Shortcut for Comparing Fractions
Date: 01/14/2003 at 07:50:56 From: Joe Subject: Comparing fractions We are going over comparing fractions in class and my teacher showed a little shortcut. He used the example of 2/3 and 3/5. He said you can cross multiply and get 9/10. The numerator (the product of the interior terms, 3 and 3) always corresponds to the second of the two fractions, and the denominator (the product of the exterior terms, 2 and 5) corresponds to the first fraction. He showed us many examples and I have tried many myself and this always works. 9 is less than 10, so 3/5 is less than 2/3. My question is WHY?
Date: 01/14/2003 at 11:20:56 From: Doctor Ian Subject: Re: Comparing fractions Hi Joe, Suppose I have two quantities, and I want to know which is larger. One way to do that is to divide one by the other, and see if I end up with something less than, greater to, or equal to 1. This is trivial when you do it with integers: 7 / 5 > 1 so 7 > 5 3 / 4 < 1 so 3 < 4 But it also works with fractions. Suppose I use '?' to represent '=', '>', or '<'. That is, I'm going to use it as a 'variable' for these relational operators. Then if I want to compare two fractions, a/b and c/d, I can do this: a/b ----- ? 1 c/d Of course, to divide by a fraction, I invert and multiply, so this is the same as a d - * - ? 1 b c If I get something greater than 1, a/b must be larger. If I get something less than 1, a/b must be smaller. If I get 1, the two fractions must be equal. Let's try it with your example: 2/3 ----- ? 1 3/5 2 5 - * - ? 1 3 3 10 -- > 1 9 So 'cross-multiplying' is just what you have to do to divide the first fraction by the second. Comparing the result to 1 tells you whether the numerator (first fraction) or denominator (second fraction) is larger. Does that make sense? You can get a better feel for what is going on if you use letters instead of numbers. For example, suppose my two fractions are a/b and (a+1)/b. I know that the second one has to be larger, right? Let's see what happens when we divide: a/b --------- ? 1 (a+1)/b a b - * ----- ? 1 b (a+1) a ----- < 1 (a+1) Let's try comparing two equivalent fractions. If we have a fraction a/b, we can make an equivalent fraction by multiplying by k/k, where k is anything other than zero: a/b --------- ? 1 (ka)/(kb) a kb - * ---- ? 1 b ka kab --- = 1 kab For fun, you might consider trying to see which is larger: a/b, or (a+k)/(b+k). That is, when you increment both the numerator and denominator of a fraction by the same amount, is the resulting fraction larger, or smaller? I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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