How to Reduce a Fraction to its Lowest Expression

The idea of reducing a fraction to its lowest terms means to take a fraction and express it in terms of its simplest possible form, by having a fraction that has the same value as the original fraction, but it all possible common factors between the numerator and denominator have been simplified. This is achieved by computing the greatest common divisor (GCD) between two numerator and denominator, and then simplify the fraction by it.

● Let us see the see the following example: Reduce this Fraction to Lowest Terms

\[\frac{165}{1575}\]

First, we compute the GCD for \(n_1 = 165\) and \(n_2 = 1575\). Let us find the prime decomposition of each of these numbers (you can use our prime decomposition calculator)

\[165 = 3 \cdot 5 \cdot 11\] \[1575 = 3^2 \cdot 5^2 \cdot 7\]

From the above: what primes do these two numbers have in common? As we can see, the common primes are 3 and 5. Looking at the exponents of these common primes in each of the numbers, we look the minimum between the two. In this case, the minimum exponent for 3 is 1, and the minimum exponent for 5 is also 1. Therefore

\[GCD = 3^1 \cdot 5^1 = 3 \cdot 5 = 15 \]

Now, all we have to do is to simplify the original fraction by 15: \[\frac{165}{1575} = \frac{165/15}{1575/15} = \frac{11}{105} \]

which corresponds to the fraction in its lowest terms, because it cannot be further simplified.

The question is: why do we reduce fractions to lowest terms? Well, the answer is easy. Like in everything in Math, simplicity is the most desirable condition. Why work with a cumbersome fraction if we can reduce to an equivalent fraction that is simpler?

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