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How do you divide fractions? Say, 2/3 divided by 3/4.

Wouldn't it be nice if dividing fractions were as easy as dividing whole numbers? Well, dividing fractions can be as easy as multiplying fractions, if we use this shortcut:

    ...invert and multiply.

Say you want to divide 2/3 by 3/4, like this:



That looks complicated. Can we start by getting rid of the denominator fraction, so we only have to deal with the numerator fraction? There's a way to do just that: if we multiply a fraction by its reciprocal, the result equals one.

In this case, the reciprocal of 3/4 is 4/3. If we multiply both the numerator and denominator by 4/3, we get:

     2/3       4/3
    _____  *  ____
     3/4       4/3

Why did we multiply by a fraction with 4/3 in both the numerator and denominator? Because this fraction equals one, and when we multiply by one, we don't change the values in the problem.

Why should you be careful to multiply by a fraction equal to one?

Are you ready? Let's multiply.

     2/3     4/3      8/9      8/9
    _____ * _____ = _______ = _____ = 8/9

     3/4     4/3     12/12      1

Where does "invert and multiply" come from? Well, since the denominator becomes 1 using our method, you wind up with just the numerator multiplied by the reciprocal of the denominator. If you understand this, you can make up a shortcut: invert the denominator fraction and multiply it by the numerator fraction.

     2/3    2   4   8
    _____ = _ * _ = _
     3/4    3   3   9
Danger: it's important to invert the denominator fraction (what you're dividing by), because otherwise your answer will come out upside-down!

It's not that you can't divide fractions using division, it's just a whole lot easier using multiplication.

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