Multiplying FractionsDate: 04/11/99 at 20:53:19 From: Derek Reed Subject: Fractions I have trouble multiplying fractions. Could you help me? Date: 04/12/99 at 12:47:39 From: Doctor Peterson Subject: Re: Fractions Hi, Derek. Rather than just remind you that to multiply two fractions you multiply the numerators and the denominators, let's think it through. Suppose I have half a cake left over from my birthday, and want to take 2/3 of it. First I'll cut the half into thirds. If I had cut the whole cake into pieces this size, there would have been twice as many pieces (6), so the pieces are each one sixth of the whole cake. Now I want to take two of these pieces; that make 2/6 of a cake, or 1/3. So 2/3 of 1/2 is 1/3. You can see it even more easily if you reverse it: 2/3 of 1/2 is the same as 1/2 of 2/3. Half of two thirds is one third. Here's a picture of how it works. I first cut the cake in two from top to bottom, and take one half: +--+--+ |//| | |//| | |//| | |//| | |//| | +--+--+ Now I'll cut that half into three pieces; while I'm at it, I'll cut the part I didn't take at the same time, showing that these pieces are really sixths: +--+--+ |//| | +--+--+ |//| | +--+--+ |//| | +--+--+ Now I'll take two of the three pieces: +--+--+ |XX| | +--+--+ |XX| | +--+--+ |//| | +--+--+ The two pieces I've taken are the product of the numerators of the fractions 2/3 and 1/2: 2 * 1 = 2. 1 +--+ |XX| 2 +--+ |XX| +--+ The six pieces the whole cake was cut into are the product of the denominators of the fractions: 3 * 2 = 6. 2 +--+--+ | | | +--+--+ 3 | | | +--+--+ | | | +--+--+ So the pieces I've taken are 2/6 of the cake: 2 1 2 * 1 2 --- * --- = ----- = --- 3 2 3 * 2 6 There's the answer yet again! Now there's one more thing to add: often you will have to simplify the answer (as in this case, which turns it into 1/3), and there's a way to make that easier. What we do is to simplify before we actually multiply. Just notice that when I wrote out the multiplied fraction above, 2 * 1 ----- 3 * 2 both numerator and denominator contain a 2. I can rearrange this as 2 * 1 2 1 1 1 ----- = --- * --- = 1 * --- = --- 2 * 3 2 3 3 3 You can do this very easily by just crossing out the 2's, which is called cancelling: / 2 * 1 1 ----- = --- 3 * 2 3 / In more complicated problems, this can save a lot of work, since otherwise you would have to multiply, then factor or divide, undoing part of your work. I hope this helps a little. If I haven't helped with whatever is the hardest part for you, please write back and give me an example of a problem you find hard and how you tried to do it; that will help me see where you need the most help. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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