When to Add or Multiply Denominators?Date: 04/02/2001 at 01:35:41 From: Laurie Subject: Differences in adding and multiplying denominators When you multiply two fractions together you multiply the numerators and then multiply the denominators. When you add two fractions you add the numerators but don't add the denominators. Why the difference? Date: 04/02/2001 at 13:00:44 From: Doctor Peterson Subject: Re: Differences in adding and multiplying denominators Hi, Laurie. Basically, they are just different things. Let's look closely at what the two operations mean, and see if we can find a simple description of what that difference is. When you multiply by a fraction, you are dividing something up into some number of pieces (the denominator, which therefore tells what kind of pieces they are, such as thirds), and taking a certain number of those (the numerator, which tells how many there are). We can diagram this by representing each of the two fractions by a cut in a different direction. For example, suppose we multiply 3/4 by 2/3. We can start with 3/4 (the shaded 3 out of 4 bars): +-----------------+ |/////////////////| +-----------------+ |/////////////////| +-----------------+ |/////////////////| +-----------------+ | | +-----------------+ When we multiply this by 2/3, we cut the whole thing into 3 parts vertically, and take 2 of them: +-----+-----+-----+ |XXXXX|XXXXX|/////| +-----+-----+-----+ |XXXXX|XXXXX|/////| +-----+-----+-----+ |XXXXX|XXXXX|/////| +-----+-----+-----+ | | | | +-----+-----+-----+ The product is the doubly-shaded (X'ed) part, which consists of 6 twelfths. Notice where these numbers come from: we divided the original whole square into 3*4 = 12 pieces, so the denominator of the result is 12 (the product of the denominators); we've selected 2*3 = 6 of these, so the numerator is 6 (the product of the numerators). We multiplied both numerator and denominator because our cuts affected both the number and size of the pieces. How about addition? Let's add 2/3 to 3/4. This time, we have a different situation. We aren't cutting each piece up; rather, we're putting pieces of two different kinds (thirds and fourths) together. Since we can't add apples and oranges, we have to turn both fractions into the same kind of piece. We can do this by using a common denominator: turning 2/3 into 8/12 and 3/4 into 9/12: +-----------------+ +-----+-----+-----+ |/////////////////| |\\\\\|\\\\\| | +-----------------+ |\\\\\|\\\\\| | |/////////////////| |\\\\\|\\\\\| | +-----------------+ + |\\\\\|\\\\\| | |/////////////////| |\\\\\|\\\\\| | +-----------------+ |\\\\\|\\\\\| | | | |\\\\\|\\\\\| | +-----------------+ +-----+-----+-----+ 3/4 2/3 is the same as +-----+-----+-----+ +-----+-----+-----+ |/////|/////|/////| |\\\\\|\\\\\| | +-----+-----+-----+ +-----+-----+-----+ |/////|/////|/////| |\\\\\|\\\\\| | +-----+-----+-----+ + +-----+-----+-----+ |/////|/////|/////| |\\\\\|\\\\\| | +-----+-----+-----+ +-----+-----+-----+ | | | | |\\\\\|\\\\\| | +-----+-----+-----+ +-----+-----+-----+ 9/12 8/12 Now that we are in a position to add the fractions, all we do is add the numerators, because the size of the pieces has already been determined: we have 17 twelfths. (You may notice that we actually did multiply the denominators together to get the common denominator; that's not always necessary, because the lowest common denominator may be smaller than the product, but you can always use the product as a common denominator.) When we did the addition itself, the denominator was not affected. Now, what's the difference between the two problems? One way to say it is that multiplication is a two-dimensional activity (like finding an area from two lengths, which is what I really did in my pictures), while addition is a one-dimensional activity (putting together two lengths, or counts). In my picture for addition, the second dimension came in only as part of the conversion to a common denominator (where I multiplied). Where two dimensions are involved, I am cutting across both numerator (the chosen part of the fraction) and denominator (the whole) at once; both numbers are therefore affected. So the multiplication and the conversion multiplied both numerator and denominator. But when I do the actual addition, I'm just setting one group of pieces next to another and counting; that adds together the numbers of pieces (numerators), but doesn't change what they are (the denominator: twelfths). To put it more briefly: addition puts together two groups of the same type (denominator) of things, while multiplication changes each of the things you multiply. If you'd like either a deeper (algebraic) or shallower answer, write back and let me know what you think. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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