and under the result write the common denominator; the fraction thus formed will be the sum of those proposed. If any mixed numbers are to be added, the fractional portions only of such numbers are to be added by the preceding rule, and the sum of the integral parts to be added afterwards. NOTE. Those fractions, of which the denominators have a factor in common, you will find it much more convenient to add two at a time, as in the examples 2, 3, and 4, below. Ex. 1. Add together the fractions , ., and .. These reduced to a common denominator are 5.6, and ; and 54 +84 +==211, the sum. 2. Required the sum of 2, 14, and lo Here, taking the fractions only, we see at once that the first and third are s and , the sum of which is 1=1}; therefore, taking in the second fraction, we have 1+1=1+ 1=id; and since the sum of the whole numbers is 2+1+1 =4, the sum of the proposed quantities is 411 3. Add together 326, 38, and 64. Here it is at once seen that the denominators 40 and 36 each have a common factor, 4; the first two fractions may 31 29 therefore be written ; so that the two deno 10 X 4' 9X4 minators will be the same, if the first be multiplied by 9 and the second by 10; for the factors of each will then be 4, 9, and 10. Consequently, += 38+388 = 388 = 1368 The third fraction, after multiplying its terms by 2, is $6 ; and 38% + = 1:87 + 185 = 100; hence the sum is The reason why I changed my into 46 was, that 50 is a more convenient number to multiply by than 25. 4. Add together of to, and 18 The denominators of the first two fractions are 4x4 and 3 x 4; these are made alike by the factor 3 for the first, and 4 for the second; using therefore these for multipliers, the first two fractions become changed into 5, 2. The denominators of the other two fractions are 53 and 5x8; they are made alike by the factor 8 for the first, and 3 for the second. Hence the last two fractions are changed into 1922 5%. The only differing denominators are now 48 and 120; that is, 24 x 2 and 24 x 5, which are made alike by the factor 5 for the first, and 2 for the second ; therefore, using these for multipliers of 23 and 8:00 the sums of the pairs of fractions above, we have, finally, 146+148=$18=1,94 111907 And this is the way in which you should proceed in working the following exercises, reducing the fractions to a common denominator, by the rule, only when no two of the denominators have a factor in common. If you are only careful always to select the greatest factor in common, your changed fractions will always have the least common denominator. A rule will presently be given for enabling you to find the greatest common factor of two numbers, however large those numbers may be. Exercises. 1. 1+3+. 9. 217 +1% +18 2. ++ 10. 11. +33 +3+370 3. +10 +11. 11. 1+11+49 +48. 4. * +5 +6. 12. 4+1 +5,38 +13 5. 27 +33 +1 13. 19 +1% +16+3 6. 8j+511+ 14. 61% +17++ 7. ++3. 15. ++++42 8. ++15 16. 215+11+315 + . 5 (56.) SUBTRACTION OF FRACTIONS. RULE. Reduce the fractions to a common denominator, which place under the difference of the changed numerators. As in addition, after the fractions are brought to a common denominator, the numérators only are added, so in subtraction, the numerators only are subtracted; the common denominator in both operations being preserved, as it is merely that which denotes what the things added or subtracted are. Ex. 1. Subtract from Here *-=-35=3 Here 73-39=71-3139; is cannot be taken from for we must therefore increase the is by a unit, or 1, borrowed from the 7, thus converting the operation into 629-312= 311. And similarly in other such cases. Exercises. 1. 34-2 3. 1-4 5. 24-13 2. 19 6. 46-311 4. 15 5 16 32 24 7. 511-41%. 8. 410-416 9. 111-19 10. 215+34–412. 11. 61+133-5%. 12. 53-65+15 13. 81-31-215. 14. 11+54–645. 15. 1016-766-215. 16. 435-21-14 (57.) MULTIPLICATION OF FRACTIONS. When anything is multiplied by a fraction, the operation performed is, in reality, something more than mere multiplication, in the sense in which that word is used when the multiplier is a whole number: thus, if anything is to be multiplied by }, the meaning is, that we are to take one-fifth of that thing once ; if it is to be multiplied by the meaning is, that we are to take one-fifth of it twice ; if by , we are to take one-fifth of it three times ; and so on. It is so far like ordinary multiplication, that it signifies repetitions, or the taking of a quantity a certain number of times: but then it is not the whole of this quantity, but only a part of it that is taken that number of times: the part to be taken is made known to us by the denominator of the multiplier; the number of times it is to be taken, by the numerator. Suppose we have to multiply by ; the meaning is, that we have to take a fifth part of į twice : now, by the fifth part of anything, is meant that thing divided by 5; in the proposed instance, it is divided by 5; but , 'is 3 divided by 7, and if there be another division by 5, the result is of course the same as if 3 were at once divided by 5 x 7, or 