Multiplying Fractions by CancellationDate: 10/09/98 at 22:28:25 From: Patrice Subject: Multiplying fractions In class, we're working on multiplying fractions, where you cross out numbers before you multiply. Whenever I look in books it tells me to just multiply straight, but that's not how I need to know it. Please explain the other way. Date: 10/10/98 at 15:40:21 From: Doctor Peterson Subject: Re: Multiplying fractions Hi, Patrice. I think what you are referring to is what we call "canceling." It's not an essential part of multiplying fractions, but an extra thing you can do to make the work easier. That's why you don't find it when you look up multiplication of fractions. Here's an example: Let's multiply 2/5 by 15/16. We can just multiply straight: 2 15 2 * 15 30 --- * -- = ------ = -- 5 16 5 * 16 80 But now we'd like to get it in simplest terms, so we have to find the greatest common divisor of 30 and 80. One way to do this is to factor each number, and then divide both numerator and denominator by each factor they share: / / 30 2 * 3 * 5 3 3 -- = ----------------- = --------- = --- 80 2 * 2 * 2 * 2 * 5 2 * 2 * 2 8 / / Notice that we have multiplied some numbers, then factored them back out and divided. Why bother making big numbers when you just have to cut them back down to size again? By canceling before you multiply, you can save the work: 3 \ // 2 15 2 * 15 3 --- * -- = ------ = --- 5 16 5 * 16 8 / \\ 8 What I did here was to see that the numerator and denominator both have a factor of 2, so I divided them by 2 (crossing out the 2 and replacing the 16 with 8). Then I saw that both also have a factor of 5, so I divided by 5 (crossing out the 5 and replacing the 15 with 3). That left me with only the 3 on top and the 8 on the bottom, so I never had to multiply anything. So canceling is just a shortcut. We can simplify before we multiply, rather than afterward, to save work. You may not think your teacher is saving you work by teaching this, but that's what's happening! For more information, try looking at: http://mathforum.org/dr.math/problems/lask1.4.98.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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