Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/590

530 numerator, and the divisor the denominator. For, let it be required to divide 76 by 9. Since 72 divided by 9 gives the quotient 8, and 4 divided by 9 gives , 76 divided by 9 must be 8f. A proper fraction is one whose numerator is less than its denominator ; as y^, -|-. An improper fraction is one whose numerator is not less than its denominator; as -|, -. A proper fraction is evidently less, and an improper frac tion not less, than unity. A mixed number is an integer or whole number with a fraction annexed to it; as 8^-, 5^. The integer and fraction here are to be considered as added together. A simple fraction is a single fraction that has both its terms whole numbers ; as ^, . A com pound fraction is a fraction of another fraction, or of a whole or mixed number; as f of |-, |- of lOf. A complex fraction is one that has a fraction or mixed number for one of its terms or for both ; as -^-, , -~. 4^ 4f 5 A fraction is said to be in its lowest terms when the numerator and denominator have no common divisor. 12. If both terms of a fraction be multiplied, or both divided, by the same number, the value of the fraction will remain unchanged. For, if unity be divided into 7 equal parts, and again into 28 equal parts, it is evident that each of the former parts will be identical with 4 of the latter. 5 of the former will thus be equal to 20 of the latter ; that is, = |f . 1 3. We multiply or divide a fraction by any mimber, if we multiply or divide the numerator by it. For it is evident that four times f- is ~, since unity is divided into 9 parts in both cases, and the number of parts taken in the one case is four times the number taken in the other. Conversely, dividing f by 4, we have . To multiply or divide either term of a fraction is the same as to divide or multiply the other. If, for instance, we divide the denominator by 3, we divide unity into one- third of the number of parts, and each of these parts must thus be three times greater than before. The fraction is therefore multiplied by 3. Or otherwise, dividing the denominator of by 3, we obtain ; and multiplying the numerator by 3, we get |-, which ( 12) is the same as . Therefore, to divide a fraction, we may multiply the de nominator, a method which must be employed when the divisor is not a measure of the numerator. 14. To reduce a fraction to its lowest terms, divide both terms by their greatest common measure. The reason of this is evident from the definitions. The value of the fraction remains unchanged ( 12). The results of opera tions in fractions should, with rare exceptions, be expressed in their lowest terms. 15. To reduce a mixed number to an improper fraction, the integer is multiplied by the denominator of the frac tion, and the numerator added to the product. This gives the numerator of the required improper fraction, and the denominator of the given fraction is its denominator. Take, e.g., 4f. If unity be divided into 8 equal parts, 4 units will make 32 of these parts, and the fraction f con tains 5 of them; therefore 4| = ^. Conversely, to re duce an improper fraction to a ivhole or mixed number, divide the numerator by the denominator ; the quotient is the integer required, and the remainder, if there is one, is the numerator of a fraction of which the given denominator is the denominator. Thus, ~ = 5| ; for, if unity be divided into 7 equal parts, 35 of these parts will be the same as 5 units ; therefore 38 parts will be 5 units and 3 parts, i.e., 3_i_K3 7 -7 16. To reduce a compound fraction to a simple one, multiply the numerators together for the numerator of the simple fraction, and the denominators together for its denominator. Thus, of | is equal to if. For, if we divide unity into 7 equal parts, and each of these again into 4, we shall have for the fraction 20 of these parts, i.e., f-g- of unity. The fourth part of this is 5 of these parts, and therefore f is 15 of them, i.e., ^ of is -||-. Mixed numbers must be put in the form of improper fractions before the multiplication. The reduction of the result to its lowest terms may be effected by removing before multiplication any factors that are common to the numerators and denominators. In such compound expressions as where one number is spoken of as one-fourth greater or less than another, the fourth is always to be taken of that number with which the comparison is made. Thus, 30 is one-fifth more than 25 (i.e., -g- of 25), but 25 is one-sixth less than 30 (i.e., ^ of 30). 17. In order to compare the values of fractions, or to add or subtract them, it is necessary to reduce them to others of equal value that have the same denominator. From the definition of a fraction we see at once that is greater than , but we do not readily see whether - is greater or less than -|. If, however, we take the equiva lents of these fractions, |-j and ^, unity being now divided into the same number of parts in both cases, we have like quantities to compare, and see that the former fraction is the greater. In practice it is usually the least common denominator that the fractions are compared by. To find this, we first find the least common multiple of the de nominators ( 10), then divkle this by the denominator of each fraction separately, and multiply both terms of that fraction by the quotient. Take, e.g., the fractions if, f, , and . The least common multiple of the denominators is 180, and dividing this by 15, 20, 36, and 45, we get 12, 9, 5, and 4. Multiplying both terms of the fractions in succession by these numbers, we have Jf, if|, i, and if . The value of the fractions has not been altered, and all have now the same denominator, 180, which we must obtain in each case, because we re-multiply the factors into which we resolved it. In practice, indeed, we merely divide 180 by 15, multiply 13 by the 12, and set down 180 under 156 at once. 18. To add fractions, reduce them to others having a common denominator, then add the numerators of these, and write the common denominator under the sum. Thus, A + 1 + w = T^T + T A + .r? 2 T7 = In each of the re- duced fractions unity is divided into 120 equal parts, and the fractions are respectively equivalent to 50, 15, and 42 of these parts ; therefore the sum of the fractions must be 50 + 15 -1- 42 = 107 of them, i.e., i|. Compound fractions must be reduced to simple ones before addition. Mixed numbers may be brought to im proper fractions, and so added ; but it is generally prefer able to add the whole numbers and the fractions separately, and then add the two results. The operation may often be shortened by first adding any of the given fractions whose denominator is considerably less than that of all the fractions; thus, to add , 4-f, -If-, f-, the sum of the first three may be found first, amounting to Iff. The addition of | to this gives 2f|i 19. To subtract fractions, reduce them to others having a common denominator, take the difference of the numer ators so found, and write the common denominator under it. The principle is precisely the same as in addition. Compound fractions must be simplified as in addition. "With mixed numbers the subtraction will generally be best effected by treating the fractions separately, borrowing and carrying, if necessary, on the principle explained in 5. In subtracting 47f| from 85 |f, for instance, borrow

ing Hi and carrying 1 gives 85^-| - 48ff - 37i|f , or