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+1=6; the next is 28, the next 496, and the next 8128. Only ten perfect numbers are yet kuowu. 216. WHOLE UNDER THE NUMBERS, CONSIDERED proper fraction. ANALYSIS. 1. Change ze to a whole or 1. Change 25$ to an immixed number. 3)76 As the denominator de notes the number of parts i 25x3+vision of 1 by 3, 25) | into which the whole, or 3 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 to 1, making (25x3+1=) 76, (22), there are evidently as many and then write the 76 over 3, thus, whores, as the number of times the 76, we evidently both multiply and numerator contains the denomina- divide 25 by 3; but as the maltiplitor; or, otherwise, since every frac- cation is actually performed, and tion denotes the division of the mu- the division only denoted, the exmerator by the denominator (129), pression becomes an improper fracwhere the numerator is greater than tion. the denominator, we have only to A whole number is changed to an periorm the division which is de- improper fraction, by writing 1 under noted. it, with a line between. 27. To change an improper | 218. To change a whole or 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 whale, or the fraction, add the numerator inixed number required. to the product, and write the sum over the denominator for the required fraction. QUESTIONS FOR PRACTICE 2. Change 25 to a mixed 2. Change 8p to an impronuinber. 3. Change 24 € to a mixed 3. Change 273 to an imnumber. 4. In 2365. shillings, how 4. In 19, s. how many many shillings? 12ths ? 5. In 24 of a week, how 5. In 3 weeks, how many many weeks? 7ths ? per fraction. proper fraction. 219. OF FRACTIONS MULTIPLICATION AND DIVISION BY WHOLE NUMBERS. divided terms. ANALYSIS. 1. 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 had Henry? lity James had; how many liad Here we have evidently to multiply James ? f by 2; but two times is ; Here we have evidently to divide but hence, to multiply s by 2, we multi-l into 2 equal parts ; ply the numerator 2 by 2, and write into 2 parts, one of them is d; then the product, 4, over 8, the denom to divide å 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, , the fraction be- denominator,' by 2, and write the comes multipled; for while the num product, 8, tinder 2, the numerator, ber of parts signified remains the thus, a, the fraction becomes dividsame, the division has rendered those ed by 2 ; for while the number of parts twice as great; and these re parts remains the same, the multiplisults, f and , 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- & and s, are evidently the same in fore value, though expressed in different Hence 220. To multiply a fraction 221. To divide a fraction by by a whole number. a whole number. RULE.—Multiply the nume RULE.—Divide the numerarator, or divide the denomina- | tor, or multiply the denominator, 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?_off by 32?-of ?-32 in 169?_36 in 198? by 36?-off by 42?-of -42 in 148 ?-9 in 11 ? 3. How many times 5 in 3. How many are 5 times 2?_3 in f ?-14 in 41?_7 1 ?-3 times f?–14 times in ý, or 5? 33?–7 times ? 4. If 5 lb. of rice cost } 4. If 1 lb. of rice cost z's of a dollar, what will 1 lb. of a dollar, what will 5 lb. cost? cost? 5. If a bushel of wheat 5. If 6 bushels of wheat cost of a dollar, what will cost of a dollar, what is it a 6 bushels cost? bushel ? by 3? MULTIPLICATION BY FRACTIONS. ANALYSIS. 222. If a load of hay be worth $12, what are of it worth? Here 12 and 3 are evidently two factors, which, multiplied together, will give the price; and since the result is the same, whichever is made the multiplier (86), we may make š the multiplicand, and proceed (220) thus, fx 12–4–8 dollars. Ans. Otherwise, since in the multiplication by a whole number, the multiplicand is repeated as many times as the multiplier conains units, if therefore the multiplier be 1, the multiplicand will be repeated one time, and the product will be just equal to the multiplicand ; if the multiplier be :), the multiplicand will be repeated half a time, and the product will be half the multiplicand; if the multiplier be }, it will be repeated one third of a time, and the product will be one third of the multiplicand, and generally, multiplying 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 s, is į of 12; and to find 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 šof $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 1 ?