41, 242. VULGAR FRACTIONS. 99 common EXAMPLES. 3. Reduce ş and it to a 5. Change , and it to common denominator. fractions having a Ans. , land 3. 3X4=12 com. denom. 6. Express and of a then 2 and is Ans. dollar in parts of a dollar of 4. Reduce 1, and to the same magnitude. a cuinmon denominator. Ang. 18 and 25 211. TO REDUCE FRACTIONS TO THEIR LEAST COM MON DENOMINATOR. ANALYSIS. 1. Reduce }, 4, 5, and 14 to their least common denominator. The common denominator found by the foregoing rule is a common multiple of the denominators of the given fractions, but wol always the least cominon multiple, and consequently not alvays the least (common denomivator. The least common multiple of the denominators, 3, 4, 8 and 12 is 24 (238), which may be divided into thirds pourths, eighths and twelfths ; for the new numerators we must therefore take such parts of 24 as are denoted by the given fractions; and this is done by dividing 24. by each of the denominators (4438, 46, 4433, and 42), and multiplying the quotients by the respective numerators, (8X18, 6X3=18, 3x5=15, and 2x11=2), and the new numerators 18, 18, 15 and 22) written over 26, the common denominator, give , and 24 for the new fractions, baving the least possible common denominator. Hence, 242. To reduce fractions of different denominators to equivalent fractions having the least common denominators. RULE.- Reduce the several fractions to their least terms (235). Find the least common multiple of all the denonsinators for a common denominator. Divide the common denominator by the denominators of the several fractions, and multiply the quotients by the respective numerators, and the products will be the new numerators' required. QUESTIONS FOR PRACTICE. 2. What is the least com- 3. What is the least com. mon denominator of }, j and mon denominator of }, } and 1 ? Ans. , , 2)2, 3, 4 4. What is the least com mon denominator of and 1, 3, 2 11? Ans. 1$, ts: Then 2x3x2=12, least 5. Express, and of a common denominator. dollar in the least possible And similar parts of a dollar. 12:2–6 and 6x1=6 Ans. $g and s. 12)(3-4 4 X 2=8 6. Reduce it is and to 12)(453 383-9 the least common denomina. Then 19, Ans. tor. Ans. 3, 123, 123 new num. 243. REDUCTION OF FRACTIONS. (52) ANALYSIS. 1. What part of a shilling 1. What part of a pound is is at of a pound? of a shilling ? Pounds are reduced to shillings by To change shillings to pounds, dimultiplying them by 20 (138), and vide them by 20 (138). 2013 4*2=14 (220), and 1= Tho (221), and To=(235). } (235). * of a pound, then, is of a shilling is, then, 24 of a pound. of a shilling DESCENDING, ASCENDING. 244. To change fractions of 245. To change fractions of & higher into those of a lover a lower into those of a higher denomination. denomination. RULE.-Reduce the numer - Rule.-Multiply the denomator to the lower denomina- inator by the number which tion by. Art. 139, and write it is required to make one of the over the given denominator. 'next higher denomination, and so on (140); and write the last product under the given numcrator. QUESTIONS FOR PRACTICE. 2. What part of a pound is 2. What part of a cwt is of a cwt. ? of a pound? 3X4X.28 336 6 6 6 3 Ang. Ane, 392 352 7X28X4784 3942 246. To reduce fractions to integers of a lower denomination, and the reverse. ANALYSIS. 1. Reduce f of a pound to 1. Reduce 78. 6d. to the shillings and pence. fraction of a pound. £3x20=60s. and 69.745.; 7s.6d._90d. £1208,240d., but is.x12=48.d., and 480.=ud. then 7s. 6d.==£3. Hence, Then £3=7s. 6d. Hence, 247. To reduce fractions to 248. To reduce integers to integers of a lower denomina-fractions of a higher denomination. tion. Rule.-Reduce the numer- RULE.