Date: 2/8/96 at 8:48:56 From: Anonymous Subject: Addition of fractions Why do fractions need common denominators in order to be added?
Date: 4/1/96 at 2:11:33 From: Doctor Jodi Subject: Re: Addition of fractions Hi there! Let me try to give you an example of why common denominators are important. Suppose I want to know how tall I am. I don't have any string, but I DO have a yardstick and a collection of short sticks. I find out that I am 1 yardstick and 3 short rulers tall. Let's say I stop here. Now, I say to myself, I am 4 sticks tall. If each stick is a meter stick, then that's 12 feet! I have friends who are over 6 feet tall, and I've heard of basketball players being 7 or so feet tall, but 12 feet tall? That's Jack-in-the-Beanstalk-GIANT tall and I know that if I were in that story I *wouldn't* be the giant. And I'm sure that I didn't eat any strange mushrooms for breakfast, so I couldn't be changing size (I hope!) like Alice. So I try again. I know that each of the short sticks is LESS than 1 yard (That's why I'm calling them short!). So if I want to measure myself in yards, I'll find that I'm this tall: 1 + first stick in yards + second stick in yards + third stick in yards. Now I need to find out how long each of the sticks is in yards. I measure the first stick and I find out that it is 1/6 of a yard. I measure the second stick and find that it is 3/4 of a yard. Finally, I measure the third stick and find that it is 15/36 of a yard. So I'm 1 + 1/6 + 3/4 + 15/36 yards tall. So, I say to myself, I'm 1 and 19/36 of a yard, right? Nope, I've made another mistake. 3/4 = 27/36, as you can see by multiplying numerator and denominator by 9. But I wound up thinking I was 19/36 tall by adding the numerators and taking the highest denominator. This method, or any other attempt to add fractions with different denominators, will give the wrong answer. Why? It's hard to give a good explanation, but I *can* compare it to trying to add the yard stick to the three smaller sticks without taking their size into account. Similarly, if you add fractions with different denominators, you're adding "sticks" of different sizes. First, you need to put them in a common system of measurement so that you can compare their sizes. This is the common denominator. Here's one way to see the difference in sizes of different fractions: Take a piece of paper and fold it equally into four pieces. Now take another piece of paper and fold it equally into two pieces. You can repeat this with as many different sizes as you want. If you are careful, you can also make 3 folds, representing 1/3, etc. You can do the same thing with pies. Suppose you were sharing two pizzas among several friends and made the mistake of adding eighths and quarters together... Would you be content with one-eighth of a pizza while your friend was getting one-quarter? Probably not.... Well, I hope that this helps you understand the importance of common deonominators. Write back if you have any more questions. -Doctor Jodi, The Math Forum
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