215. VULGAR FRACTIONS. 89 rates the parts of the denominalor signified by the fraction. As there are no limits to the number of parts into which a thing, or whole, may be divided, it is evident that it is possible for every number to be a numerator, or a denominator of a fraction. Hence the variety of fractions must be unlimited. 2. Fractions are of two kinds, Vulgar and Decimal, which differ in the form of expression, and the modes of operation. 3. A Vulgar Fraction is expressed by two numbers, called the nume--rator and denominator, written the former over the latter, with a line betweeu, as $, the former before the latter, as 3-8=. 4. A Decimal Fraction, or a Decimál, is a fraction which denotes parts of a unit which become ten times smaller by each successive division (113), and is expressed by writing down the numerator only, . (See Part H. Seci. III). A decimal is read in the same manner as a vulgar fraction; thus -0.5'is read 5 tenths, 0.25 25 hundredths, and it is put into the form of a vul. gar fraction hy drawing a line under it, and writing as many ciphers under the line as there are figures in the decimal, with a l at the left hand; thus, 0,5 becomes , 0.25, , and 0.005, Toto VULGAR FRACTIONS 215. 1. A proper fraction is one whose numerator is less than its denominator; as 3, 4, 3, &c. (23). 2. An improper fraction is one whose numerator is greater than its denominator; as, , , , &c. (24). 3. The numerator and denominator of a fraction are called its terms (30). 4. A compound fraction is a fraction of a fraction; as, $ of j. 5. A mixed number is a whole number and a fraction written together, as 124, and 6$ (23). 6. A common divisor, or common measure, of two or more numbers, is a number which will divide each of them without a remainder. 7. The greatest common divisor of two or more numbers, is the greatest number which will divide those numbers severally without a remainder. 8. Two or more fractions are said to have a common denomi, nator, when the denominator of each is the same number (25) 9. A common multiple of two or more numbers is a number, which may be divided by each of those numbers without a remainder. The least common multiple is the least number, which may be divided as above. io. A prime number is one which can be divided without a remainder, only by itself, or a unit. 11. An aliquot part of any number, is such part of it, as being taken a certain number of times, will exactly make that pumber. 12. A perfect number is one which is just equal to the sum of all its aliquot parts. The smallest perfect number is 6, whose aliquot parts are 3, 2, and 1, and 3+2+I3; the next is 28, the next 496, and the next 8128. Only ten perfect numbers are yet known. 216. UNDER THE WHOLE NUMBERS, CONSIDERED FORM OF FRACTIONS. proper fraction As the denominator des 25x3+1=76 vision of 1 by 3. 3 ANALYSIS. 1. Change 76 to a whole or 1. Change 25$ to an immixed number. 3)76 notes the number of parts devoles the di254 | into which the whole, or unit, is divided, and the nu (129); if now we merator' shows how many of those multiply 25 by 3, and add the proparts are contained in the fraction duct 1o 1, making (25x3+1=) 76, (22), there are evidently as many and then write the 76 over 3, thus, wholes, as the number of times the 76, we evidently both multiply and numerator contains the denomina- divide 25 by 3; but as the multiplitor; or, otherwise, since every frac- cation is actually performed, and tion denotes the division of the nu- the division only denoted, the exmerator by the denominator (129), pression becomes an improper fracwhere the numerator is greater than tion. áte denominator, we have only to A whole number is changed to an perform the division which is de- improper fraction, by writing I under uotod. it, with a line between. 217. To change an improper 218. To change a whole of fraction to an equivalent whole mired number to an equivalent or mixed number. improper fraction. 'Rule.- Divide the numera- RULE.—Multiply the whole tor by the denominator, and the number by the denominator of quotient will be the whole, or the fraction, add the numerator mixed number required. to the product, and write the sum over the denominator for the required fraction QUESTIONS FOR PRACTICE 2. Change 26 to 8 mixed 2. Change 87 to an imprenumber. 3. Change 24 8 to a mixed 3. Change 27g to an ine number. 4. In 44s. shillings, how 4. In 1970s. how many many shilling3 ? 12ths? -5. In 24 of a week, bow 5. In 34 weeks, how many many weakna 7ths ? per fraction. proper fraction. 219, 220, 22). VOLGAR FRACTIONS 01 219. MULTIPLICATION AND DIVISION OF FRACTIONS BY WHOLE NUMBERS. ANALYSIS i. James had ß of a peck of 1. Henry had of a peck of plums, and "Henry had twice as plums, which were twice the quan, many; how many bad Henry? lily James had; bow many lad Here we have evidently to multiply James ? by 2; but two times f is ; Here we liave evideiilly to divide hence, to multiply by 2, we multi-into 2 equal parts ; but divided ply the numerator 2 by 2, and write into 2 parts, one of them is $; then the product, 4, over 8, the denom- in divided by 2, we must divide inator; or, otherwise, if we divide 8, the numerator by 2, and write the the denominator, by 2, and write quotient, 1, over 4, the denominator; the quotient, 4, under 2, the nu- or, otherwise, if we multiply 4, the merator, thus, ş, the traction be- denominator,' by 2, and write the comes multipled; for while the num product, 8, under 2, the numerator, ber of parts signified remains the thus, &, the fraction becomes dividsame, the division has rendered those ed hy 2; for while the number of parts twice as great; and these re- parts remains the same, the multiplisulls, and ļ, are evidently the cation has rendered the parts only same in value, though differing in half as great; and these results, the magnitude of the terms. There- 4 and , are evidently the same in fore value, though expressed in different terms. Hence 220. To multiply a fraction 221. To divide a fraction by by a whole number: a whole number. RULE.