tiplier 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 3, is of 12; and to find of 12, we must first find 3 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 812, or the product of $12 by 3, 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. 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, 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 com pound 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 farm, sold of his share; what part of the farm did he sell? Ans. 4. What part of a mile is of 3 of a mile ? Ans. 5. Change of 3 of 4 of 3. What part of a foot is | 4 of to a single fraction. of of a foot ? 225. Ans.. 6. Multiply 31 by 77. 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 doiJar, 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 doliars. 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. 227, 228. 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. 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 3 by 4, as 3x4 12 (220) 5 5 is divided by 3 by multiplying the denominator, 5, 12 (121); 12 of a dollar then the is price and 12 5x3 15 of one bushel: Hence, 15 228. To multiply a fraction by a fraction. 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 a new denominator. NOTE.-In practice it will be most convenient to invert the divisar, and then proceed as in Art. 224. 2. In 7 how many times 3. In 22 how many times 5. If g of a yard cost of a dollar, what is that a yard? Ans. $1.773. 6. If of a piece of cloth of of an eagle, 2?? buy for 229. Ans. 1-3 bush. is the whole piece worth? Ans. 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 230. To enlarge the terms of a fraction RULE.-Multiply both the terms of the fraction by the number which denotes how many times the terms are to be enlarged. 231. To diminish the terms of a fraction RULE.-Divide both the terms of the fraction by such a number as will divide each without a remainder. QUESTIONS FOR PRACTICE. 1. What is the express- 1. What is the express. ion for in terms which are 10 times as large ?for the terms being increased 9 times? ion for 18 in terms 10 times less for the terms being diminished 9 times ? 232. OF THE GREATEST COMMON DIVISOR OF TWO NUMBERS. ANALYSIS. 1. If the two terms of a fraction be 8 and 38. what is the greatest number that will divide them both without a remainder? 8) 38 (4 32 6)3(1 2)6(3 6 It is evident that the greatest common divisor of 8 and 38 cannot exceed the smallest of them. We will therefore see if 8, which divides itself and gives 1 for the quotient, will divide 33; if it will, it is manifestly the greatest common divisor sought. But dividing 38 by 8 we obtain a quotient 4 and a remainder 6; hence 8 is not a common divisor. Again, it is evident that the common divisor of 8 and 38 must also divide 6, because 38-4 times 8 plus 6; hence a number which will divide 8 and 6 will also divide 8 and 38; we will therefore see if 6 which divides itself will divide 2. But dividing 8 by 6 we have a quotient 1, and remainder 2; hence 6 is not a common divisor. Again, for the reason above stated, the common divisor of 6 and 8 must also divide the remainder 2; and by dividing 6 by 2, we find that 2, which divides itself, divides 6 also; 2 is therefore a divisor of 6 and 8, and it has been shown that a number which will divide 6 and 8, will also divide 8 and 38. Hence 2 is the common divisor of 8 and 38, and it is evidently the greatest common divisor, since it is manifest from the method of obtaining it that? will divide by it, and a number will not divide by another greater than itself. Therefore, 233. To find the greatest common divisor of two numbers. RULE -Divide the greater number by the less, and the divisor by the remainder, and so on, always dividing the last divisor by the last remainder, till nothing remains; then will the last divisor be the common divisor required. 1. What is the most simple expression, or the least terms of ? The terms of a fraction are diminished, or made more simple, by division, (230). Now if we divide 43 so long as we can find any number greater than 1 which will divide them both without a remainder, the fraction will evidently be diminished to the least terms which are capable of expressing it, since the two terms now contain no common factor greater than unity. Thus 2)=36, 2) 745-18. 2) 1 and 2) = least terms. Or if we find the greatest common divisor of the two terms, 48 and 272, we may evidently reduce the fraction to its lowest terms at once by dividing the two terms by it. By Art. 233 we find the greatest com. mon divisor to be 16, and 16)= least terms as before. Hence, 48 212 24 235. To reduce a Fraction to its least terms. 34 RULE.-Divide both the terms of the fraction by the greatest common divisor, and the quotient will be the fraction in its least terms. QUESTIONS FOR PRACTICE. 2. What are the least 5. Reduce terms of 4? Ans. . 272 3. What are the least terms of 88? Ans. ¿. 4. What are the least terms of 17? Ans.. least terms. to its Ans. |