ADDITION OF VULGAR FRACTIONS. RULE. When the given fractions have a common denominator, the sum of the numerators placed over the common denominator will be the answer :—when the fractions have not a common denominator, reduce them to a common denominator, and then proceed as before: when mixed numbers are given, to the sum of the fractions add the sum of the integers:—when the fractions are of different denominations, find their values, and add them together as in Compound Addition. Rule. Fractions which have different denominators are dissimilar, and consequently cannot be incorporated with one another; but when they are reduced to a common denominator, they are made like parts of the same thing, and their sum or difference will then evidently be expressed by the sum or difference of their numerators; whence the reason of the rules both for Addition and Subtraction is manifest. RULE. When the fractions have a common denominator, the difference of the numerators placed over the common denominator is the answer:when the fractions have not a common denominator, reduce them to a common denominator, and take the difference as before: when mixed numbers are given, first subtract the fractions, (after having reduced them to a common denominator,) in which 5 process if the numerator of the subtrahend exceeds that of the minuend, subtract it from the denominator, and to the difference add the numerator of the minuend, and carry one to the figure in the unit's place of the subtrahend :—when the fractions are of different denominations, find their values, and subtract them as in Compound Subtraction. RULE. Multiply all the numerators of the given fractions toge ther for the numerator, and all the denominators together for the denominator of the product. Note. To multiply a fraction by an integer, either divide the denominator or multiply the numerator by the integer. 2. To multiply an integer by a mixed number, multiply by the integer and fraction separately, and add the products. 3. When both factors are mixed numbers, reduce them to improper fractions, and then work by the rule. 2. Mult. 8 by 52. 123=312=483. Ans. Remark. To multiply one fraction by another, is to take such a part or parts of the former as the latter, expresses; the processs is therefore truly expressed by a compound fraction: thus, x is the same as of; hence the rule for the multiplication of the former is the same as that for the reduction of the latter to a simple fraction. ៖ 1. Mult. by . by §. EXERCISES. 4. Mult. by by RULE. Invert the terms of the divisor, and then multiply the dividend by the inverted divisor, as in Multiplication. Note. Mixed numbers and integers must be reduced to improper fractions be fore the rule can be applied. 2. When the numerator and denominator of the dividend can be exactly di vided by the numerator and denominator of the divisor respectively, that method of finding the quotient is preferable to the general rule. 3. To divide a fraction by an integer, either divide the numerator, or multiply the denominator, by the integer. 4. To divide an integer by a fraction, multiply by the denominator, and divide the product by the numerator. 5. To divide an integer by a mixed number, proceed as directed in Rule 2d, page 17. EXAMPLES. 1. Divide by . 2. Divide by . 3. Divide 51 by 23. 3×31. Ans. Ans. 2. Ans. = Remark. The truth of the rule may be shown thus: in the first example, it is plain, that 4 is but is only of 4, con45 Vop sequently will be contained 5 times as often in as 4 is. Thereforex5 is the answer, which agrees with the rule; and the same way of reasoning may be applied in every instance, with the DECIMAL FRACTIONS. 23 141 &c. A DECIMAL FRACTION is that which has for its denominator 1, with one or more ciphers annexed to it; as, 1, 10' 1'000' The denominator of a decimal is not written, like that of a vulgar fraction, under the numerator; but it is expressed by pointing off from the right of the numerator as many figures as there are ciphers in the denominator, supplying the deficiency, when there is any, by ciphers on the left. The point placed before a decimal, and by which it is distinguished from an integer, is called the decimal point. The notation of decimals proceeds on the same principles as that of integers, and is a continuation of the same scale; the first place after the decimal point being 10th parts, the second 100th parts, the third 1000th parts; and so on, decreasing towards the right in a tenfold proportion. Thus, 3, 03, 003, 0003, are respectively the decimal expressions for %, 180, 100, 10800; thus, also, 4, 29, 373, 15.71 are the expressions for %, and 15,7%, respectively. 29 From this general view, it is plain, 1. That ciphers on the left of a decimal diminish its value 10 times for every cipher. 50 2. That ciphers on the right of a decimal do not alter its value; thus, 5, 50, 500, are respectively equal to, o, 100, each of which is equal to . 3. That decimals may be reduced to a common denominator by adding ciphers to the right, where it is necessary, till all the decimals have an equal number of places: thus, 3, 41, and ·323, when reduced to a common denominator, become 300, 410, and 323, that is, 300 410 and 323 1000, 1000, 1000. REDUCTION OF DECIMALS. CASE I. To reduce a vulgar fraction to an equivalent decimal. RULE. Suppose ciphers annexed to the numerator as decimals, and then divide it by the denominator; if there be not so many figures in the quotient as there were ciphers annexed, supply the defect by writing ciphers before it. EXAMPLES. 1. Reduce to a decimal. 2. Reduce to a decimal. 8)7.000 .875 Ans. 400)1:0000 • 0025 Ans. Remark. The reason of the rule may be shown thus: in the first example, represents the quotient of 7 divided by 8; but 7=7888, and 78888-875-875; and the same reasoning may be applied in every case. When a vulgar fraction is reduced to a decimal, if there is no remainder, the decimal is called a finite decimal; if the quotient repeats the same figure in infinitum, it is called a single repetend; if it repeats two or more figures in infinitum, it is called a compound repetend or circulate; if the figures repeat from the decimal point, it is called a pure repetend; if there are figures before those which repeat, it is called a mixed repetend, and the figures before those which repeat are called the finite part of the decimal. It is usual to set a point over a single repetend, and a point over the first and last figures of a compound repetend: it is also convenient to place the decimal point near the upper part of the figures, to prevent it from being confounded with any other necessary point. CASE II. To reduce numbers of lower denominations to equivalent decimals of higher denominations. RULE. Write the given numbers in their order, under each other, the least name being uppermost, for dividends; on the left of these, and separated from them by a perpendicular line, write, opposite to each dividend, such a number for a divisor as will reduce it to the next higher name: then divide, beginning with the least name, and annex the quotient to the next dividend, as a decimal part of it; next divide this mixed number by its divisor, writing the quotient as a decimal part of the next dividend; and so on. The last quotient is the decimal required. Note. When there is only one number to be reduced, the process is the same as in Case II. of Reduction of whole numbers. EXAMPLES. 1. Reduce 13/64 to the decimal | 2. Reduce 2d. to the decimal of of a £. 4 1 2013-520833 a £. 12)2. 20).1666 .0083 Remark. The reason of the rule may be shown thus: in the first example, d. being reduced to a decimal is 25; consequently 64d. is = 6·25d. =§3§d. =79%/; which, being reduced to a 625 =1200 = |