8. Find value of — (1.) :3 x .3. (2.) 9.001 ~ 27.06. (3.) 0.403 x .009. (4.) 17 .017 ~ 100. (5.) :3 x .005 64. (6.) (-4)2 x (032) 9. Find the quotient of— (1.) 79.4 by 397. (2.) 5.928 by 4742.4. (3.) 28 by .007. (4.) •6426 by 2:8. (5.) (-24)2 by 9.6. (6.) 1:806 by (1.9)? 10. Given the quotient •00073, the dividend 124:1, find the divisor when there is no remainder. 11. What is the value of (1.) 2 - •815} = { '201 + •039 – •002}} (2.) {(693)2 – (-307)} = {.693 – •307}? 12. If I add .061 to a certain number, and then divide the result by 290, I get .0009 for a quotient; what is the number? CHAPTER II. THE TREATMENT OF FRACTIONS CONSIDERED AS RATIOS. 5. A Fraction is a part or parts of a whole. It is generally expressed by two numbers, the one placed above the other and separated by a line. The lower number expresses the number of equal parts into which the whole quantity has been divided, and the upper number, how many of those parts are taken. Thus is a fraction, and tells us that unity has been divided into 5 equal parts, and that we have taken 3 of those parts. The fraction is read three fifths ; each of the equal parts into which unity has been divided being called a fifth. The denominator of a fraction is the lower number, and therefore shows the number of equal parts into which we have divided the unit. The numerator is the upper number, and tells us how many of these equal parts are taken. When the numerator is less than the denominator, the quantity expressed is actually less than a whole. The quantity is therefore a real or proper fraction. Again, when Thus , 4, 4, 6, , 44 of 3 of , are all included the numerator and denominator are both integral numbers, the fraction is termed a simple fraction. Thus 2, 3, ti, are both proper and simple fractions. It is however usual to include in the term fraction every expression which contains one or more simple fractions, with or without integral numbers. Thus 2. 4 12 6 3 41 u 21 ' 21' 71 in the term fraction. They are, moreover, called vulgar fractions to distinguish them from decimals, which, as will be shown further on, may be looked upon as fractions, according to the above definition, whose denominators are powers of 10, and not expressed but understood. It is convenient to classify fractions as follows: (1.) A proper fraction is one whose numerator is less than its denominator, as , }, 13, (2.) An improper fraction is one whose numerator is not less than its denominator, as £, , ti, (3.) A simple fraction is one whose numerator and denominator are both integral numbers, as ķ, , 1. (4.) A mixed number is a fraction expressed by an integer and a simple fraction, as 21, 43, 3. (5.) A complex fraction has its numerator, or denominator, 3 or both, in a fractional form, as 4, is 57' 69' 11 2} , (6.) A compound fraction is a fraction of a quantity whicli is itself fractional, as of 2}, 24 of 6 75 6. In the preceding article we have spoken of fractions in the ordinary way. We will now approach them from a different point of view. By the term ratio we understand the result of the com. parison of two quantities with regard to magnitude. There are two kinds of ratios-ratio by difference or subtraction, 2 and ratio by quotient or division. Thus we may consider how much one quantity exceeds another, or we may consider how many times one quantity contains another. The former kind is called the arithmetical ratio, and the latter the geometrical ratio. We shall speak only of the latter. DEFINITION.—The ratio between two quantities is that multiple, part, or parts which the former is of the latter. It is evident that a ratio can exist only between quantities of the same kind ; thus, we may compare 12 horses and 6 horses, but not 9 men and 4 miles. And if the quantities are reduced to the same denomination, we may treat the quantities as abstract, just as we find the quotient of one concrete quantity by another, by reducing them both to the same denomination, and dividing as if they were abstract quantities. Now, according to what has been stated above, the ratio of 12 to 7, or, as it is usually written, 12:7, is obtained by dividing 12 by 7; and this is the