# 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

Let’s see how our Algebraic math solver solves this and similar expressions. Click on "Solve Similar" button to see more examples.

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)

Let’s see how our Algebraic math solver solves this and similar expressions. Click on "Solve Similar" button to see more examples.

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)

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