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).
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.
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.
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.
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!
For example, (1/x)/(x/6) is easy to simplify with inverse multiplication. 1/x × 6/x = 6/x2. Here, there is no need to use an alternate method. However, (((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, inverse multiplying, and reducing the result to simplest terms is likely to be a complicated process. In this case, the alternate method below may be easier.
This is easier to understand with an example. Let’s try to simplify the complex fraction we mentioned 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).
In our example, we would multiply our complex fraction, (((1)/(x+3)) + x - 10)/(x +4 +((1)/(x - 5))), by ((x+3)(x-5))/((x+3)(x-5)). We’ll have to multiply through the numerator and denominator of the complex fraction, multiplying each term by (x+3)(x-5). First, let’s 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
The denominator of our complex fraction, (((1)/(x+3)) + x - 10)/(x +4 +((1)/(x - 5))), is x +4 +((1)/(x-5)). We’ll multiply this 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
Using the numerator and denominator we found above, we can construct a fraction that’s equal to our initial complex fraction but which contains no fractional terms. The numerator we obtained was x3 - 12x2 + 6x + 145 and the denominator was x3 + 2x2 - 22x - 57, so our new fraction is (x3 - 12x2 + 6x + 145)/(x3 + 2x2 - 22x - 57)
