How to Simplify Complex Fractions: 9 Steps

2024 Author: Samantha Chapman | [email protected]. Last modified: 2024-01-15 13:47

Complex fractions are fractions where the numerator, denominator, or both contain fractions themselves. For this reason, complex fractions are sometimes called "stacked fractions". Simplifying complex fractions is a process that can range from easy to difficult based on how many terms are present in the numerator and denominator, if any of them are variable, and, if so, the complexity of the terms with variable. See step 1 to get started!

Steps

Method 1 of 2: Simplify Complex Fractions with Inverse Multiplication

Step 1. If necessary, simplify the numerator and denominator into single fractions

Complex fractions are not necessarily difficult to solve. In fact, complex fractions in which both the numerator and denominator contain a single fraction are often very easy to solve. So, if the numerator or denominator of your complex fraction (or both) contains multiple fractions or fractions and whole numbers, simplify so that you get a single fraction in both the numerator and denominator. This step requires the calculation of the Minimum Common Denominator (LCD) of two or more fractions.

• For example, suppose we want to simplify the complex fraction (3/5 + 2/15) / (5/7 - 3/10). First, we will simplify both the numerator and denominator of our complex fraction into single fractions.

  • To simplify the numerator, we will use the LCD equal to 15 by multiplying 3/5 by 3/3. Our numerator will become 9/15 + 2/15, which is equal to 11/15.
  • To simplify the denominator, we will use the LCD equal to 70 by multiplying 5/7 by 10/10 and 3/10 by 7/7. Our denominator will become 50/70 - 21/70, which is equal to 29/70.
  • Thus, our new complex fraction will be (11/15)/(29/70).

  

  Step 2. Flip the denominator to find its inverse

  By definition, dividing one number by another is the same as multiplying the first number by the inverse of the second. Now that we have a complex fraction with a single fraction in both the numerator and denominator, we can use this division property to simplify our complex fraction! First, find the inverse of the fraction in the denominator of the complex fraction. Do this by reversing the fraction - putting the numerator in place of the denominator and vice versa.

  • In our example, the fraction in the denominator of our complex fraction (11/15) / (29/70) is 29/70. To find the inverse, we simply reverse it by obtaining 70/29.

    Note that if your complex fraction has an integer as a denominator, you can treat it as if it were a fraction and invert it the same way. For example, if our complex function were (11/15) / (29), we could define its denominator as 29/1, and thus its inverse would be 1/29.

    

    Step 3. Multiply the numerator of the complex fraction by the inverse of the denominator

    Now that you've got the inverse of your fraction in the denominator, multiply it by the numerator to get a single simple fraction! Remember that to multiply two fractions, you simply multiply the whole - the numerator of the new fraction will be the product of the numerators of the two old ones, the same for the denominator.

    In our example we will multiply 11/15 × 70/29. 70 × 11 = 770 and 15 × 29 = 435. Thus, our new simple fraction will be 770/435.

    

    Step 4. Simplify the new fraction by finding the greatest common divisor (M. C. D

    ). We now have a single simple fraction, so all that remains is to simplify it as much as possible. Find the M. C. D. of the numerator and denominator and divide both by this number to simplify them.

    A common factor of 770 and 435 is 5. So if we divide the numerator and denominator of our fraction by 5, we get 154/87. 154 and 87 no longer have common factors, so we know we have found our solution!

    Method 2 of 2: Simplify Complex Fractions Containing Variables

    

    Step 1. Whenever possible, use the inverse multiplication method of the previous method

    To be clear, potentially all complex fractions can be simplified by reducing the numerator and denominator to simple fractions and multiplying the numerator by the inverse of the denominator. Complex fractions that contain variables are not an exception, but the more complicated the expression containing the variable, the more complicated and time-consuming it is to use the inverse multiplication method. For "simple" complex fractions containing variables, inverse multiplication is a good choice, but for fractions with many terms containing variables, both in the numerator and denominator, it may be easier to simplify with the method described below.

