Multiplying Fractions, Dividing Common Factors: Why Does It Work?Date: 06/10/2010 at 14:10:42 From: Jim Subject: Multiplying Fractions A student's former teacher taught her students that if the numerator of one fraction and the denominator of the other fraction had a common factor, then they could divide each by that factor before multiplying. For example, say we have 4/6 x 8/9 Since the 6 and 8 are multiples of 2, you could divide both by two, creating the following result: 4/3 x 4/9. When multiplied, 4/6 x 8/9 has the same result as 4/3 x 4/9. My question is ... why does this work? What is the underlying math principle? My kids and I would love to know! Thanks for your help. Date: 06/10/2010 at 15:37:00 From: Doctor Ian Subject: Re: Multiplying Fractions Hi Jim, Let's start with a particularly simple example, where one numerator is the same as the other denominator: 2 3 - * - 3 5 The first thing to notice is that when we follow the rule for multiplying fractions, we get 2 3 2 * 3 - * - = ----- 3 5 3 * 5 Now, because multiplication is commutative, we can change the order of the operands, so we could rewrite that as 2 3 2 * 3 3 * 2 - * - = ----- = ----- 3 5 3 * 5 3 * 5 And we can "un-multiply" that to get 2 3 2 * 3 3 * 2 3 2 - * - = ----- = ----- = - * - 3 5 3 * 5 3 * 5 3 5 But 3/3 is the same as 1, and multiplying by 1 is the same as doing nothing: 2 3 2 * 3 3 * 2 3 2 2 2 - * - = ----- = ----- = - * - = 1 * - = - 3 5 3 * 5 3 * 5 3 5 5 5 So here, the numerator and denominator just cancel out. Does this make sense? The important idea here is that when multiplying fractions, we can group factors in the numerator and denominator any way we want. And if we have something that appears in both the numerator and denominator, we can "separate that out" to get n/n, or 1. That's really what we're _doing_ when we cancel. We're separating out n/n pairs, which "turn into 1," and thus disappear. This really comes into its own when we break things into prime factors before multiplying. For example, 4 35 -- * -- = ? 15 72 We can break the numerators and denominators up into prime factors: 4 35 2*2 5*7 -- * -- = --- * --------- 15 72 3*5 2*2*2*3*3 Those all just get multiplied together ... 4 35 2*2*5*7 -- * -- = ------------- 15 72 2*2*2*3*3*3*5 ... and if we align them, we can spot pairs that can be canceled: 4 35 2 * 2 * 5 * 7 -- * -- = ------------------------------ 15 72 2 * 2 * 2 * 3 * 3 * 3 * 5 We could go to the trouble of separating them ... 4 35 2 2 5 7 -- * -- = - * - * - * ------------- 15 72 2 2 5 2 * 3 * 3 * 3 ... but in practice, we don't bother. We would just cross them out, and multiply what's left: x x x 4 35 2 * 2 * 5 * 7 7 7 -- * -- = ------------------------------ = ------- = -- 15 72 2 * 2 * 2 * 3 * 3 * 3 * 5 2*3*3*3 54 x x x This is a lot easier than doing the multiplication, then reducing it by finding common factors. Still making sense? If so, we're almost done! Let's look at your example: 4 8 - * - = ? 6 9 When we note that 6 and 8 have a common factor ... 4 2*4 --- * --- = ? 2*3 9 ... we recognize that we could separate those out, but we just toss the common factor, and keep the rest: 4 4 --- * --- = ? 3 9 It's the same result as writing everything out, and moving everything around -- but a lot quicker. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 06/11/2010 at 08:08:27 From: Jim Subject: Thank you (Multiplying Fractions) After hearing me drone on and on about the importance of the commutative property, my 7th graders really connected with this explanation. They are now making up their own examples trying to "stump" each other. This is also giving them invaluable practice working with factors. Thanks so much! |
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