the denominator of each fraction, placing the quotient below the line under the denominator which gave it. Lastly, we multiply the two terms of each fraction by the corresponding number below the line, and have the fractions sought. We need not multiply each denominator by the number below the line, because we already know the result. Also, when the scholar becomes skilful, he may omit the placing of the numbers below the line. 16 12 6 3 To reduce,,, and to a com. denom., we place the numbers thus: 15 and having found (185) that the least com. denom. is 48, we divide this successively by 3, 4, 8, and 16, and place the quotients 16, 12, 6, and 3, each under the number which gave it. Lastly, we multiply the two terms of each fraction, or rather, the numerator only, by the number below the line, and have 38, 39, 42, and 19 for the required fractions. 36 We easily see (190) that each of the numbers 16, 12, 6, and 3, respectively, shows the number of times that the new common generic unit is less than that of the fraction under which the number stands; and that in multiplying the numerator by this number, we have a number of parts in each new fraction, which is just as many times greater as the number of times that the parts are less in value. Wherefore, the value of the fraction is not altered. Examples. 55, 1. §, 11, and, are equal to §8, 88, and 38. ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF VULGAR FRACTIONS AND MIXED NUMBERS ;—ALSO DIVISION BY FACTORS. gar Fractions. be added have a com. denom., pul place the sum over the com. denom. The fraction thus formed, whether proper or improper, is the sum of the given fractions. The numerators, that is to say, the fractions, being each composed of units of the order signified by the com. denom., their sum is a number of units of the same order; and, for this reason, we give it the same denominator But when they have not a common denominator, (198,) we first reduce them to a com. denom., (200 or 201,) and proceed as before. Thus, to find the sum of 1, 2, and 3, we first reduce these fractions to a com. denom., (200,) and have for their equivalents, 28, and 18. Then to find their sum we say, Hence, the sum of 4, 5, and § is 212 Σ 445 12 Let the expressions in the following examples be written in the same manner: The letters a b c represent the new numerators, s their sum, and d the common denominator. 5. j + 3 + 1⁄2 + ¿§ = §; also, 11 + 3 + 28 = 1. 1+1+2+} d 203. Mixed numbers may be added either by first changing them to improper fractions, and then proceeding as above, or by first finding the sum of the integral parts; then, that of the fractional parts; and lastly, the sum of the two sums. Examples. 1. To add 51, 63, and 79, we may proceed thus: 51= {}; 63 = 27; 78 = ; (193,) therefore, 5 + 6 + 7§ 44+54 +61 = Or thus: 5+ 6+7=18; the sum of the integral parts: then, of the fractional parts; and lastly, 18 +17=197, as before. The student may pursue the method which seems most convenient, according to circumstances. The one also proves the other. 2. 2} + 1}+3§ + §7=8. 5. + + 8 + 1928+ 627=91,498. Subtraction of Fractions. 204. When the given fractions have a com. denom., we subtract the numerator of the less from that of the greater, and place the remainder over the com. denom.; thus, &— 3 = 4. The numerators being each composed of units of the order signified by the com. denom., their difference is evidently a number of units of the same order; for which reason we give it the same denominator. When the fractions have different denominators, we reduce them to a com, denom., as in addition; because we cannot subtract units of different orders; after which we proceed as before. 19 5 255 {་=ias;8༔— ༔༔= z;7༔༔— 1༔༔= Subtraction of Mixed Numbers. 205. We may reduce the mixed numbers to improper fractions; and, when the denominators are not alike, bring them to a common denominator, and subtract as above. For example: = if, from 7, we would subtract 2, we say 71 = 57, and 23 = 17; therefore, 7}—23=57—17-399-136: 399-136 56 But, when the integral part of one or both numbers is very great, the following method will be found more convenient: If the fractional parts have different denominators, we first reduce them to a com. denom. We then write the less mixed number under the greater, and, having drawn a line underneath, we subtract the numerator of the lower fraction from that of the upper one, and place the remainder over the com. denom. for the difference of the fractions. Lastly, we find the difference of the whole numbers, as usual. When the numerator of the lower fraction is the greater of the two, we subtract it from the sum of the terms of the upper fraction; and, having placed the remainder over the com. denom., we carry one to the unit figure of the lower number, and proceed as usual. As the denominator is a unit, and the numerator the fraction, when we add the denominator to the numerator, we add a unit to the fraction; therefore, having added a unit to the upper number, we add a unit to the lower, by which means both numbers are increased alike, which cannot affect the difference. Examples. 1. To subtract 2 from 73, we first find that and are equivalent to and ; we then place the numbers thus: 7336 22 42 and, as 24 is greater than 7, we say 56-2432; and 32 + 7=39, which we place over 56. Then, having added a unit to the upper number, we add a unit to the lower number 2, saying, 3 from 7, four. Wherefore, 71—2%=43%. the denominator of each fraction, placing the quotient below the line under the denominator which gave it. Lastly, we multiply the two terms of each fraction by the corresponding number below the line, and have the fractions sought. We need not multiply each denominator by the number below the line, because we already know the result. Also, when the scholar becomes skilful, he may omit the placing of the numbers below the line. 2 7 16 12 6 5 3 To reduce,,, and to a com. denom., we place the numbers thus: and having found (185) that the least com. denom. is 48, we divide this successively by 3, 4, 8, and 16, and place the quotients 16, 12, 6, and 3, each under the number which gave it. Lastly, we multiply the two terms of each fraction, or rather, the numerator only, by the number below the line, and have 38, 39, 48, and for the 1 required fractions. 48 32 36 We easily see (190) that each of the numbers 16, 12, 6, and 3, respectively, shows the number of times that the new common generic unit is less than that of the fraction under which the number stands; and that in multiplying the numerator by this number, we have a number of parts in each new fraction, which is just as many times greater as the number of times that the parts are less in value. Wherefore, the value of the fraction is not altered. Examples. 1., 11, and, are equal to §8, 35, and 28. 609 37 4. 8, 8, 13, 29, 87, " 5. 3, 14, 12, 31 2 16 19 24 51 SECTION IX. ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF VULGAR FRACTIONS AND MIXED NUMBERS ;—ALSO DIVISION BY FACTORS. Addition of Vulgar Fractions. 202. WHEN the fractions to be added have a com. denom., we add all the numerators, and place the sum over the com. |