And as they are now all of the same denomination, their sum or collective value is evidently the sum of the several numerators referred to that denomination, or 886 , a fraction in its lowest terms. When the result, as in this case, is a fraction greater than unity, its value will be more readily discerned, if the numerator be divided by the denominator, and the quotient and remainder be set out in the form of a mixed number, thus, 437. The numerator comprises 4 whole repetitions of the 60 equal parts expressed by the denominator, and 37 parts over. 60 To add a fraction to a whole number the rule is the same as that just given for reducing a mixed number to a simple fraction. (See page 70.) 35 To illustrate the subtraction of fractions, let it be proposed to ascertain the difference of and On reduction to a common denominator, it appears that is the greater fraction, being equal to 185, while is only 128; and the difference of these numerators placed over the common denominator, gives as the answer to the question, Every other instance of subtraction may be treated 6 in a similar manner. 15 from 1g 37 842 113 40 From the sum of,, and, take the difference between 198 and Answ. 20 MULTIPLICATION AND DIVISION OF VULGAR FRACTIONS.-To multiply fractions by one another. RULE.-Multiply the several numerators together for the numerator of the product and the denominators together, for the denominator of the product. Examples.-(1) multiplied by is ៖ 1 2x5 7x9=38 4x 6 (2), and multiplied together=5x11x8x153688=278 6600 It is advisable in all cases, before multiplying, to see whether any of the numerators and denominators have an obvious common measure, and if so, to divide by that measure and use the quotients instead of the original numbers. This step will abridge the labour of calculation, and generally yield the result at once in its lowest terms. Thus, in example (2), although each of the fractions is by itself in its lowest terms, yet when connected with the others so as to form one compound fraction, it will be readily perceived that a considerable reduction may be effected. One of the numerators, 4, and one of the denominators, 8, are divisible by 4: numerator 6 and denominator 15 are divisible by 3, and the result of the division of 6 and 8, giving the same number, 2, in the line of numerators and denominators respectively, may be cancelled. No further reduction can be made, and we have accordingly, 4× 6 ×7×13 6× 7 ×13 5x11x8x155×11× 2 × 15' = = 7×13 91 2× 7 × 13 In performing these operations it saves repetition to draw a line through the divided numbers, and to write the quotients over or under the proper dividends, thus, 4× 6 ×7×13 But, the work, if exhibited in this manner on an examination paper, may be objected to as unsightly or immethodical. A fraction of a fraction, is the product of one of these fractions multiplied by the other. rds of Zths, is Zths taken 3rds of a time. Similarly, Zths of 3rds, is rds taken ths of a time. The result in both cases is the same, as it is obtained by multiplying the numerators and denominators respectively into each other, and forming a new fraction of the products. Whenever it is given as a question to find what fraction one fraction is of another, or of several others, it should be borne in mind that the process to be gone through is always that of multiplying these fractions together. 14 9x3 27 The reason of the general rule for the multiplication of fractions may be illustrated briefly as follows. If a fraction be multiplied by a whole number, as for instance by 2, the product must be as many repetitions of the fraction as there are units in the multiplier: twice is +7=1. But if only a third part of this product is to be taken, that is, a third part of twice, or of, then the result of the multiplication by 2, will have to be divided by 3, and to divide by 3 is to multiply the denominator by 3, giving or 14 for when a number, as 14 is divided first by 9-as denoted by the fraction 14-and then by 3, the result must be the same as if the number were at once divided by 27, the product of 9 and 3. This is well known principle in the division of whole numbers. Now, the steps of the above reasoning show, that in taking ths 3rds of a time, and in arriving at the result, we multiply the numerators together and also the denominators, and it will be readily inferred that a similar process applies to every other case. To multiply a fraction by a whole number, or a whole number by a fraction.— RULE.-Consider the whole number as a fraction having 1 for its denominator, and apply the rule for the multiplication of fractions. Example: 13 multiplied by is 13 × 8 = 101 18/1 1 1 Here the number 13 is taken 8ths of a time, reducing it to 95; or the fraction is taken 13 times, raising it to 95 In an operation of this kind, it must be obvious that there is no necessity actually to write the denominator 1 under the whole number. The usual rule, therefore, is to multiply the numerator of the fraction by the whole number, and let the denominator stand: thus 13 x = 13 x 8 = 10 = 9,5 If the denominator be divisible without remainder by the whole number, it is better simply to make that division, than first to multiply the numerator, and then have to divide the denominator by the same number, in order to bring the fraction to its lowest terms. If the numerator were multiplied in this case, as the rule directs, the process would be Hence it is clear that division of the denominator, when practicable, is the shorter method. To represent a whole number as a fraction with any given denominator. RULE.Multiply the denominator by the whole number: the product will be the numerator of the fraction required. Example: To represent 5 as a fraction with the denominator 8. 5 is which, when both its terms are multiplied by 8, is not altered in value. To divide one fraction by another. RULE.