35; so that a fifth part of is; and, consequently, two-fifths must be ; we thus see, that is 395, so that the product of the two fractions is nothing more than the product of their numerators divided by the product of their denominators; and, moreover, that multiplied by ), and ths of $ths, or şths of ths, are one and the same in meaning. This must be kept in remembrance. Although only a single instance is here taken for illustration, yet you will easily see, that the same reasoning would 2 apply to any two fractions whatever, and therefore that the following rule must be true for all cases. RULE. Multiply the numerators together, and you will get the numerator of the product. Multiply the denominators together, and you will get the denominator of the product. Note 1. If a multiplier be a whole number, it may be considered as a fraction with 1 for denominator: if a multiplier be a mixed number, it must be converted into an improper fraction. 2. It is very likely, that in a row of fractions to be multiplied together, there may be found factors in the numerators equal to factors in the denominators; if SO, cancel them, or omit them in the multiplications; for there is no use in preserving common factors in numerator and denominator of the product, which ought always to be in the lowest terms. And, on this account, if a fraction is to be multiplied by a whole number, it is better to divide the denominator by the number, enever the denominator exactly contai it. Ex. 1. What is the product of 1, 24, and ? Reducing the mixed number to an improper fraction, we 2 9 2 X3 X3 X 2 bave Х . 3 7 3 x 2 x 2x7 As here the factors, 2 and 2, furnished by the numerators, are the same as the factors of 4 in a denominator, these comanon factors are cancelled ; also, since there is a factor 3 in the numerator 9, the same as a denominator, these two 3's are also cancelled; so that, after the cancellings, there remains only 3 in the numerator, and 7 in the denominator; and, therefore, the product is found without actually multiplying at all. 2. Multiply of 3} by of 21. 4 16 5 2 x 16 32 ģ*5*7 = 111 2 3x7 21 In working this example, I would recommend you to proceed thus: having written down all the given fractions, with the signs of multiplication between them, as above, and having put the sign of equality, draw the line that is to separate the resulting numerator and denominator, before putting anything above or below; then, looking at the first numerator, 4, examine the row of denominators; the last of this row, 2, you find to be a factor of 4, therefore expunge or cancel this factor, putting only the other factor 2 above 3 Х 3 3 5 the line of separation, and draw your pen through the denominator 2, to remind you, when you come to it, that it is done with: then, looking at the denominator 9, of the first fraction, and glancing at the row of numerators, you see a 3, which being a factor of 9, you put only the other factor 3 below the line of separation, and draw your pen through the numerator 3, to show that it is cancelled. The next numerator, 16, has no factor belonging also to the uncancelled denominators; this 16, therefore, you put down as it is, against the number before put in the numerator's place; and as the denominator 5 of the second fraction is cancelled by the numerator 5 of the fourth, you draw your pen through both, and pass on to the next fraction, and you see that the only uncancelled number remaining is the denominator 7; you write this, therefore, in the denominators place, against the number already there, and put the multiplication sign between the two; and you thus have the numerator and denominator of the product free from useless factors; that is, you get the resulting fraction in its lowest terms. 3. Multiply }, 31, 5 and of together. 2 13 5 13 x3 39 x Х Х ī 45. 2 X4 8 2 Here the first numerator 2 cancels a factor 2, in the next denominator 4, but as the other factor 2 remains uncancelled, you write the uncancelled 2 below the 4, which, with the numerator 2, is then crossed out: the first denominator 3 is also cancelled, with the fourth numerator, so that nothing as yet is put on the right of the sign =. Passing then to the numerator 13, you see that nothing below cancels it; you therefore put 13 in the numerators place, and the uncancelled 2 below it. Passing now to the third numerator, 5, you see that it is cancelled by the last denominator; there remains, therefore, only the uncancelled denominator of the fourth fraction, and the uncancelled numerator of the fifth. In most books on Arithmetic, the cancelled figures, in examples of this kind, are printed with the cancelling marks across them; but this gives the work an unsightly appearance ; and although I recommend this plan to you for your own private convenience, yet, in presenting your work to the inspection of another, I would not recommend it to be shown all defaced by these scratches : when they have served your purpose, they should be removed, or an undefaced copy of the work taken. |