![]() ![]() Where the entire numerator is enclosed within parentheses. We must be especially careful with binomial numerators. When subtraction is involved, it is helpful to change to standard form before adding. ![]() The sum of two or more fractions with common denominators is a fraction with the same denominator and a numerator equal to the sum of the numerators of the original fractions. The sum of two or more arithmetic or algebraic fractions is defined as follows: SUMS AND DIFFERENCES OF FRACTIONS WITH LIKE DENOMINATORS When the fractions in a quotient involve algebraic expressions, it is necessary to factor wherever possible and divide out common factors before multiplying. ![]() Some quotients occur so frequently that it is helpful to recognize equivalent forms directly. In symbols,Īs in multiplication, when fractions in a quotient have signs attached, it is advisable to proceed with the problem as if all the factors were positive and then attach the appropriate sign to the solution. That is, to divide one fraction by another, we invert the divisor and multiply. The quotient of two fractions equals the product of the dividend and the reciprocal of the divisor. ![]() That is, we obtain the reciprocal of a fraction by "inverting" the fraction. In general, the reciprocal of a fraction is the fraction. In the above example, we call the number the reciprocal of the number. To solve this equation for q, we multiply each member of the equation by. To find, we look for a number q such that. This is precisely the same notion as that of dividing one integer by another a ÷ b is a number q, the quotient, such that bq = a. In dividing one fraction by another, we look for a number that, when multiplied by the divisor, yields the dividend. Similarly,īecause 3 is not a factor of the entire numerator 3y + 2. Use whichever form is most convenient for a particular problem.Ĭommon Errors: Remember that we can only divide out common factors, never common terms! For example,īecause x is a term and cannot be divided out. In algebra, we often rewrite an expression such as as an equivalent expression. Very often, fractions are more useful in this form. Note that when writing fractional answers, we will multiply out the numerator and leave the denominator in factored form. We now multiply the remaining factors of the numerators and denominators to obtain Solution First, we must factor the numerators and denominators to get When the fractions contain algebraic expressions, it is necessary to factor wherever possible and divide out common factors before multiplying. A positive sign is attached if there are no negative signs or an even number of negative signs on the factors a negative sign is attached if there is an odd number of negative signs on the factors. If a negative sign is attached to any of the factors, it is advisable to proceed as if all the factors were positive and then attach the appropriate sign to the result. Now, multiplying the remaining factors of the numerators and denominators yields Solution First, we divide the numerator and denominator by the common factors to get The same procedures apply to fractions containing variables. The product of two fractions is a fraction whose numerator is the product of the numerators and whose denominator is the product of the denominators of the given fractions.Īny common factor occurring in both a numerator and a denominator of either fraction can be divided out either before or after multiplying. The product of two fractions is defined as follows. OPERATIONS WITH FRACTIONS PRODUCTS OF FRACTIONS ![]()
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