Inverting and Multiplying When Dividing FractionsDate: 9/28/95 at 0:30:35 From: Anonymous Subject: dividing fractions I have always been taught that to divide fractions I must invert and multiply. Why does this algorithm work? Is there any other way to divide fractions? Date: 9/28/95 at 2:30:58 From: Doctor Andrew Subject: Re: dividing fractions Division can be defined as x = a / b if and only if b * x = a. This says that if b times x equals a, then x equals a divided by b. It also says that if x equals a divided by b, then b times x equals a. "If and only if" is an expression that mathematicians like to use a lot so they can say one sentence instead of two. Note: when I write a/b I mean a fraction and when I write a / b I mean division. So if we have a fraction p/q and we want to divide it into a number n the result is the number x such that p/q * x = n Let's solve this for x. Start by multiplying both sides by q (make sure q isn't zero) px = n * q. now divide both sides by p (p had better not be zero): x = n * q / p = n * q/p So, n / (p/q) = n times the reciprocol of p/q (which is q/p.) You can divide fractions any way you want so long as the statement x = a / b if only if b * x = a is always true. For instance, instead of multiplying the reciprocols you could, if you were dividing two fractions, divide the numerators and the denominators separately like this: a c a / c - / - = ----- b d c / d But if (a/c) is not an integer (say 0.12623456 and not a number like 10) or (b/d) is not an integer, then you're going to have a fraction with decimal numbers, which is just not allowed. The reciprocol method allows you to keep integers in the numerator and denominator of the quotient (that's what the result is called) when you divide. I hope this helps. Please send us any more questions or ask me to clarify anything here that might be confusing. - Doctor Andrew, The Geometry Forum |
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