the new fractions as in the preceding manner. Thus, if it is proposed to add,,, these fractions are changed into 45, 40 48, of which the sum is 33, which is reduc 60 60 60' ed to 213, (85.) EXAMPLES FOR PRACTICE. 5 3, 1. Add 4, 3, 4, 7, and together. 2. Add 1, 1, 1, 4, and I together. together. Ans. 3.7. 67 168 4. Add 54, 63, and 4 together. Ans. 17 24. 5. Add 1,, and 9 together. 6. Add 1, 67, 2, and 7 together. Ans. 1673. 120 Ans. 1170. SUBTRACTION OF FRACTIONS. 103. If the two proposed fractions have the same denominator, the numerator of one is subtracted from the numerator of the other, and to the remainder the common denominator is affixed. If it is required to subtract & from the remainder will be, which is reduced to 3, (93.) 104. If we wish to subtract 47 from 95, as cannot be taken from, we borrow 1 from 9, which reduced to eighths and added to, make, from which taking }, there will remain §. Then taking 4 from the 8 which remains after 1 was borrowed, we have in all 4g, or 43. 105. If the fractions have not the same denominator, they must be reduced to a common denominator, (90) and (91.) Subtraction is then performed as in the preceding Thus, to take from, these fractions are changand; then subtracting 8 from 9 there remains case. ed to As it has been seen (88) and (89), that the value of a fraction is not changed by multiplying or dividing both of its terms by the same number; we are naturally led to examine what effect is produced when the same quantity is added to, or subtracted from, both terms of the fraction. Let be a given fraction; by adding 3 to each of its terms, the fraction becomes, or . But this last fraction is greater than the first, as may be proved by observing, that the fractions and are equivalent to and 7 I for, if the numerator or denominator of a fraction is multiplied or divided by the same number, the value of the fraction is not changed. But, 7 is greater than the fraction is then greater than the fraction 2. For, a fraction is augmented or diminished, by augmenting or diminishing its denominator, the numerator remaining the same. Hence it is plain, that by adding the same number, 3, to both terms of the fraction 4, (equivalent to) it is augmented and becomes or 7. 4 This is a general property. For, as the value of a fraction is not changed, by adding the numerator and denominator, each to itself; both terms of the fraction may be doubled without altering its value. But, the denominator being always greater than the numerator, in order that the value of the fraction may remain the same, a greater number must be added to the denominator than to the numerator. Consequently, by adding the same number to both terms, we add too little to the denominator; the denominator then is too small, and the fraction too large, according to what has been said above. Hence it is plain, that a fraction is augmented by adding the same number to both its terms. And reciprocally, a fraction is diminished when the same number is subtracted from both its terms. The multiplication and division of one fraction by another is deduced from the preceding principles. H 2 MULTIPLICATION OF FRACTIONS. 103. To multiply a fraction by a fraction, the numera tor of one must be multiplied by the numerator of the other, and the denominator by the denominator. For ex ample, to multiply by, we multiply 2 by 4, which gives 8 for the numerator; multiplying, in like manner, 3 by 5, we have 15 for the denominator, and consequently for the product. The reason for this rule will be more plain if we recollect, that in the multiplication of one number by another, the multiplicand is taken as many times as there are units in the multiplier. Then, to multiply by, is to take 4 times the fifth part of . But in multiplying the denominator 3 by 5, we change the thirds to fifteenths, that is, into parts five times smaller. And in multiplying the numerator 2 by 4, we take these new parts four times; hence we take four times the fifth part of Therefore is multiplied by . 107. If whole numbers are to be multiplied by a fraction, or a fraction by whole numbers, the whole numbers are written as a fraction, by giving them unity for a denominator. Thus, if 9 is to be multiplied by, the operation becomes to multiply by 4, which, according to the rule already given, produces 6. This product is reduced to 54. Hence, when a whole number is multiplied by a fraction, or a fraction by a whole number, the operation is reduced to multiplying the numerator of the fraction by the whole number. 108. If whole numbers are joined to fractions, before multiplication, the whole numbers must be reduced, each to the kind of fraction which accompanies it. For example, if 12 is to be multiplied by 93, we change (86) the multiplicand into, and the multiplier into 39; and multiply 3 by according to the rule in article (106), which gives 2457, or 12217. 63 39 20' 109. In dividing one fraction by another, the terms of the fraction which is the divisor must be inverted; the fraction which is the dividend is then multiplied by the fraction thus inverted. For Example. To divide by, the fraction is inverted and becomes; we then multiply by 3, according to the rule in article (106,) and have 1 or 12 for the quotient of divided by 3. 2 The reason of this rule may be seen by observing, that to divide by is to seek how many times contains 3. But it is evident, since the divisor is 2 thirds, that it will be contained in the dividend three times as often as there are 2 whole numbers in it. It is then necessary first to divide by 2, and afterwards to multiply by 3; and this operation consists in taking three times the half of the dividend, or to multiply by which is the divisor inverted. 1 110. If a fraction is to be divided by a whole number, or a whole number by a fraction, the whole number must be written as a fraction by giving to it unity for a denominator. Thus if 12 is to be divided by, the operation becomes to divide 2 by 5, and this, according to the rule, is to multiply 12 by, which gves 84 or 16. In like manner, if the fraction is to be divided by 5, the operation becomes to divide 3 by, that is, to multiply aby, which gives. Hence, when a fraction is to be divided by a whole number, the operation requires the denominator to be multiplied by the whole number. 111. If whole numbers are joined to fractions, the whole numbers must be reduced, each to the kind of fraction which accompanies it. For Example, if 543 is 38 to be divided by 123 we change the dividend into 273, and the divisor into 3. The operation then is to divide 273 by 3, that is, (109,) to multiply 273 by, which gives 13, or 4596. or 112. After what has been said in article (96) it is easy to see how the value of a fraction can be found. For example, if the value of of a pound is required; since of one pound is the same as the seventh of 5 pounds (96,) we reduce the 5 pounds to shillings (57,) and divide 100 shillings which is the result, by 7, which gives 14 shillings for the quotient and 2 shillings for remainder. The 2 shillings are reduced to pence, and we divide 24 pence by 7, and have 3 pence. Thus the of a pound, are 14s. 3 d. If the 4 of 24 pounds are required, we can, as in the last case, take the of one pound, and multiply 24 by that result; but it is more convenient to multiply, at first, by 24 pounds, which gives (107) 12° of a pound. After estimating the value of this fraction, we find £17. 2s. 104d. for the answer. The principle explained in this article may be expressed in the following rule for finding the |