-Multiply the denominator of the divisor by the numerator of the dividend for a new numerator; and the numerator of the divisor by the denominator of the dividend, for a new denominator. Or, more simply, invert the fraction to be used as divisor, that is, make the numerator and denominator change places, and proceed as in the multiplication of fractions. 3 divided by & is 3 × 7 After inverting the divisor, it is of advantage before multiplying to see whether the numerators and denominators have an obvious common measure, and if so, to divide by that measure, and use the quotients in place of the original numbers. (See MULTIPLICATION OF FRACTIONS, page 71.) Thus, in Example (2) × is evidently reducible to x = 1% If the denominators of the given fractions are equal, all that is requisite is to place the numerator of the dividend over the numerator of the divisor, for the quotient or answer. Example. How often does 1 contain 13 ? 19 divided by 1 is 19 × 17 = 1; = 1}. Answer 14 times. As to the reason of the rule for division: let the fractions in example (1) be reduced to a common denominator, and they become respectively 21 and 28. The question now is, how often does contain ? Plainly, as often as 21 contain 20, that is ths of a time, the answer obtained by operating in the manner directed by the rule. 20 To divide a fraction by a whole number, or a whole number by a fraction. RULESuppose the whole number to be a fraction with the denominator 1, and apply the rule for the division of one fraction by another. Example (1.) (2.) 18 divided by 7 is 18 × = 105. 7 divided by 13 is 7 × 18 = 105 = 81• As the writing down of 1 as denominator may be dispensed with, the rule in this case is generally stated as follows. To divide a fraction by a whole number, multiply the denominator by that number, and retain the same numerator. To divide a whole number by a fraction; invert the fraction; then multiply the denominator by the given number, and retain the denominator as before. But it will be found better in practice to treat the whole numbers as if they were fractions, and to proceed by the one general rule for division, as above exemplified. F In dividing a fraction by a whole number, the numerator should always be divided when that can be done without remainder, in preference to multiplying the denominator. Thus, the division of 3 by 12 is at once effected by dividing the numerator by 12: otherwise in place of the simple operation, 3 ÷ 12 = &› there would be incurred the useless labour of first multiplying by a number and then dividing by it, as follows:— The management of vulgar fractions in general is greatly facilitated by the recollection of these two principles 1. Division of the numerator is equivalent to multiplication of the denominator by the same number. 2. Division of the denominator is equivalent to multiplication of the numerator by the same number. COMPLEX FRACTIONS.-Those which have fractions or mixed numbers in one or both terms, are reduced to the form of simple fractions by the process of division. Thus to reduce the complex expression. 22 3 41 to a simple equivalent fraction, we write the divisor 41 as 21, and then proceed to The same result would follow from multiplying the fractions 22 and 21 by 15, the least common multiple of the denominators 3 and 5 : find the quotient of 22 divided by 23, = 22 × 71 = 110 = 143 23 15 × 22, or 22 × 5 = 110 : 15 x1, or 21 × 3 = 63. This is generally the best method of reducing complex fractions. APPLICATION OF THE RULES OF FRACTIONS TO COMPOUND QUANTITIES.-In the treatment of questions occurring under this head, a few varied examples will probably afford sufficient guidance without the aid of a system of rules, which would consist of little else than a repetition of previous parts of the chapter. The process to be pursued in each case will, at once, suggest itself on referring to the general theory of fractions. Example (1.) of £1 is What is the value of of a pound sterling? The operation, therefore, is to divide £5 by 7. the value of 13 of £24 16s. 101d.? 15 13 of any quantity is of 13 times that quantity. We, therefore, have to multiply the sum by 13 and divide the product by 15. To divide by is to multiply by 11, that is, to multiply by 11 and divide by 9. (4.) What fraction of a pound is 16s. 43d.? *16s. 43d. reduced to farthings = 787. 960 99 =960. The fraction sought, therefore, is 787, for the unit or pound being divided into 960 equal parts, the given sum contains 787 of such parts. (5.) What fraction of 9s. 5d. is half-a-crown? One shilling is of a pound; consequently of one shilling is of of a pound, or x = Answer. 180 of a pound. of of a gallon? of 2 = And of } = 1. Answer. (7.) What fraction of a bushel is 1 gallon is of a bushel. (8.) Reduce 38. 71d. to the fraction of £1 3s. 41d. ? 3s. 71d. 87 halfpence. £1 3s. 41d. =561 halfpence. It is unnecessary to reduce the given sums to farthings, as there is no lower denomination in the question than halfpence. (9.) Reduce £7 9s. 6d. to the fraction of £13 4s. 6d. Here the lowest denomination consists of sixpences, and we have, £7 9s. 6d. = 299 sixpences, £13 4s. 6d. = 529 sixpences. And 2 divided in both terms by 23 = Answer. 529 (10.) Add together of a pound, of 6s. 8d., of a crown, and of When a question is proposed which requires the application of several rules in fractions, it will be found greatly conducive to clearness and ease of working, to state the question by means of the arithmetical signs, so that the successive steps of the process may be readily apprehended. Thus, if it be given to divide 2 by the sum of 31, 3, and 8; to subtract of of from the quotient, and to multiply the difference by 67, it will be of advantage to represent the various operations as follows |