 16§, 8|~ 9$, 3J, and 1£? Ans. 40^. 22. What is the sum of 371}£, 614£§, and 81f? Ans. 1068§ J. 23. Add f of 18T\, and || of $• of 6^ together. Ans. 12||f 24a Add | of 18, and ^ ofof 7^- together. 229i To add any two fractions, whose numerators are alike. Ex. 1. Add i to J. Ans. ^. Oferation. \\re first find the Sum of the denominators, 5 -\- 4 = 9 sum of the denoin Product of the denominators, 4x5 = 20 »natorJ, ,whic1} .is 9, and then their product, which is 20; and the 9 being written as a numerator of a fraction, and the 20 as its denominator, the result, -fa, 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 f to f. Ans. 1^. OPERATION. Sum of the denominators X by one of the numerators, (4 -J- 5) X 3 27 ^ Product of the denominators, 4 x 5 20 2' By multiplying the sum of the denominators by one of the numerators 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 £J = 1-^, 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. EXAMPLKS. 3. Add T^to i, i to i, i to i, I to i, i to i, i to I, i to f 4. Add ft to VT to i, ft to i, ft to |, ft to ft to f 5. Add ft to £, ft to ft to ^, ft to £, ft to $, -ft to j. 6. Add itoi,i to ito{,i to £, i to i to |, £ to i. 7. Add A to i, £ to f | to i, i to | to i, | to I i to ^. 8. Add ) to J, f to f to } to |, f to J, | to i, $ to 9. Add i to £, £ to i to i, i to J, 4 to J, J to f i to |. 10. Add | to ft, | to ft, | to ft, | to ft, | to ft, f to ft, 11. Add | to f, | to I, t to ft, § to f to f, § to ft, f to 12. Add I to ft, ^ to 3 £ to % to £ to ^ to ft. 13. Add f to f, f to ft, f to ft, ft to ft, ft to ft, ft to ft. 14. Add | to ft, ft to ft, ft to ft, ft to ft, ft to ft, ft to ft. 15. Add ft to ft, ft to ft, ft to ft, ft to ft, ft to ft. SUBTRACTION OF COMMON FRACTIONS. 230i Subtraction of Fractions is the process of finding the ditference between two fractions. Note. — When the fractions express different fractionnl units, they require to be brought to those of the same kind before the subtraction can be performed. To subtract one fraction from another. Ex. 1. From \% take -ft. Ans. ft = Operation. The fractions both being twelfths, having 12 T£ — ft = ft. 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 ft as the difference required. 2. From \% take f Ans. f f. Operation. The given frac 7 7 common denominator. tions not express i i 7 v i n r~n i 'nP the same kind 7 i !0 A = A At new numerators. of fractional unit, 7 1 1 X 4=44) we reduce them 2 6 dif. of numerators. to a common de-- - , nominator, and 7 7 common denominator, thus make the fractional parts all of the same kind. We next find the difference of the new numerators, which we write over the common denominator, and obtain 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. Examples. 3. Subtract -jfy- from Ans. 4. Subtract T\ from -j-f. Ans. T9S. 5. From ,§f take 6. From take Ans. 7. From f J take . Ans. fy. 8. From f £ take \\. 9. Subtract from f £. Ans. 4. 10. Subtract T'j6T from T3r\. Ans. 11. Subtract -^q- from -^y. Ans. j^. 12. Subtract from T^6b. Ans. -ft^. 13. Subtract T${fo from ^ftfe. Ans. ^y. 14. From +f take -fa. 15. From take -fa. Ans. J J. 16. From ff take ^. Ans. $f$. 17. From 4§ take 73F. Ans. §§. 18. From £f take Ans. 19. From ?_4 take 20. From take j^. Ans. -fifa. 21. From ^ take -fa. Ans. ^. 22. From & take Ans. 23. From 7£ take f of 9. Ans. 1?V 24. What is the value of f of 8} — § of 5? 25. What is the value of i of 3 — £ of 2? Ans. 231 i To subtract a proper fraction or a mixed number from a whole number. Ex. 1. From 7 take 3f. Ans. 3f. Operat.o*. Since we have no fraction from which to subtract From 7 tne I' we must ', or 1** e(lua'' I't0 tne minuend, m, qs and say | from f leaves f. We write the f below la * the line, and carry 1 to the 3 in the subtrahend, and Rem. 3$ subtract as in subtraction of simple whole numbers. The result will be obtained, if we |