4 5. A mixed number is a whole number and a fraction written together, as, 124, and 63.(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 denominator, 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 these numbers without a remainder. The least common multiple is the least number, which may be divided as above. 10. 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 Lumber. 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. 216. WHOLE NUMBERS CONSIDERED ANALYSIS. 1. Change to a whole or mixed number. notes the number of parts 251 into which the whole, or u nit, 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 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 UNDER THE 1. James had 3 of a peck of plumbs, and Henry had twice as many; how many had Henry? 1. Henry had of a peck of plumbs, which were twice the quantity James had; how many had James? Here we have evidently to divide into 2 equal parts; but divided into 2 parts, one of them is ; then to divide by 2, we must divide the numerator Here we have evidently to multiply by 2; but two times is; hence to multiply by 2, we multiply | the numerator 2 by 2, and write the product, 4, over 8, the denominator; or oth-by 2 and write the quotient 1 erwise, if we divide 8, the denominator, by 2, and write the quotient, 4, under 2, the numerator, thus, &, the fraction becomes multiplied; for while the number of, the fraction becomes diparts signified remains the same, the division has ren over 4, the denominator; or, otherwise, if we multiply 4, the denominator, by 2, and write the product, 8, under 2, the numerator,thus, vided by 2, for while the number of parts remains the dered those parts twice as great; and these results, and, are evidently the same in value,though differing in the magnitude of the terms. Therefore 220. To multiply a fraction by a whole number. RULE.-Multiply the nume rator, or divide the denominator, of the fraction by the whole number, the result will be the product required. same, the multiplication has rendered the parts only half as great; and these results, 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 2. How many times 24 of by 24 ? of § by 32 ?— in 2-32 in 190 ?-36 in of 3 by 36 P-of by 42P-188 ?-42 in 146 ?-9 in 27? of by 3 ? 3. How many are 5 times P-3 times & ?-14 times -7 times ? 4. If 1lb. of rice cost of a dollar, what will 5 lb. cost? 5. If a bushel of wheat cost of a dollar, what will 6 bushels cost? 3. How many times 5 in P-3 in ?—14 in ?— 7 in f, or 5? 4. If 5 lb. of rice cost of a dollar, what will 1 lb. cost? 5. If 6 bushels of wheat cost $, 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 multiplier,(86) we may make the multiplicand, and proceed(220) thus, x12= 28 doll. 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 mul tiplier be 1, the multiplicand will be repeated one time, and the product will be just equal to the multiplicand; if the multiplier be 2, 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; 88 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. 224. MULTIPLICATION OF ONE FRACTIONAL QUANTITY BY ANOTHER. 1. A person owning of a gristmill, sold of his share; what part 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 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. 5. What part of a foot is of of a foot? 225. Ans. T 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 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 the fraction, thus: 4 times 6 are 24; 6 dollars then, are 24 fourths, or quarters of a dollar; and dividing 2 to 4 fourths by 3 fourths, (the numerator the quotient, 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 denominater of the fraction, and divide the product by the numerator. |