How Can Division Result in an Increase?Date: 10/29/2003 at 17:03:55 From: Jen Subject: division What is the logic behind dividing an integer by a decimal and getting a larger number? For example, 100 / 0.9185 = 108.87 I just can't understand the logic! It seems like the answer should be a smaller number. Date: 10/29/2003 at 23:30:12 From: Doctor Ian Subject: Re: division Hi Jen, Well, remember that a decimal is just a fraction. For example, 0.23 = 23/100 0.1542 = 1542/10,000 and so on. So what happens when we divide by a fraction? We multiply by the reciprocal: 8 3 ----- = 8 * - 2/3 2 So if we divide by something where the denominator is smaller than the numerator, we'll end up multiplying by something where the numerator is larger than the denominator. Does that make sense? Here's another way to think about it, using the idea that division means breaking things into pieces. Suppose I have something 6 inches long, and I divide it into pieces that are 2 inches long. How many pieces do I get? 1 +----+----+ +----+----+ +----+----+ | | | | | | | | | +----+----+ +----+----+ +----+----+ 6 / 2 = 3 Okay, now what if I divide it into pieces that are only 1/2 inch long? I'm going to end up with 12 pieces, right? 1 - 2 +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ | | | | | | | | | | | | | | | | | | | | | | | | +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+ 6 / (1/2) = 12 So we divided by something less than one, and ended up with more than our original result. But the clearest way to see the logic of this is to remember what we _mean_ by division. That is, a division is just another way of representing a multiplication. For example, when we say that 3 * 4 = 12 two other ways to say exactly the same thing are 12 / 3 = 4 and 12 / 4 = 3 So suppose we have something like 6 / (1/2) = ? This is the same as saying that 6 = ? * (1/2) Now, if this is true, then the value of '?' had better be larger than 6, right? Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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