Simplifying Fractions and Reducing to Lowest Terms
How to simplify a fraction
A fraction can be simplified by dividing both the numerator and the denominator by the same number. $$ \frac{a}{b} \sim \frac{a:k}{b:k} $$
The simplified fraction is an equivalent fraction of the original, since both represent the same value.
Example. The fraction 8/4 $$ \frac{8}{4} $$ can be simplified by dividing both the numerator and the denominator by 2. $$ \frac{8:2}{4:2} = \frac{4}{2} $$ The result is a fraction equivalent to the original. $$ \frac{8}{4} \sim \frac{4}{2} $$ The fractions 8/4 and 4/2 are equivalent because they evaluate to the same number $$ \frac{8}{4} = 8:4 = 2 $$ $$ \frac{4}{2} = 4:2 = 2 $$
A fraction is said to be in lowest terms when the numerator and the denominator share no common divisors other than 1, that is, when they are coprime.
To reduce a fraction to lowest terms, divide both the numerator and the denominator by their greatest common divisor (GCD).
$$ \frac{a}{b} \sim \frac{a:\gcd(a,b)}{b:\gcd(a,b)} $$
Note. To reduce the fraction $$ \frac{9}{12} $$ to lowest terms, first compute the greatest common divisor of the numerator and the denominator, $$ \gcd(9,12)=3 $$ Then divide both the numerator and the denominator by the GCD (3). $$ \frac{9:\gcd(9,12)}{12:\gcd(9,12)} = \frac{9:3}{12:3} = \frac{3}{4} $$ The result is the fraction in lowest terms, since it cannot be simplified any further. $$ \frac{9}{12} \sim \frac{3}{4} $$
The simplification of fractions follows from the invariant property of fractions.
A practical example
Consider the fraction
$$ \frac{32}{28} $$
Both numbers are divisible by 2.
To simplify the fraction, divide both the numerator and the denominator by their common divisor, 2.
$$ \frac{32:2}{28:2} $$
This produces an equivalent fraction, called "simplified" because the numerator and denominator are smaller.
$$ \frac{16}{14} $$
This fraction can be simplified further, since both the numerator and the denominator are still divisible by 2.
So, divide both the numerator and the denominator by 2 again.
$$ \frac{16:2}{14:2} $$
The result is another simplified equivalent fraction.
$$ \frac{8}{7} $$
This final fraction is in lowest terms because the numerator (8) and the denominator (7) have no common divisors other than 1.
The numbers 8 and 7 are coprime (or relatively prime).
Note. To reduce a fraction to lowest terms without performing all the intermediate steps, simply compute the greatest common divisor of the numerator and the denominator.
Example 2
Reduce the fraction
$$ \frac{32}{28} $$
Compute the greatest common divisor of the numerator (32) and the denominator (28).
$$ \gcd(32,28) = 4 $$
Note. To compute the greatest common divisor, factor both numbers into prime factors $$ 32 = 2^5 $$ $$ 28 = 2^2 \cdot 7 $$ Then multiply the common prime factors, taking each with the smallest exponent. In this case, 32 and 28 share only one prime factor (2), and the smallest exponent is 22. Since there are no other common factors, the greatest common divisor is 22, that is, 4. $$ \gcd(32,28)=4 $$
Now divide the numerator (32) and the denominator (28) by their greatest common divisor (4).
$$ \frac{32:\gcd(32,28)}{28:\gcd(32,28)} $$
$$ \frac{32:4}{28:4} $$
$$ \frac{8}{7} $$
The final result is the equivalent fraction reduced to lowest terms.
Note. This is the same result obtained in the previous example by simplifying step by step. In this case, however, the fraction is reduced to lowest terms in a single step after computing the greatest common divisor. When the numbers in the numerator and denominator are large, this method significantly reduces the amount of computation and helps avoid careless errors.
And so on.
