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 known. denotes the division of 1 by 3, (129); if now we multiply 25 by 3, and add the product to 1, making (25×3+1=)* 76, and then write the 76 over 3, thus, unit, is divided, and the numerator shows how many of those parts are contained in the fraction (22), there are evidently as many wholes, as the number of times the, we evidently both multiply and numerator contains the denominator; or, otherwise, since every fraction denotes the division of the numerator by the denominator (129), where the numerator is greater than the denominator, we have only to perform the division which is denoted. divide 25 by 3; but as the multiplication is actually performed, and the division only denoted, the expression becomes an improper frac tion. A whole number is changed to an improper fraction, by writing 1 under it, with a line between. 218. To change a whole or mixed number to an equivalent improper fraction. RULE.-Multiply the whole number by the denominator of the fraction, add the numerator to the product, and write the sum over the denominator for the required fraction. QUESTIONS FOR PRACTICE 2. Change 25 to a mixed number. 3. Change 24 to a mixed number. 4. In 236s. shillings, how many shillings? 5. In 24 of a week, how many weeks? 2. Change 8 to an improper fraction. 3. Change 27 to an improper fraction. 4. In 19s. how many 12ths? 5. In 33 weeks, how many 7ths? 219. MULTIPLICATION AND DIVISION OF FRACTIONS BY WHOLE NUMBERS. ANALYSIS. 1. James had of a peck of plums, and Henry had twice as many; how many had Henry? Here we have evidently to multiply by 2; but two times hence, to multiply by 2, we ply the numerator by 2, and write the product, 4, over 8, the denominator; or, otherwise, if we divide 8, the denominator, by 2, and write the quotient, 4, under 2, the numerator, thus, 2, the fraction becomes multipled; for while the number of parts signified remains the the division has rendered those parts twice as great; and these results, and, are evidently the same in value, though differing in the magnitude of the terms. There same, fore 1. Henry had of a peck of plums, which were twice the quantity James had; how many had James? is; 220. To multiply a fraction by a whole number. RULE.-Multiply the numerator, or divide the denomina- | tor, of the fraction by the whole number; the result will be the product required. and, are evidently the same in value, though expressed in different terms. Hence 221. To divide a fraction by a whole number. RULE.-Divide the numerator, or multiply the denominator, of the fraction by the whole number; the result will be the required quotient. QUESTIONS FOR PRACTICE. 2. What is the product of by 24?-of by 32?-of by 36?-of by 42?—-of by 3? 3. How many are 5 times -3 times ?-14 times ?-7 times ? 4. If 1 lb. of rice cost of a dollar, what will 5 lb. cost? 5. If a bushel of wheat of a dollar, what will cost 6 bushels cost? 2. How many times 24 in 12 ?-32 in 160 ?-36 in 188 ? 42 in 126 ?-9 in 27? 3. How many times 5 in 2-3 in ?-14 in 1?—7 in, or 5? 4. If 5 lb. of rice cost of a dollar, what will 1 lb. cost? 5. If 6 bushels of wheat cost of a dollar, what is it a bushel? MULTIPLICATION BY FRACTIONS. ANALYSIS. 222. If a load of hay be worth $12, what are of it worth? Here 12 and 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 the multiplicand, and proceed (220) thus, 12-24-8 dollars. Ans. Otherwise, since in the multiplication by a whole number, the multiplicand is repeated as many times as the multiplier contains 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 , is 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 4×2-8; $8 then are 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 4 multiplied by ?-of 7 multiplied by ?-of 9 by ? of 17 by ? 3. If a barrel of rum cost $24, what cost of it? Ans. $18. 4. What cost 18 bushels of corn, at of a dollar a bushel? Ans. $6. 224. MULTIPLICATION 5. If a bushel of pears cost 75 cents, what cost of them? Ans. 15 cts. 6. What is the product of 16 by ?-256 by 1 ?—of 12 by? NOTE. It will be observed from the above examples, that multiplication by a proper fraction gives a product which is less than the multiplicand (121). OF ONE FRACTIONAL QUANTITY BY ANOTHER. son 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) to be the same as multiplying 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 farm, sold of his share; what part of the farm did he sell? Ans. 3. What part of a foot is of a foot? of Ans. 1 4. What part of a mile is g of of a mile? of 5. Change Ans. 6 24=1• of of of to a single fraction. 6. Multiply by 21. 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 4. Now in order to find how many times in 6, we reduce 6 to 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 of a dollar may be had in 24 or 6 dollars. Hence, 226. To divide a whole number by a fraction. RULE.-Multiply the number to be divided by the denominator of the fraction, and divide the product by the numerator. QUESTIONS FOR PRACTICE. 2. In 7 shillings, how many of a shilling? times Ans. 28. 6. In a pound of tobacco, how many quids, each weighing of an ounce? Ans. 394-101. 7. How many are 7÷? 8÷? 2÷32? NOTE.-Here it will be seen that division by a fraction, gives a w tient larger than the dividend. 227. DIVISION OF ONE FRACTIONAL QUANTITY BY AN OTHER. 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 2. But to divide a number by a frac tion, we multiply it by the denominator, and divide the product by the numerator (226); hence, we must multiply by 4, as 3×4 12 (220), and 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. 12 12 5X3 15 RULE.-Multiply the numerator of the dividend by the denominator of the divisor for a new numerator, and the denominator 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. QUESTIONS FOR PRACTICE. 2. In 7 how many times Ans. 45. 3. In 22 how many times 22? Ans. 418-1. 4. At of a dollar a bushel for oats, how many can I buy for of a dollar? 229. Ans. 1-3 bush. 5. If of a yard cost of a dollar, what is that a yard? Ans. $1.773. of a piece of cloth 6. If be worth of of an eagle, what is the whole piece worth? Ans. 12 eag. ALTERATION IN THE TERMS OF A FRACTION 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 the fraction by the same number, and therefore do not alter its value. Hence, |