    • For example, (1 / x) / (x / 6) is easy to simplify with the use of inverse multiplication. 1 / x × 6 / x = 6 / x². Here, there is no need to use an alternative method.
    • While, (((1) / (x + 3)) + x - 10) / (x +4 + ((1) / (x - 5))) is more difficult to simplify with inverse multiplication. Reducing the numerator and denominator of this complex fraction to single fractions, and reducing the result to a minimum is probably a complicated process. In this case the alternative method shown below should be simpler.

    

    Step 2. If inverse multiplication is impractical, start by finding the lowest common denominator between the fractional terms of the complex function

    The first step in this alternative simplification method is to find the LCD of all the fractional terms present in the complex fraction - in both its numerator and denominator. Usually, one or more of the fractional terms have variables in their denominator, the LCD is simply the product of their denominators.

    This is easier to understand with an example. Let's try to simplify the complex fraction named above, (((1) / (x + 3)) + x - 10) / (x +4 + ((1) / (x - 5))). The fractional terms in this complex fraction are (1) / (x + 3) and (1) / (x-5). The common denominator of these two fractions is the product of their denominators: (x + 3) (x-5).

    

    Step 3. Multiply the numerator of the complex fraction by the LCD you just found

    Then we will have to multiply the terms of the complex fraction by the LCD of its fractional terms. In other words, we will multiply the complex fraction by (LCD) / (LCD). We can do this since (LCD) / (LCD) = 1. First, multiply the numerator by itself.

    • In our example, we will multiply our complex fraction, (((1) / (x + 3)) + x - 10) / (x +4 + ((1) / (x - 5))), by ((x +3) (x-5)) / ((x + 3) (x-5)). We should multiply it by both the numerator and the denominator of the complex fraction, multiplying each term by (x + 3) (x-5).

      • First, we multiply the numerator: (((1) / (x + 3)) + x - 10) × (x + 3) (x-5)

        • = (((x + 3) (x-5) / (x + 3)) + x ((x + 3) (x-5)) - 10 ((x + 3) (x-5))
        • = (x-5) + (x (x² - 2x - 15)) - (10 (x² - 2x - 15))
        • = (x-5) + (x³ - 2x² - 15x) - (10x² - 20x - 150)
        • = (x-5) + x³ - 12x² + 5x + 150
        • = x³ - 12x² + 6x + 145

      

      Step 4. Multiply the denominator of the complex fraction by the LCD as you did with the numerator

      Continue multiplying the complex fraction by the LCD you found, proceeding with the denominator. Multiply each term by the LCD:

      • The denominator of our complex fraction, (((1) / (x + 3)) + x - 10) / (x +4 + ((1) / (x - 5))), is x +4 + ((1) / (x-5)). We will multiply it by the LCD we found, (x + 3) (x-5).

        • (x +4 + ((1) / (x - 5))) × (x + 3) (x-5)
        • = x ((x + 3) (x-5)) + 4 ((x + 3) (x-5)) + (1 / (x-5)) (x + 3) (x-5).
        • = x (x² - 2x - 15) + 4 (x² - 2x - 15) + ((x + 3) (x-5)) / (x-5)
        • = x³ - 2x² - 15x + 4x² - 8x - 60 + (x + 3)
        • = x³ + 2x² - 23x - 60 + (x + 3)
        • = x³ + 2x² - 22x - 57

        

        Step 5. Form a new simplified fraction from the numerator and denominator you just found

        After multiplying your fraction by your (LCD) / (LCD) and simplifying similar terms, you should be left with a simple fraction with no fractional terms. As you may have understood, by multiplying the fractional terms in the original complex fraction by the LCD, the denominators of these fractions cancel out, leaving terms with variables and integers in both the numerator and denominator of your solution, but no fraction.

        Using the numerator and denominator found above, we can construct a fraction which is equivalent to the starting one, but which does not contain fractional terms. The numerator we obtained was x³ - 12x² + 6x + 145 and the denominator was x³ + 2x² - 22x - 57, so our new fraction will be (x³ - 12x² + 6x + 145) / (x³ + 2x² - 22x - 57)

        Advice

        • Write down each step you take. Fractions can be easily confusing if you try to solve them too quickly or in your head.
        • Find examples of complex fractions online or in your textbook. Follow each step until you can solve them.