Complex fractions - addition, subtraction, multiplication and division

Combined Operations and Complex Fractions

  In the previous sections we discussed the addition and subtraction of fractions as well as their multiplication and division. In all cases the final answer was cine fraction in its simplest form. Here we shall confront the four operations in one problem and still require the final answer to be one fraction in its simplest form.

  When there are no symbols of grouping in the problem, multiplications and divisions are performed first in the order in which they appear in the problem. Only after all the multiplications and divisions have been dune are the additions and subtractions performed.

EXAMPLE  Perform the indicated operations and simplify:

        5/(2x+1)-(2x+6)/(x^2-4x+3) ÷ (2x^2+5x-3)/(2x^2-3x+1)

Solution   

       When there are symbols of grouping, as in the problem

       (x-(4x)/(x+2))(3+12/(x-2)

       we have the choice of performing the multiplication first or of combining the terms within parentheses. Combining the terms within parentheses is easier, as illustrated it the following examples.

EXAMPLE  (x-(4x)/(x+2))(3+(12)/(x-2))

Solution    (x-(4x)/(x+2))(3+(12)/(x-2))== (x(x+2)-4x)/(x+2)*(3(x-2)+12)/(x-2)== (x^2+2x-4x)/(x+2)*(3x-6+12)/(x-2)== (x^2-2x)/((x+2))*(3x+6)/((x-2))== (x(x-2))/((x+2))*(3(x+2))/((x-2))== 3x

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EXAMPLE   Perform the indicated operations and simplify:

      (x-3/(2x-3)) ÷ (x+9/(2x+9)

SOLUTION  (x-3/(2x-3)) ÷ ( x+9/(2x+9) )== (x(2x-3)-9)/((2x-3)) ÷ (x(2x+9)+9)/(2x+9) == (2x^2-3x-9)/((2x-3)) ÷ (2x^2+9x+9)/((2x+9))== ((2x+3)(x-3))/(2x-3) * (2x+9)/((2x+3)(x+3))== ((x-3)(2x+9))/((2x-3)(x+3))

Since    (a+b) ÷ (c+d) can be written as (a+b)/(c+d), we can write

      (3-11/x+6/x^2) ÷ (3+4/x-4/x^2) in the form

      (3-11/x+6/x^2)/(3+4/x-4/x^2) which is a complex fraction.

  Given a complex fraction, we can either simplify the problem as it is, in fraclion form, or write it in division form and simplify. Sometimes a complex fraction can hc easily Simplified by multiplying both numerator and denominator by the least common multiple of all the denominators involved.

EXAMPLE  Simplify (3-11/x+6/x^2)/(3+4/x-4/x^2)

Solution    The LCM of the denominators id x^2.

        (3-11/x+6/x^2)/(3+4/x-4/x^2)== (x^2/1(3/1-11/x+6/x^2))/(x^2/1(3/1+4/x-4/x^2))== (3x^2-11x+6)/(3x^2-4x-4)== ((3x-2)(x-3))/((3x-2)(x+2))== (x-3)/(x+2)

EXAMPLE  Simplify (x-2)/(x+2-4/(x-1)

Solution    The LCM of the denominators id (x-1).

        (x-2)/(x+2-4/(x-1))== ((x-1)(x-2))/((x-1)/1((x+2)/1-4/(x-1)))== ((x-1)(x-2))/((x-1)(x+2)-4)== ((x-1)(x-2))/(x^2+x-2-4)== ((x-1)(x-2))/(x^2+x-6)== ((x-1)(x-2))/((x+3)(x-2))== (x-1)/(x+3)

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EXAMPLE  Simplify (x+3+6/(x-4))/(x+5+18/(x-4))

Solution    The LCM of the denominators id (x-4).

        (x+3+6/(x-4))/(x+5+18/(x-4))== ((x-4)/1((x+3)/1+6/(x-4)))/((x-4)/1((x+5)/1+18/(x-4)))== ((x-4)(x+3)+6)/((x-4)(x+5)+18)== (x^2-x-12+6)/(x^2+x-20+18)== (x^2-x-6)/(x^2+x-2)== ((x-3)(x+2))/((x+2)(x-1))== (x-3)/(x-1)