Multiplying and Dividing FractionsDate: 01/23/2001 at 13:05:33 From: Adom and Jamie Subject: Multiplying and dividing fractions We were just trying to figure out how to understand division of fractions and multiplication of fractions. It is weird because when I divided 1/2 by 1/2 on the calculator, I got 1, but when I multiplied them, I got 1/4.... I want to be able to explain in a drawing. HELP! Adom and Jamie Date: 01/25/2001 at 21:12:05 From: Doctor Ian Subject: Re: Multiplying and dividing fractions Hi guys, Let's start from the beginning. When you multiply by a whole number, you replicate something some number of times: * * * x 4 = * * * * * * * * * * * * And when you divide by a whole number, you cut something into some number of pieces, and throw away all but one of them: * * * / 4 = * * * * * * * * * * * * When you multiply by a fraction, you do BOTH of these things. For example, to multiply by 3/4, you divide by 4 and then multiply by 3: * * * * * x (3/4) = * * * * * x 3 = * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * or you multiply by 3 and then divide by 4: * * * * * x (3/4) = * * * * * / 4 = * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Either way you end up with the same result. So there are no new ideas here, just a couple of old ideas bunched together. If the fraction is less than 1, you lose more in the division than you gain in the multiplication. (For example, when you multiply a dollar by 3/4, you break the dollar into 4 quarters, and triple one of them, leaving you with 3 quarters.) If the fraction is greater than 1, you gain more in the multiplication than you lose in the division. (For example, when you multiply a dollar by 5/4, you break the dollar into 4 quarters, and quintuple one of them, leaving you with 5 quarters.) If the fraction is equal to 1, the multiplication and the division cancel each other out, and you end up where you started. If it helps, you can think of 1/4 * 1/4 as being the same thing as 1 * 1/4 * 1/4 So, you start with a whole something; cut it into four pieces, and keep one; then cut _that_ into four pieces, and keep one. Think about doing that with a pizza. How many pieces of the final size would you need to make a whole pizza? You'd need 16 of them, right? Conveniently, 1/4 * 1/4 is 1/16. How about 2/3 * 3/2? Again, you start with a whole something; cut it into three pieces, and double one of them. Then you take those two pieces, cut each one in half, and triple each one of them. What do you end up with? Six pieces, each of which is 1/6th of the original thing. That is, you end up where you started. You can actually do this with paper and scissors, and that's not a bad idea if it isn't yet clear how this works. If you don't absolutely understand what it means to multiply something by a fraction, you are going to have great difficulty in every math class that you take from now on. Okay, so what about dividing by a fraction? Well, there is no good way that I know of to illustrate that with the kinds of pictures that you can use for multiplication. But perhaps that's not such a big deal, because division is primarily just another way of looking at multiplication. That is, once we know something like 24 = 6 * 4 this is really exactly the same piece of information as 24 / 6 = 4 and 24 / 4 = 6 isn't it? In fact, we DEFINE division this way. We say that a / b = c WHENEVER c * b = a That's the definition of division. That's what division MEANS. So how does this apply to fractions? Well, now that you know how to multiply fractions, you understand why 9 * 2/3 = 6 right? We divide 9 by 3 to get 3, and multiply that by 2 to get 6; or we multiply 9 by 2 to get 18, and divide by 3 to get 6. Either way, this isn't a surprising fact. Well, because of how we've DEFINED division, if we say 9 * 2/3 = 6 this is really exactly the same piece of information as 6 / 9 = 2/3 and 6 / (2/3) = 9 Make sure you understand why this is true. If it helps, look at these patterns again, and match up the letters with the numbers: b * c = a <-------> a / b = c, and a /c = b So, what do we have to do to get 6 from 9? Well, we could cut it in half, and then triple what we get; or we could triple it, and take half of what we get. In other words, we get from 6 to 9 in this way: 6 * (3/2) = 9 But we also know that 6 / (2/3) = 9 And when two things are equal to the same thing, they must be equal to each other, right? That means 6 * (3/2) = 6 / (2/3) So when you want to divide by a fraction, you invert the fraction and multiply instead. This is really all that's going on. Note that I've used some particular numbers: 6, 9, 2/3, etc. - but you can work it out using only letters, and you'll see that so long as we accept the DEFINITION of division, we have to invert and multiply in order to divide by a fraction. If we did anything else, we'd get crazy results. In a sense, this goofy rule for dividing by fractions is the price we pay to keep the rest of math running smoothly. I hope this helps. Write back if there is any part of this that wasn't crystal clear, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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