b 3. Reduce 74 and ax+- to improper fractions. с C 4. Reduce 3 and 2-2 to improper fractions. 3- x2 y b x2+2xy + y2+a x+y 7. Reduce 4+2x+ to an improper fraction. с The converse of this operation must be true, and, therefore, to reduce an improper fraction to a mixed quantity, we have the following RULE.-Divide the numerator by the denominator, as far as possible, and set the remainder, (if any), over the denominator for the fractional part; the two joined together with their proper sign, will be the mixed quantity sought. (ART. 35.) A fraction is an expression for unperformed division. Thus, 2 divided by 5, is written . The double of this is, 3 times is §, &c. That is, to multiply a fraction by any number, we multiply the numerator of the fraction by the number, without changing the denominator. The nature of division is the same, whatever numbers represent the dividend and divisor. Hence, for the sake of simplicity, let us consider the result of dividing 24 by 6. Here 24 is the dividend and 6 the divisor, and the division expressed and unperformed, must be written 24, and the value of this expression, or quotient, is 4. Now observe, that we can double the quotient by doubling 24, or by taking the half of 6. We can find 3 times the value of this quotient, by multiplying the numerator 24 by 3, or by dividing the denominator 6 by 3. Hence, to multiply a fraction by a whole number, we have the following rule: RULE.-Multiply the numerator by the whole number; or, when you can, divide the denominator by the whole number. (ART. 36.) When we multiply a fraction by its denominator, we merely suppress the denominator. Thus, multiply by 3, the result is 1, the numerator of the fraction; multiply by 5, and we have 2, the numerator for the product. 7. Multiply by 20. Ans. 6a-2x. (ART. 37.) As a fraction is an expression for unperformed division, we may express the division of 31⁄2 by 5%, in the following form: But this is certainly a complex fraction; so are 3 and 7 5 81 complex fractions; hence complex fractions may be defined thus: A complex fraction is one in which the numerator or denominator, or both, are fractions or mixed quantities. To simplify a complex fraction, we multiply both numerator and denominator by the denominators of the fractional parts: or by their product, or by their least common multiple. For example, let us simplify the fraction 3. If we mul5/ tiply both numerator and denominator by 2, the numerator will contain no fraction, and the result will be 7 10/1/ Multiply numerator and denominator of this fraction by 3, and the denominator will contain no fraction; and the final result will 21 be a simple fraction, equal in value to the complex fraction. 34 But we could have arrived at this result at once, by multiplying both terms by 6, the product of 2.3. Hence, the rule just given. b 6. Divide by, that is, simplify the complex fraction Here the division is expressed, but unperformed, and by ad the rule to simplify the fraction, we find its value to be bc From this result we can draw a rule for dividing one fraction by another; and the rule here indicated, when expressed in words, is the rule commonly found in Arithmetic. |