—Reduce the given ator to the next lower denom- number to the lowest denomiination, and divide by the de- nation mentioned for a numernominator; if there be a re- ator, and a unit of the higher mainder, reduce it still lower, denomination to the same for and divide as before:; the sev- a denominator of the fraction eral quotients will be the an- required, gwer. QUESTIONS FOR PRACTICE. 2. In of a day, how many 2. What part of a day are hours. 8 hours ? 3 In $ of an hour, how 3. What part of an hour many minutes and seconds ? are 6m. 40s.? 4. In g of a mile, how many 4. What part of a mile arg rods? 120 rods? S. In yo of an acre, how 5. What part of an acre wany toods and rods ? are 1 rood and 80 rods ? 249. ADDITION OF FRACTIONS. ANALYSIS 1. What is the sum of f of a dollar and of a dollar? As both the fractions are 9this of the same unit, the magnitude of the parts is the same in both the number of parts, 3 and 4, may therefore be added as whole numbers, and their sum, 7, written over 9, thus, fi ex presses the sum of two given fractions 2. What is the sum off of a yard and f of a yard ? As the parts denoted by the given fractions are not similar, we cannot add them by adding their numerators, 3 and 2, because the answer would be neither į nor ; but if we reduce them to a common denominator, % becomes 4, and g, (240). Now each fraction denotes parts of the same unit, which are of the same magnitude, namely, 24ths; their numerators, 8 and 9, may therefore be added; and their sum, 17, being written pver 24 we have di of a yard for the sum of Ķ and g of a yard. 250. To add fractional quantities. RULE.—Prepare them, when necessary, by changing compound fractions to single ones (224), mixed numbers to improper fractions (218), fractions of different integers to those of the same (247, 248), and the whole to a common denominator (240); and then the sum of the numerators written over the.common denominator, will be the sum of the fractions required. QUESTIONS FOR PRACTICE. 3. What is the sum of 1 6. What is the sum of $ and 5 of a dollar ? mile, of a yard, and of a 4+š=+= Ans. foot ? 4. What is the sum of ģ Ans. 660yds. 2ft. Iin, and of a cwt. ? 7. What is the sum of % of 6%, of }, and 74 ? 5. What is the sum of $ Ans. 13117 of a week and 7 of a day? 8. What is the sum of po #tab=list=ftw.= $, and ? W 14h. Ans. Ans. 44 Ans. 335 251, 252. VULGAR FRACTIONS. 103 251. SUBTRACTION OF FRACTIONS.: ANALYSIS. 1. What is the difference between ro of a dollar and it of a dollar ? By evidently expresses 2 tonths more than 3 tenths ; t then is the differenee. 2. What is the difference between % of a yard and f of a yard? Here we cannot subtract & from f, for the same reason that we could not add them (49). We therefore reduce them, to a common denominator, (298) 21), and then the difference of the numerators (9—831), written over 24, the common denominator, gives 24 for the difference of the fractions. RULE. Prepare the fractions as for addition (250), and then the differ. cnce of the numerators written over the common deuominator will be the difference of the fractions required. QUESTIONS FOR PRACTICE. 3. What is the difference 6. From 961 take 144. between } and ? Ans. 811+ ==12, Ans. 7. From 19 take g. 4. From take Ize 8. From 7 weeks take 970 5. From af take of ho days. Ans. 5w. 4d. 7h. 12m. Ans. 184 Ans. * Ans. 252. RULE OF THREE IN VULGAR FRAC TIONS. RULE.—Prepare the fractions by reduction, if necessary, and state the question by the general rule (198); invert the first term, and then maltiply all the numerators together for a new numerator, and all the denominators together for a new denominator; the new numerator, written over the new denominatot, will be the answer required. QUESTIONS FOR PRACTICE. 1. If oz. cost £7, what is fyd. wide, will line 13 vill loz, cost? yards of cloth that is 24 yde £ wide ? 131= and 24= 3X3XF=£*=£1 1s. 94. *:52 :: $x*X= Ans. 1264=44yds. 6in. Ans. 2. How much shalloon that |