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Complex Fractions: More Examples (page 2 of 2)

  • Simplify the following expression:
    • [ 3 + 9/(x - 3) ] / [ 4 + 12/(x - 3) ]

    Can I start by hacking off the x – 3's? Can I cancel the 4 with the 12? Or the 3 with the 9 or the 12? (Hint: No!)

    The common denominator for this complex fraction would be x – 3, so I'll multiply through, top and bottom, by that.



It is highly unusual for a complex fraction to simplify this much, but it can happen. In this case, the "except for x equal to 3" part is rather important, since the original fraction is not always equal to 3/4. Indeed, it is not even defined for x equal to 3 (since this would cause division by zero).

  • Simplify the following expression:
    • [ (y / x) - (x / y) ] / [ (x + y) / xy ]


    Can I start off by canceling like this:


    NO! NO! NO!

    I can only cancel factors, not terms, so the above cancellations are not proper.

    The first thing I need to do is multiply through, top and bottom, by the common denominator of xy.



  • Simplify the following expression:
    • [ (1 / t) - 1 ] / [ (1 / t) + 1 ]

    Can I start by canceling off the 1's or the 1/t's? (Hint: No!)

    I'll multiply through, top and bottom, by the common denominator of t.


    (1/t – 1)/(1/t + 1) = (1 – t)/(1 + t)

    Can I cancel off the t's now? Or cancel off the 1's? (Hint: No!) I can only cancel off factors, not terms, and nothing factors here, so this is as simplified as it gets. The final answer is:

      (1 - t) / (1 + t) for t not equal to zero or -1

    (Why the restrictions? )

When working with complex fractions, be careful to show each step completely. Don't try to skip steps or do everything in your head. And don't get careless with cancellation; remember that you can only cancel factors, not terms. If you remember this, and do your work clearly, you should be fairly successful with these problems.

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Cite this article as:

Stapel, Elizabeth. "Complex Fractions: More Examples." Purplemath. Available from Accessed


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