Canceling and Reducing When Multiplying Several FractionsDate: 01/25/2008 at 09:56:07 From: Renee Subject: cancellation in multiple fractions I need help when multiplying three or more fractions. I know how to cancel diagonally when the fractions are right next to each other. I know how to cancel the numerator and denominator of a fraction, too. I have seen people cancel using the numerators and denominators that aren't near each other. How do they do that? Can you double cancel a number? How do you keep it all straight then? Thanks a million for your help! Date: 01/25/2008 at 10:26:10 From: Doctor Ian Subject: Re: cancellation in multiple fractions Hi Renee, The key idea is that multiplication is commutative. Suppose you have something like 4 3 25 - * -- * -- 5 10 30 You can break everything into prime factors, 2*2 3 5*5 --- * --- * ----- 5 2*5 2*3*5 Now, to multiply fractions, we just multiply the numerators and denominators separately, right? So this is the same as 2*2 * 3 * 5*5 ------------------- 5 * 2*5 * 2*3*5 So now, we're just reducing a fraction. And we can do that by identifying prime factors that appear in both the numerator and denominator: 2 * 3 * 5*5 ------------------- Cancel a 2 5 * 5 * 2*3*5 3 * 5*5 ------------------- Cancel another 2 5 * 5 * 3*5 5*5 ------------------- Cancel a 3 5 * 5 * 5 1 ------------------- Cancel two 5's 5 So the result of the multiplications is 1/5. Does this make sense so far? Remember that when you "cancel" two identical numbers you are making each number into 1. That's why there is a 1 left in the numerator after all the numbers that were there have canceled. Here's how I'd do it in practice: 4 3 25 - * -- * -- Our original problem. 5 10 30 2 3 25 - * -- * -- I canceled a 2 from the 4 and the 10. 5 5 30 1 3 25 - * -- * -- I canceled a 2 from the 2 and the 30. 5 5 15 1 3 5 - * -- * -- I canceled a 5 from the 5 and the 25. 1 5 15 1 3 1 - * -- * -- I canceled the two 5's. 1 1 15 1 1 1 - * -- * -- I canceled a 3 from the 3 and the 15. 1 1 5 1 1 1 - * -- * -- Nothing left to cancel. 1 1 5 As I said, the key idea is that since the numerators and denominators are going to get multiplied anyway, it doesn't matter whether the common factors are next to each other, or far away. What I'm doing here is realizing that I really have one big fraction that I need to reduce, rather than three fractions that I need to multiply. Does that make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 02/06/2008 at 10:18:52 From: Renee Subject: Thank you (cancellation in multiple fractions) Thanks a lot Dr. Ian! I actually understand the concept for the first time! Realizing that its all really just one BIG fraction comprised of prime factors - and you're actually reducing as you go along rather than just at the end...sure makes sense to me now. Whoaaa, you're good! |
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