Multiplying Mixed NumbersDate: 05/28/98 at 16:56:50 From: Devon Dunphy Subject: mult. fractions Could you please explain how to multiply fractions? For example: 2 4/7 x 5 1/6 Thank you, Devon Dunphy Date: 06/09/98 at 21:58:08 From: Doctor Peterson Subject: Re: mult. fractions Hi, Devon. Fractions can be a lot of fun once you've made friends with them, but they can be a little scary when you first meet them. These two look pretty ugly, so we'd better straighten them up before we try to work with them. In order to give you a chance to practice with your own problem, I won't do exactly your problem, but one like it: 2 3/7 x 3 2/3 To make them look neater, we first change the mixed numbers into improper fractions. You probably remember how to do that. You want to turn 2 3/7 into some number of sevenths, so you remember that 2 is just 14 sevenths, and add them: 3 14 3 17 2 --- = --- + --- = --- 7 7 7 7 The same way, 2 9 2 11 3 --- = --- + --- = --- 3 3 3 3 Note: When you're used to it, you can try my way, which is to start at the denominator, multiply by the number clockwise from it (the whole number) and add the next number clockwise (the numerator). This way, we get: 3 x 3 + 2 = 11 by following the arrow: +--------> | + 2 -> 11 | 3 --- -- | x 3 -> 3 +----- But that's just a trick to avoid extra writing. Okay, so regardless of how you change the mixed numbers into improper fractions, our problem is really: 20 11 --- x --- 7 3 Now how do you multiply fractions? The "rule" says you multiply the numerators to get the numerator of the product, and you multiply the denominators to get the denominator of the product: a c a x c - x - = ----- b d b x d So in this case, we multiply 20 x 11 and 7 x 3: 20 11 20 x 11 220 --- x --- = ------- = --- 7 3 7 x 3 21 Often, you'll find something to cancel out to put the result in lowest terms; I recommend doing that before you actually multiply anything, to save work. In this case, we don't find anything to cancel, so we'll try the cancelling trick in the next example. Now I never like to do something just because of a rule, so I'll let you in on the secret behind that rule. Let's take a simpler example, 2 3 - x - 3 5 I can draw a picture of 2/3 by dividing a rectangle into three (the denominator) slices and coloring in two (the numerator) of them: +--------------+ |//////////////| |//////////////| +--------------+ |//////////////| |//////////////| +--------------+ | | | | +--------------+ Now I want 3/5 of what I colored in. To do that, I can cut the whole thing into five slices and take three of the pieces that are colored in: +--+--+--+--+--+ |XX|XX|XX|//|//| |XX|XX|XX|//|//| +--+--+--+--+--+ |XX|XX|XX|//|//| |XX|XX|XX|//|//| +--+--+--+--+--+ | | | | | | | | | | | | +--+--+--+--+--+ You can see I've cut the original rectangle into 3 x 5 = 15 pieces (that's the denominator of the result, which is the product of the denominators), and I've picked 2 x 3 = 6 of them (that's the numerator, which is the product of the two numerators). So 3/5 of 2/3 is 6/15. I can rearrange the six pieces to simplify the fraction and show that it's really 2/5: +--+--+--+--+--+ |XX|XX| | | | |XX|XX| | | | +--+--+--+--+--+ |XX|XX| | | | |XX|XX| | | | +--+--+--+--+--+ |XX|XX| | | | |XX|XX| | | | +--+--+--+--+--+ So when you follow the rule, you're really just cutting and choosing, then cutting and choosing again. To check our example, we can use the rule. In this case, we multiply 2 x 3 for the numerator, and 3 x 5 for the denominator: 2 3 2 x 3 6 - x - = ----- = -- 3 5 3 x 5 15 Now we can reduce the fraction to lowest terms, dividing the numerator and denominator by 3 to get 2/5. That's not really part of multiplying, but it makes the result easier to work with. Something that often saves work, though, is to simplify your fraction before you actually do the multiplication. That's because simplifying means looking for factors you can cancel, and the factors are easier to see before you multiply them. In this example, you can see the two 3's sitting there, and cancel them without ever having to multiply at all: / 2 3 2 x 3 2 - x - = ----- = - 3 5 3 x 5 5 / See if that helps you work out your problem, and let us know if you need more. -Doctor Peterson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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