3. Add 4. Add 5. Add EXAMPLES. to,to,to,to,to,to,to. to,to,to,to, to, to 4. to, to, to, to, to 1, To to 4. 10 10 6. Add to, to, to, to, to, 69 to 4, † to §. 6 6 to, to, to, to, to, 3 to 4. to, to 3, & to 4, & to §, & to §, & to 1. to §, to fr, to 3, fr to f, fr to f, 9 11 139 to, to, to, to fo, fo to fr, fo to fs. 11. Add to, 12. Add 13. Add 14. Add 15. Add to f. to 8, 8 to, to, to fr, sto 1. SUBTRACTION OF COMMON FRACTIONS. 230. SUBTRACTION of Fractions is the process of finding the difference between two fractions. NOTE. 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 44, 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 3ğ. OPERATION. From 7 Take 3 Rem. 33 Ans. 38. 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 below 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 Subtract the number denoting the numerator from that denoting the denominator, under the remainder write the denominator, and adding one to the whole number in the subtrahend, subtract the sum from the minuend. NOTE. When the subtrahend is a mixed number, we may reduce it to an improper fraction, and change the whole number in the minuend to a fraction having the same denominator, and then proceed as in Art. 230. Ex. 1. From 8 take 4. FIRST OPERATION. From 83 Take 44 Rem. = = 359 Ans. 333. We first reduce the fractional parts to a com815 mon denominator, and obtain as their equivalents and. Now, since we cannot take from we add 1, equal to, to the in the minuend, and obtain . From taking, we have left, which we write below the line, and carry 1 to the 4 in the subtrahend, and subtract from the 8 above as in subtraction of simple whole numbers. 33 SECOND OPERATION. = 59 295 From 83 35 In this operation, we reduce the mixed numbers to improper fractions, and these fractions to a common denominator. We 33 then subtract the less fraction from the greater, and, reducing the remainder to a mixed number, obtain 33, as before. Hence, in performing like examples, we may Rem. Reduce the fractional parts, if necessary, to a common denominator, and subtract the fractional parts and the whole numbers separately. Increase the fractional part of the minuend, when otherwise it would be less than the subtrahend, before subtracting, by as many parts as it takes to make a unit of the fraction (Art. 208), and carry 1 to the whole number of the subtrahend before subtracting it. Or Reduce the mixed numbers to improper fractions, then to a common denominator, and subtract the less fraction from the greater. 14. From a hogshead of wine there leaked out 7 gallons; what quantity remained? Ans. 55gal. 15. A man engaged to labor 30 days, but was absent 57 days; how many days did he work? 16. From 144 pounds of sugar there were taken at one time 17 pounds, and at another 28 remains? pounds; what quantity Ans. 971 lb. 17. A man sells 97 yards from a piece of cloth containing 34 yards; how many yards remain ? Ans. 24дyd. 18. The distance from Boston to Providence is 40 miles. A, having set out from Boston, has travelled of the distance; and B, having set out at the same time from Providence, has gone of the distance; how far is A from B? Ans. 287m. 19. From of a square yard take of a yard square. 233. To subtract one fraction from another, when their numerators are alike. Ex. 1. From take 4. 7 3 7 X 3 = = OPERATION. Ans. 4, difference of the denominators. 21, product of the denominators. We first find the product of the denominators, which is 21, and then their difference, which is 4, and write the former for the denom、 inator of the required fraction, and the latter for the numerator. By this process the fractions are reduced to a common denominator, and their difference found. Hence, to subtract one fraction from another, whose numerators are a unit, we may Write the difference of the denominators over their product. We multiply the difference of the denominators by one of the numerators for a new numerator, and the denominators together for a new denominator, by which process the fractions are reduced to a common denominator, and the difference of their numerators is found. Hence, when the given fractions have their numerators alike and greater than a unit, we may Write the product of the difference of the given denominators, by one of the numerators, over the product of the denominators. EXAMPLES. 3. Take 4. Take 5. Take from 1, 1, 1, 1, 1, 4; Ts from T, TE, T'S. 6. Take 8. Take 9. Take 10. Take from 7, 1, 1, 1, 1, 6, 4, §, §. 7. Take from ; from 2, 3; † from §. 19. Take 19 from 19. 13 from 12. from 40; 19 from 19; 18 from 18; 20. Take from 12; 2 from 42; 12 from 12; MISCELLANEOUS EXAMPLES IN ADDITION AND SUBTRACTION OF FRACTIONS. 1. A benevolent man has given to one poor family ‡ of a cord of wood, to another of a cord, and to a third of a cord; how much has he given to them all? Ans. 24 cords. |