Ans. &, &, QUESTIONS FOR PRACTICE. 2. What is the least com 3. What is the least com. mon denominator of t, and mon denominator of }, { and * ? 2)2, 3, 4 Ans. 14, 15: 4. What is the least com mon denominator of } and 1, 3, 2 ig? 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=61 Ans. $g and . 12)/3-4 4x2=8 6. Reduce it is and to 12)(4=3 3x3-9 the least common denomina Then f , Ans. tor. Ans. 233, 23, 43897 nei num. 243. REDUCTION OF FRACTIONS. (52) ANALYSIS. 1. What part of a shilling 1. What part of a pound is is 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). 2015(= 4x20=34 (220), and 2= (221), and Tho=3 (235). $ (235). zof a pound, then, is & of a shilling is, then, z4 of a pound. of a shilling. DESCENDING. ASCENDING. 244. To change fractions of 245. To change fractions of a higher into those of a lower 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 nu. merator. QUESTIONS FOR PRACTICE. 2. What part of a pound is 2. What part of a cwt. is of of a cwt. ? of a pound? 3X4X.28 336 6 6 6 3 Ang. Ans. 392 392 7X28X4784392 80 20 of 28 3. Reduce is of a pound 3. Reduce 40 d. to the frac. to the fraction of a penny. tion of a pound. 4. What part of a pound is 4. What part of a guinea of a guinea? is of a pound ? Ans. 7 20-140-5 140 5. What part of a rod is 5. What part of a mile is žy of a mile? 2 rods? 6. What part of a minute 6. What part of an hour is is of an hour ? 15 of a minute ? 7. What part of a pwt. is 7. What part of a pound is Toolb. Troy? of a pwt. ? 246. To reduce fractions to integers of a lower denomination, and the reverse. ANALYSIS. 1. Reduce g of a pound to 1. Reduce 78. Od. to the shillings and pence. fraction of a pound. £3x20=60s. and Ses=795.; 7s.6d.30d. £1_203.240d.; but s.X 12—42.d., and 434.5cd. then 7s. 6d.-£4=£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. tioil. 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 cral quotients will be the an- required. swer. QUESTIONS FOR PRACTICE. 2. In of a day, how many 2. What part of a day are hours ? 8 hours ? 3. In j 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 are rods? 120 rods ? 5. In 1e of an acre, how 5. What part of an acre many roods and rods? are 1 rood and 30 rods ? 249. ADDITION OF FRACTIONS. ANALYSIS. 1. What is the sum of f of a dollar and of a dollar ? As hoth the fractions are this 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, }, ex• presses the sum of two given fractions 2. What is the sum of 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 é nor ; but if we reduce them to a common denominator, & becomes , 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 over 24 we have 37 of a yard for the sum of and 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 numeratcrs written over the common denominator, will be the sum of the fractions required. QUESTIONS FOR PRACTICE. 3. What is the sum of } 6. What is the sum of and į of a dollar ? mile, of a yard, and of a +=isti=15, Ans, foot ? 4. What is the sum of s Ans. 660yds, 2ft. Iin. and of a cwt. ? 7. What is the sum of 1 of 67, of , and 74 ? 5. What is the sum of Ans. 1311 of a week and of a day? 8. What is the sum of 7, *+ze=to=.= 4, and ? Ans. 395 2d 14h. Ans. Ans. 24 251. SUBTRACTION OF FRACTIONS. ANALYSIS. 1. What is the difference between 3 of a dollar and to of a dollar ? in evidently expresses 2 tenths more than 3 tenths ; it then is the difference. 2. What is the difference between % of a yard and g of a yard ? Here we cannot subtract from , for the same reason that we could not add them (49). We therefore reduce them to a common denominator, (2987 29), and then the difference of the numerators (9—8=1), 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 difference of the numerators written over the common denominator will be the difference of the fractions required. QUESTIONS FOR PRACTICE. 3. What is the difference 6. From 961 take 14. between f and ? Ans. 8111 =m1=1, Ans. 7. From 48 take ţ. 4. From take I. Ans. 194 8. From 7 weeks take 976 5. From take of 4. days. Ans. 5w.4d. 7h. 12m. Ans. 1 Ans. 4. 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 new denominator; the new numerator, written over the new, denominator, will be the answer required. QUESTIONS FOR PRACTICE. 1. If {oz. cost £7, what is gyd. wide, will line 131 will loz, cost? yards of cloth that is 24 yds. £ wide ? ::: 1 131=5 and 215 }XfXt=£*=£1 ls. 94. 1:52:: 4x5X= Ans. 1264=44yds. 6in. Ans. 2. How much shalloon that OZ. Then, 3. If | gallon cost £3, what 5. A lends B $48 for of will g tun cost? a year; how much must B of of of zote tun. lend A is of a year to bal. 2016: 5 :: 5. Ans. £140. ance the favor? 4. If my horse and chaise Ans. $86.40. be worth $175, and the value 6. A person owning of a of my horse be that of my farm, sells of his share for chaise, what is the value of £171; what is the whole each? farm worth? Ans. £380. 1:175:: 1 : $105 horse. 1:175:: : $70 chaise. MISCELLANEOUS. Tor miscellaneous exercises, let 5. Two thirds and of a the pupil review Section IV. Part I. and also the following articles : 51, person's money amounted to 52, 55, 56, 57, 58, and 59. $760; how much had he ? 1. In an orchard I the trees Ans. $600. bear apples, peaches, 1 6. A man spent of liis plums, 30 pears, 15 cherries, life in England, 1 in Scotland, and 5 quinces; what is the and the remaining 20 years, in whole number of trees? the United States : to what 1+1+1=niet= age did he arrive ? Fi and 12=50 1; then 50 Ans. 48 years. X 12—600, Ans. 2. One half,} of a school, 7. A pole is in the mud, and 10 scholars, make up the 4 in the water, and 12 feet school : how many scholars out of the water; what is its are there? Ans. 60. length ? Ans. 70 feet. 3. There is an army, to 8. There is a fish whose which if you add }, }, and head is 1 foot long, his tail itself, and take away 5000, as long as his head and half the sum total will be 100000; the length of his body, and what is the number of the his body as long as his head whole army ? and tail both; what is the Ans. 50400 men. length of the fish? 4. Triple, the half, and the Ans. 8 feet. fourth of a certain number are equal to 104; what is that 9. What number is that number? whose 6th part exceeds its 8th part by 20? Ans. 480. Ans. 2715 |