—Multiply the nume- Rulb.-Divide the numerarator, or divide the denomina- tor, or multiply the denomina tor, of the fraction by the tor, of the fraction by the whole whole number; the result will | number; the result will be the be the product required. required quotient. QUESTIONS FOR PRACTICE. 2. What is the product of 2. How many times 24 in by 24?-of ý by 32?_of 2 ?-_-32 in 189?-36 in 188? f by 36?_off by 42?-of - 42 in 1767-9 in 27? 1 by 3 ? 3. How many tiines 5 in 3. How many are 5 times 1 ?-3 in 4 ?-14 in 41?_7 16?-3 times ?_14 times in B, or 5? #?–7 times ? 4. If 5 lb. of rice cost $ 4. If 1-lb. of rice cost z's of a dollar, what will like of a dollar, what will 5 lb. cost? cost? 5. If a bushel of wheat 5. If 6 bushels of wheat vost is of a dellar, wbat will cost of a dollar, what is it a 6 bushels cost? bushel ? MULTIPLICATION BY FRACTIONS. ANALYSIS. 222. If a load of hay be worth $12, what are of it worth? Here 12 and g are evidently two factors, which, multiplied together, will give the price; and since the result is the same, whichever is made the muluplier (86), we may make š the multiplicand, and proceed (220) thus, fx 12=438 dollars. Ans. Otherwise, since in the multiplication by a whole number, the multiplicand is repeated as many times as the multiplier conlains units, if therefore the multiplier be 1, the multiplicand will be repeat. ed one time, and the product will be just equal to the multiplicand; if the inultipher be 1, the multiplicand will be repeated half a time, and the product will be half the multiplicand; if thie multiplier be }, it will be repeated one third of a time, and the product will be one third of the multiplicand, sand generally, multipljing by a fraction is taking out such a part of the multiplicand as the fraction is part of a unit. Hence the product of 12 by 3, is of 12; and to find 3 of 12, we must first find } of 12, by dividing 12 by 3, and then multiply this third by 2; thus, 12;3=4, and 4X2=8; $8 then are g of $12, or the product of $12 by }, as by the former method. Therefore, 223. To multiply a whole number by a fraction. RULE.—Divide the whole number by the denominator of the fraction, and multiply the quotient by the numerator,—or multiply the whole number by the numerator, and divide the product by the denominator. QUESTIONS FOR PRACTICE. 2. What is the product of 5. If a bushel of pears 4 multiplied by } ?-of 7 mul- cost 75 cents, what cost of tiplied by 1?-of 9 by $?- them? Ans. 15 cts. of 17 by ?? 6. What is the product of 3. If a barrel of rum cost 16 by $ ?-256 by ?-of 12 $24, what cost of it? by ? Ans. $18. NOTE.-It will be observed from 4. What cost 18 bushels the above examples, that multiplicar of corn, at of a dollar a tion by a proper fraction gives a product which is less than the multibashel ? Ans. $6. plicand (121) 224. MULTPLICATION OF ONE FRACTIONAL QUANTITY BY ANOTHER. nagon owning of a gristmill, sold of his share ; what part of the whole mill did he sell ? Here we wish to take out of %, which has been shown (222) 1o be the as multiplying by }; but to multiply by a fraction, we must divido the multiplicand by the denominator, and multiply the quotient by the aur merator ; f is divided by 3, by multiplying the denominator 1 by 3 (19! 225, 26. VULGAR FRACTIONS. 93 and the quotient is is; and it is multiplied by 2. by multiplying tho numerator, 3, by 2 (2:20), and the product is equal to the part of tbe mill sold. Hence, To mulliply a fraction by a fraction, or to change a compound fraction to a single one. Rule.-Multiply the numerators together for a new nume. rator, and 'the denominators together for a new denominator QUESTIONS FOR PRACTICE (56). 2. A man owning # of a 4. What part of a mile is farm, sold 4 of his share ; of of a mile ? what part of the farm did he Ans. Et sell? 5. Change of } of of! 3. What part of a foot is of it to a single fraction. of th of a foot ? 6. Multiply 34 by it Ans. t Aus. Tom Ans. Is 225. DIVISION BY FRACTIONS. 1. In 6 dollars, how many times of a dollar ? Here we wish to divide $6 imo parts, each of which shall be of a dol 'lar, or in other words, divide 6 by: Now in order to find how many times in 6, we reduce 6 to 4ths, by multiplying it by 4, the denominator of tho Traction, thus : 4 times 6 are 24; 6 dollars, then, are 24 fourths, or quarters of a dollar; and dividing 24 fourths by 3 fourths (the numerator), the quo tient, 8, is evidently the number of times of a dollar may be had in 2 or 6 dollars. Hence, 226. To divide ia iphole number by a fraction. ROLE.--Multiply the number to be divided by the denomina tor of the fraction, and divide the product by the numerator. QUESTIONS FOR PRACTICE. 2. In 7 shillings, how many 6. In a pound of tobacco, times of a shilling? how many quids, each weigh Ans. 25 ing is of an ounce ? -3. In 17 bushels of wheat, Ans. 3.44 =1014 bow many tiines % of a bush- 7. How many are 7-13 8+4 ? 2:42? 4. In 1 gallon of wine, how many times is of a gallor? Nore.-Here it will be seen that divisieu hy a fraction, gives a que Ans. 4:317 times. tieut larger ihau the dividend. -5. In 5 eagles, of & dollar ? Aqs. 200, Ans. how many |