-of 7 mul- cost 75 cents, what cost of tiplied by 1 ?-of 9 by 1?-them ? Ans. 15 cts. of 17 by ? 6. What is the product of 3. If a barrel of rum cost 16 by } ?—256 by 1 ?-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 multiplicaof corn, at } of a dollar a tion by a proper traction gives a product which is less than the multibushel ? Ans. $6. plicand (121). 224. MULTIPLICATION OF ONE FRACTIONAL QUANTITY BY ANOTHER. nwson owning of a gristmill , sold f of his share ; what paic of the whole mill did he sell? Here we wish to take out of 4, which has been shown (222) to be the same as multiplying å by }; but to multiply by a fraction, we must divide the multiplicand by the denominator, and multiply the quotient by the numerator; } is divided by 3, by multiplying the denominator 4 by 3 (121), and the quotient is ; and is multiplied by 2, by multiplying the numerator, 3, by 2 (220), and the product is equal to the part of the mill sold. Hence, To multiply a fraction by a fraction, or to change a compound fraction to a single one. RULE.-Multiply the numerators together for a new numerator, 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 } of his share'; of şof a mile ? what part of the farm did he Ans. 2=1 sell ? 5. Change 1 of 1 of 4 of 3. What part of a foot is off to a single fraction. of Tł of a foot ? Ans. quz 6. Multiply by 11. Ans. & Ans. is 225. DIVISION BY FRACTIONS. 1. In 6 dollars, how many times of a dollar ? Here we wish to divide $6 into parts, each of which shall be of a dollar, or in other words, divide 6 by . Now in order to find how many times in 6, we reduce 6 tó 4ths, by multiplying it by 4, the denominator of the fraction, 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 i of a dollar may be had in 2 or 6 dollars. Hence, 226. To divide a whole number by a fraction. RULE.-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 ing is of an ounce? 3. In 17 bushels of wheat, Ans. 394 =1015 how many times şof a bush 7. How many are 7 = 1? el? Ans. 85. 8+4 ? 2:32? 4. In 1 gallon of wine, how NOTE.Here it will be seen that many times it of a gallor? division by a fraction, gives a puun Ans. 1 =17 times. tient larger than the dividend. 5. In 5 eagles, how many of & dollar? Ans. 200. Ans. 46 227. DIVISION OF ONE FRACTIONAL QUANTITY BY AN OTHER. 5 12 ANALYSIS. 1. If of a bushel of wheat cost of a dollar, what is that per bushel ? To find the cost per bushel, we must divide the price by the quantity (154), that is, we must divide by s. But to divide a number by a fraction, we multiply it by the denominator, and divide the product by the numerator (226); hence, we must multiply by 4, as 3X4 12 (220), and 12 12 is divided by 3, by inultiplying the denominator, 5, by 3, as, (121); 1 of a dollar then is the price of one bushel. Hence, 228. To divide a fraction by a fraction., RULE.—Multiply the numerator of the dividend by the denoininator of the divisor for a new numerator, and the denoninator of the dividend by the numerator of the divisor, for & new denominator. Note.-In practice, it will be most convenient to invert the divisor, and then proceed as in Art. 224. Ans. QUESTIONS FOR PRACTICE. 2. In 74 how many times 5. If of a yard cost of ?? a dollar, what is that a yard? 3. In 2 how many times Ans. La=$1.777 Ans. 118=1. 6. If 4 of a piece of cloth 4. At of a dollar a bush- be worth of of an eagle, el for oats, how many can I what is the whole piece worth? buy for 8 of a dollar ? Ans. 44 eag. Ans. 133 bush. 229. FRACTION ALTERATION IN THE TERMS OF A WITHOUT ALTERING ITS VALUE. ANALYSIS. A fraction is multiplied by multiplying its numerator, and divided by multiplying its denominator (219); hence if we multiply both the terms of a fraction at the same time by any number, we both multiply and divide the fraction by the same number, and therefore do not alter its value. Again, a fraction is divided by dividing its numerator, and multiplied by dividing its denominator (219); hence if we divide both the terms of a fraction at the same time by any number, we both divide and multiply tbø fraction by the same number, and therefore do not alter its value. Hence, |