Fractions and Common DenominatorsDate: 06/04/2001 at 01:54:54 From: AT Subject: Common Denominators Why do you need a common denominator to add fractions, but not to multiply them? I have tried to answer this question by using the distributive property and the definition of division. I have always been taught that you need a common denominator to add fractions. I have never been told axiomatically why this has to be done. Also, why don't you need a common denominator to multiply fractions? Date: 06/04/2001 at 12:47:52 From: Doctor Peterson Subject: Re: Common Denominators Hi, AT. Let's approach fractions algebraically. A fraction is essentially a division problem stated but not worked out; rather than divide 3 by 4 to get 0.75, we just write 3/4. This is much like algebra, where we write a/b and do not evaluate it, because we don't know the values. The reason we don't evaluate a fraction is that it has been found easier to manipulate the fraction as an expression (and then evaluate it at the end if necessary) rather than to evaluate it first, especially if we want an exact answer. To multiply two fractions, we have to manipulate the expression (a/b) * (c/d) until it is in a valid fractional form, one number "over" another. Let's do it: (a/b) * (c/d) = a * 1/b * c * 1/d = a * c * 1/b * 1/d = (ac) * 1/(bd) = (ac) / (bd) That was easy because multiplication and division work well together; division is a form of multiplication (by the reciprocal), and commutes. Now let's add two fractions, manipulating (a/b) + (c/d) until it looks like a fraction. We can't apply a commutative property, since addition and multiplication or division don't commute, but must use the distributive property (which relates addition and multiplication) instead. In particular, we have to choose a new denominator for the answer. It turns out that we can use (bd) as a denominator, and convert both fractions to that denominator by multiplying by 1: (a/b) + (c/d) = (a/b)(1) + (c/d)(1) = (a/b)(d/d) + (c/d)(b/b) = (ad)/(bd) + (bc)/(bd) Now we can apply the distributive property to make this a single fraction: = (ad) * 1/(bd) + (bc) * 1/(bd) = (ad + bc)* 1/(bd) = (ad+bc)/(bd) (In working with actual fractions, we can save work by finding the LEAST common denominator; using variables, we don't have that issue to worry about.) So why did we need a common denominator in the latter case, but not in the former? In both cases we needed a new denominator; but in multiplication, it arose by the commutative property, and is hardly noticeable as a common denominator. In doing our addition, we had to get the common denominator first so we could factor it out. Here are a couple explanations we have given at a lower level, which may help: Common Denominators http://mathforum.org/dr.math/problems/curr2.8.96.html When to Add or Multiply Denominators? http://mathforum.org/dr.math/problems/laurie.04.02.01.html - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 06/04/2001 at 22:00:46 From: Doctor Ian Subject: Re: Common Denominators Hi AT, If you have something like 3/7 + 2/7 = ? you can use the distributive property to simplify the expression on the left: 3/7 + 2/7 = 3*(1/7) + 2*(1/7) = (3 + 2)*(1/7) = 5*(1/7) = 5/7 Without common denominators, you can't do this. That's why you need to have common denominators to add. If you have something like 3/4 * 5/6 it's really just a series of multiplications and divisions: ((3 / 4) * 5) / 6 which is why you don't need to have common denominators. 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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