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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



  • 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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