Finding Common DenominatorsDate: 06/25/2001 at 11:21:01 From: Mike Aldridge Subject: Common denominators Dear Dr. Math, Can you please tell me the best way to find a common denominator? I am having a very hard time understanding this. Date: 06/25/2001 at 14:04:30 From: Doctor Ian Subject: Re: Common denominators Hi Mike, When you want to add two fractions with different denominators, the trick is to change each fraction into something with the same meaning, but a different appearance. The way you do that is by finding a creative way to multiply each fraction by some form of 1. Suppose you wanted to add 10 francs to 14 marks. One way to do that would be to convert them to a common currency... say, the euro. The conversion factor for each currency would be derived this way: 1 franc = f euros 1 mark = m euros f euros m euros 1 = ------- 1 = ------- 1 franc 1 mark As an American, I have no idea what the values of f and m would be, but let's assume that we could look them up. Your addition would look like this: f euros m euros 10 francs + 14 marks = 10 francs * ------- + 14 marks * ------- 1 franc 1 mark = 10f euros + 14m euros = (10f + 14m) euros This gives you the same result, because multiplying by 1 can't change anything. With a fraction, the easiest 'conversion factor' to use is the denominator of the other fraction. So in a case like 3 1 - + - 8 4 you would multiply the first fraction by 4/4, and the second fraction by 8/8, to get: 3 4 1 8 3 * 4 1 * 8 12 + 8 20 - * - + - * - = ----- + ----- = -- + -- = -- 8 4 4 8 8 * 4 4 * 8 32 32 32 Now, note that this fraction can be reduced to 10/16, and then to 5/8. Why is that? It's because the original denominators, 4 and 8, share some common factors. In this particular case, we _could_ have noticed that 4 goes into 8 twice, and simply converted everything into 8ths: 3 1 2 3 1 * 2 3 + 2 5 - + - * - = - + ----- = ----- = -- 8 4 2 8 4 * 2 8 8 But it's a trade-off; if you do more work up front to find the _lowest_ common denominator, then you won't have to reduce the fraction after the addition. On the other hand, you can save time up front by finding the _obvious_ common denominator, and then deal with the subsequent reduction. My own feeling is that it's easier to reduce a fraction than to find the lowest common denominator, but that's a personal preference. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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