Cancelling FractionsDate: 06/03/2003 at 10:33:34 From: Chris Breeze Subject: Cancelling Dear Dr.Math, I am reading a book on algebra but there is something I really fail to understand concerning cancelling and fractions. The author states that cancelling is division too and then shows the following without any explanation of how it's done: 1 x x 1 2 2 2 x x x 3 * 3 * 3 * 3 * 4 * 4 * 4 2 2 2 2 2 x x x x x 1 1 1 1 1 Reading through your archives I more or less understand cancelling, but I still don't understand this example. I would be very grateful if, unlike the author of the textbook, you could explain what is going on here. Date: 06/03/2003 at 12:31:44 From: Doctor Peterson Subject: Re: Cancelling Hi, Chris. Did you leave out the fraction bar? I think the problem is meant to be a simplification of the fraction 3 * 3 * 3 * 3 * 4 * 4 * 4 ------------------------- 2 * 2 * 2 * 2 * 2 The work (corrected slightly from what you wrote) is: 1 1 x x 2 2 2 x x x 3 * 3 * 3 * 3 * 4 * 4 * 4 ------------------------- 2 * 2 * 2 * 2 * 2 x x x x x 1 1 1 1 1 The order doesn't matter; each number is crossed off and replaced with the quotient. There should be no remainders involved; everything has to divide exactly. For example, when we divide 4 by 2, the quotient is 2, and that is what we write. Let's take it one step (well, a few steps, to save writing) at a time: 3 * 3 * 3 * 3 * 4 * 4 * 4 ------------------------- 2 * 2 * 2 * 2 * 2 First we see that each of the 4's can be divided evenly by 2; so we (three times) divide a 4 and a 2 by 2, replacing them by 2 and 1 respectively: 2 2 2 x x x 3 * 3 * 3 * 3 * 4 * 4 * 4 ------------------------- 2 * 2 * 2 * 2 * 2 x x x 1 1 1 Now we still see some 2's on top and some 2's on the bottom, so we divide (twice) a 2 on the top and a 2 on the bottom by 2, leaving 1 in each case: 1 1 x x 2 2 2 x x x 3 * 3 * 3 * 3 * 4 * 4 * 4 ------------------------- 2 * 2 * 2 * 2 * 2 x x x x x 1 1 1 1 1 What's left is really just 3 * 3 * 3 * 3 * 1 * 1 * 2 3 * 3 * 3 * 3 * 2 ------------------------- = ----------------- = 162 1 * 1 * 1 * 1 * 1 1 In real life, where the "x"s would be slashes across the numbers themselves (which I can't type), the first step would look more like 2 2 2 3 * 3 * 3 * 3 * / * / * / ------------------------- / * / * / * 2 * 2 1 1 1 and the second step like 1 1 / / 2 3 * 3 * 3 * 3 * / * / * / ------------------------- / * / * / * / * / 1 1 1 1 1 (Each slash would have a crossed-out number under it.) Does that help? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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