How to Simplify Complex Fractions

Step-by-Step Guide

1. 1

   Step 1: If necessary

   Complex fractions aren't necessarily difficult to solve.
   
   In fact, complex fractions in which the numerator and denominator both contain a single fraction are usually fairly easy to solve.
   
   So, if the numerator or denominator of your complex fraction (or both) contain multiple fractions or fractions and whole numbers, simplify as needed to obtain a single fraction in both the numerator and denominator.
   
   This may require finding the least common denominator (LCM) of two or more fractions.
   
   For example, let's say we want to simplify the complex fraction (3/5 + 2/15)/(5/7
   - 3/10).
   
   First, we would simplify both the numerator and denominator of our complex fraction to single fractions.
   
   To simplify the numerator, we will use a LCM of 15 by multiplying 3/5 by 3/3.
   
   Our numerator becomes 9/15 + 2/15, which equals 11/15.
   
   To simplify the denominator, we will use a LCM of 70 by multiplying 5/7 by 10/10 and 3/10 by 7/7.
   
   Our denominator becomes 50/70
   - 21/70, which equals 29/70.
   
   Thus, our new complex fraction is (11/15)/(29/70).
2. 2

   Step 2: simplify the numerator and denominator into single fractions.

   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 obtained a complex fraction with a single fraction in both the numerator and the denominator, we can use this property of division to simplify our complex fraction! First, find the inverse of the fraction on the bottom of the complex fraction.
   
   Do this by "flipping" the fraction
   - setting its numerator in the place of the denominator and vice versa.
   
   In our example, the fraction in the denominator of the complex fraction (11/15)/(29/70) is 29/70.
   
   To find its inverse, we simply "flip" it to get 70/29.
   
   Note that, if your complex fraction has a whole number in its denominator, you can treat it as a fraction and find its inverse all the same.
   
   For instance, if our complex fraction was (11/15)/(29), we can define the denominator as 29/1, which makes its inverse 1/29. , Now that you've obtained the inverse of your complex fraction's denominator, multiply it by the numerator to obtain a single simple fraction! Remember that to multiply two fractions, we simply multiply across
   - the numerator of the new fraction is the product of the numerators of the two old ones, and similarly with the denominator.
   
   In our example, we would multiply 11/15 × 70/29. 70 × 11 = 770 and 15 × 29 =
   435.
   
   So, our new simple fraction is 770/435. , We now have a single, simple fraction, so all that remains is to render it in the simplest terms possible.
   
   Find the greatest common factor (GCF) of the numerator and denominator and divide both by this number to simplify.
   
   One common factor of 770 and 435 is
   5.
   
   So, if we divide the numerator and denominator of our fraction by 5, we obtain 154/87. 154 and 87 don't have any common factors, so we know we've found our final answer!
3. 3

   Step 3: Flip the denominator to find its inverse.

4. 4

   Step 4: Multiply the numerator of the complex fraction by the inverse of the denominator.

5. 5

   Step 5: Simplify the new fraction by finding the greatest common factor.

Detailed Guide

About the Author