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Thread: Simplify the complex fraction

  1. March 6th 2009, 06:13 PM #1
    bryang
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    Simplify the complex fraction

    I'm trying to simplify this complex fraction:

    \frac{1-\frac{x}{y}}{1-\frac{x^2}{y^2}}

    but I don't know what to do with the ones in front of the fractions. This is as far as I have gotten. The common denominator is y^2 so I need to multiply them by y^2, but how?

    \frac{(1-\frac{x}{y})\frac{y^2}{1}}{(1-\frac{x^2}{y^2})\frac{y^2}{1}}
    These are mixed numbers right? To multiply a mixed number by a fraction, you need to change the mixed number to an improper fraction, but how do I do it when I don't know how much x and y are worth?
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  2. March 6th 2009, 06:41 PM #2
    Prove It
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    Quote Originally Posted by bryang View Post
    I'm trying to simplify this complex fraction:

    \frac{1-\frac{x}{y}}{1-\frac{x^2}{y^2}}

    but I don't know what to do with the ones in front of the fractions. This is as far as I have gotten. The common denominator is y^2 so I need to multiply them by y^2, but how?

    \frac{(1-\frac{x}{y})\frac{y^2}{1}}{(1-\frac{x^2}{y^2})\frac{y^2}{1}}
    These are mixed numbers right? To multiply a mixed number by a fraction, you need to change the mixed number to an improper fraction, but how do I do it when I don't know how much x and y are worth?
    It'll help you if you turn the mixed number numerators and denominators into improper fractions.

    Note that \frac{y}{y} = 1 and \frac{y^2}{y^2} = 1.

    So \frac{1-\frac{x}{y}}{1-\frac{x^2}{y^2}} = \frac{\frac{y}{y} - \frac{x}{y}}{\frac{y^2}{y^2} - \frac{x^2}{y^2}}

    = \frac{\frac{y - x}{y}}{\frac{y^2 - x^2}{y^2}}

    = \frac{y - x}{y} \div \frac{y^2 - x^2}{y^2}

    = \frac{y - x}{y} \times \frac{y^2}{y^2 - x^2}

    = \frac{y^2(y - x)}{y(y^2 - x^2)}

    = \frac{y(y - x)}{y^2 - x^2}

    = \frac{y(y - x)}{(y - x)(y + x)}

    = \frac{y}{y + x}.
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