Canceling Instead of Multiplying and DividingDate: 09/25/2003 at 23:54:31 From: Chris Subject: multiplying fractions Find the value of the product 3/5 x 5/7 x 7/9 x 9/11 x 11/13 x 13/15 = ? Date: 09/26/2003 at 11:27:41 From: Doctor Ian Subject: Re: multiplying fractions Hi Chris, The first thing is to make sure you know how to multiply two fractions, e.g., 3/5 * 5/7 If you're not sure how to do that, see Multiplying and Dividing Fractions http://mathforum.org/library/drmath/view/58080.html Next, you would multiply the first two, then multiply the product by the third, and multiply _that_ product by the fourth, and so on, just as you would with integers: 2 * 3 * 4 * 5 * 6 ...... . .............................. = 6 6 * 4 * 5 * 6 ..... . ........................... = 24 24 * 5 * 6 ...... . ....................... = 120 120 * 6 720 However, having said all that, in this particular case, there is an interesting pattern that becomes more apparent when you write the fractions using horizontal bars: 3 5 7 9 11 13 - * - * - * -- * -- * -- 5 7 9 11 13 15 Let's look at the first pair: 3 5 - * - 5 7 To multiply two fractions, you multiply the numerators, and multiply the denominators: 3 5 3 * 5 - * - = ----- 5 7 5 * 7 But we get the same result if we switch denominators: 3 5 3 * 5 - * - = ----- 7 5 5 * 7 Does that make sense? But now one of the terms is just 1! So 3 5 3 5 3 - * - = - * - = - 5 7 7 5 7 The quick way to do this is to notice that in 3 5 - * - 5 7 we're going to be multiplying by 5 (in the numerator), and dividing by 5 (in the denominator). So we can just save ourselves some time and trouble by forgetting about the two operations, since they just cancel each other out. So when we see something like 3 5 - * - 5 7 we can just 'cancel' the 5's to get x 3 5 3 - * - = - 5 7 7 x Can you see how this will allow you to find the product of all those fractions without actually doing any multiplication? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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