Reducing Fractions to Lowest Terms
A clear guide to simplifying fractions step by step
A fraction is reduced to lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD). $$ \frac{a}{b} \sim \frac{a:\gcd(a,b)}{b:\gcd(a,b)} $$
When you reduce a fraction to lowest terms, you obtain an equivalent fraction that cannot be simplified any further.
In this final form, the numerator and the denominator are coprime (relatively prime), which means they have no common divisors other than 1.
Note. Reducing a fraction to lowest terms gives its simplest form. This result follows directly from the invariant property of fractions and the definition of the greatest common divisor.
A step-by-step example
Let’s walk through the process by reducing the fraction
$$ \frac{56}{36} $$
First, compute the greatest common divisor (GCD) of the numerator (56) and the denominator (36) by factoring both numbers into primes:
$$ 56 = 2^3 \cdot 7 $$
$$ 36 = 2^2 \cdot 3^2 $$
Next, identify the common prime factors and take each with the smallest exponent.
Here, the only common prime factor is 2, and the smallest exponent is 22. Therefore, the greatest common divisor is 4.
$$ \gcd(56,36)=4 $$
Now divide both the numerator and the denominator by the GCD:
$$ \frac{56:\gcd(56,36)}{36:\gcd(56,36)} $$
$$ \frac{56:4}{36:4} $$
$$ \frac{14}{9} $$
The result is the equivalent fraction in lowest terms:
$$ \frac{56}{36} \sim \frac{14}{9} $$
This fraction cannot be simplified any further because 14 and 9 share no common divisors other than 1.
So, the complete simplification process is:
$$ \frac{56}{36} \sim \frac{28}{18} \sim \frac{14}{9} $$
Note. Relatively prime numbers (coprime numbers) are not the same as prime numbers. A prime number is divisible only by 1 and itself. Relatively prime numbers, instead, are two integers that share no common divisors other than 1. For example, 14 and 9 are not prime, but together they are relatively prime.
Once you understand this process, you can apply it to any fraction.
