Reduced fraction

Revision as of 22:41, 14 September 2006 by JBL (talk | contribs)

A reduced fraction, also known as a fraction written in lowest terms, is a rational number written in the form $\frac a b$ where $a$ and $b$ are integers with no common divisors (that is, which are relatively prime).

Given an unreduced fraction, one may reduce it by cancelling common factors of the numerator and denominator in accordance with the rules of arithmetic. For example, $\frac {15}{27}$ is not a reduced fraction because both 15 and 27 are divisible by 3. So in order to reduce, we write $\frac{15}{27} = \frac{3 \cdot 5}{3\cdot 9} = \frac5 9$, and 5 and 9 are relatively prime, so this fraction is reduced.

Sometimes, it might take several steps to reduce a fraction (because we don't notice all the common factors of the numerator and denominator, for example). For instance, in reducing the fraction $\frac{357}{273}$, we might first notice that the numerator and denominator are divisble by 3 and so reduce to $\frac{357}{273} = \frac{3\cdot 119}{3\cdot 91} = \frac{119}{91}$. We have now reduced our original fraction, but it can be reduced further: both 119 and 91 are divisble by 7, and the fraction reduces again to $\frac{17}{13}$. In this case, we call the intermediate steps partially reduced and the final, reduced fraction fully reduced.

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