same thing as dividing unity or 1 into 7 equal parts, and computing how much 12 of such parts amount to. It hence follows that the fraction is properly expressed by the ratio 12:7. The first term of a ratio is called the antecedent, and the second is called the consequent; and hence we may consider a fraction as a ratio, the numerator being the antecedent of the ratio, and the denominator the consequent. When the antecedent is equal to the consequent, the ratio is said to be a ratio of equality; and it is said to be a ratio of less or greater inequality according as the antecedent is less or greater than the consequent. Thus, 6 : 6 is a ratio of equality. 3:4 is a ratio of less inequality. 11:9 is a ratio of greater inequality. The student will therefore have no difficulty in assenting to the following definitions : (1.) A proper fraction is a ratio of less inequality. (2.) An improper fraction is a ratio of equality or of greater inequality (3.) A simple fraction is a ratio whose terms are integers. Thus, t = 3:5 is a simple fraction, (4.) A mixed number is a ratio of greater inequality, whose antecedent has been actually divided by its consequent, and the result expressed as an integer and simple fraction. Thus, 24 = = 17 : 7. (5.) A complex fraction is a ratio, whose antecedent, or consequent, or both, are not integers. Thus 3} 297 Thus, 13, ñ = respectively to 31:17, 2: 77, 91:5 are complex fractions. (6.) A compound fraction is an expression containing two or more ratios to be compounded together. Thusof & contains the ratios 3:4 and 7:9 to be compounded ; 22 of of contains the ratios 24:7, 3:61, 74:9 to be compounded. 17. A fraction whose numerator and denominator are multiplied or divided by the same quantity is not altered in value. Suppose, for example, we multiply the numerator and denominator of the fraction each by 4, we get = 12. Now the ratio of 3: 7 is, from the definition of a ratio, four times as small as the ratio (3 x 4): 7 or 12:7; and the ratio 12 : 7 is, for the same reason, four times as great as the ratio 12 : 17 x 4) or 12 : 28. It therefore follows that the ratio 3:7 is exactly equal to the ratio 12 : 28, and consequently y = ir Again, suppose we divided each term of the fraction 15 by 3, we get 19 = %. Now the ratio of 15 : 27 is, from the definition of a ratio, three times as great as the ratio (15 ; 3): 27 or 5:27; and, again, the ratio 5 : 27 is three times as small as the ratio 5: (27 ; 3) or 5 : 9. It therefore follows that the ratio 15 : 27 = the ratio 5 : 9, and consequently 1h = . Cor.–An integer may be expressed as a fraction with any given denominator. For we may consider an integer as a ratio whose conse, quent is 1, and we may multiply each term of this ratio by. any given number without altering its value. I X7 1 X Thus–6 = f = x1 = 42. Or, 6 = f = 9X9 = 54. 8. To multiply a fraction by a whole number, we may either multiply the numerator by the number, or divide the denominator by it. Ex. f x 3 = 8*3 = 24; or we may proceed thus-- The ratio 8: 9 will be evidently increased three times if we multiply its antecedent by 3; this follows from the definition of a ratio. We thus get the ratio (8 x 3) : 9 or 24 : 9. It therefore follows that ş 3 = 24. The ratio 8 : 9 will be evidently increased three times if we make the consequent three times as small; and we thus get the ratio 8: (9 ; 3) or 8:3; and hence it follows that 8 x 3 = . It may be remarked that the two results, 24 and ļ, are of exactly the same value (Art. 7), since the latter may be obtained from the former by dividing each of its terms by 3. In actual practice we sometimes pursue the first method, and sometimes the second. If the denominator of the given fraction contains the multiplier as a factor, it is more convenient to use the second method, thus :— 13 x 3 = izt; = 1 On the other hand, when the denominator does not contain the multiplier as a factor, we use the first method, thus : (1.) ?. * 5 = 7* 5 = 35. (2.) 4 x 10 = 9X19. Here we see that the numerator and denominator have a common factor 5, and therefore, by Art. 7, if we divide them both by it, we have : 's x 10 = 9*= tf. 13 13 13 55 |