How to reduce fractions to lower terms (simplify). Equivalent, irreducible fractions. Prime factors. The greatest common factor, GCF. Examples
Fractions reducing to lower terms (simplifying). Equivalent fractions
Let's learn by an example, let's simplify the fraction: 12/16
Numerator of the fraction. The number that is above the fraction bar, 12, is called the numerator of the fraction;
Denominator of the fraction. The number that is below the fraction bar, 16, is called the denominator of the fraction;
The value of the fraction. Fraction 12/16 shows us in how many equal parts the number above the fraction bar, 12, is being divided: into 16 equal parts. Thus, the value of the fraction is calculated as:
12 ÷ 16 = 0.75
We notice that the two numbers, the numerator and the denominator, are dividing themselves without any remainder by 2, so we divide them by the same number, 2:
12/16 = (12 ÷ 2)/(16 ÷ 2) = 6/8
The value of the fraction 6/8 is calculated as:
6 ÷ 8 = 0.75
We notice that the value of the fraction 6/8 is equal to that of the fraction 12/16, namely 0.75
Reduced (simplified) fraction, Equivalent fraction. The new fraction, 6/8, is equivalent to the original one, 12/16, that is, it represents the same value or proportion of the whole, and it was calculated out of the original fraction by reducing it (simplifying it): both the numerator and the denominator of the fraction were divided by the number 2.
Common factor (divisor). The number 2 that was used to divide the two numbers that make up the fraction is called a common factor or a divisor of the numerator and the denominator of the fraction.
The reduced fraction has now a numerator that is equal to 6 and a denominator that is equal to 8.
We also notice that the two new numbers, the new numerator and the new denominator, 6 and 8, are also dividing themselves without any remainder by 2 (2 is a common factor of 6 and 8), so we divide them again by 2:
6/8 = (6 ÷ 2)/(8 ÷ 2) = 3/4
The value of the fraction 3/4 is calculated as:
3 ÷ 4 = 0.75
The new fraction, 3/4, is a reduced fraction (a simplified fraction) and an equivalent of the fractions 12/16 and 6/8
Irreducible fraction. Fraction 3/4 is also called an irreducible fraction, in another words it could no longer be reduced or simplified, it is in its simplest form, the numbers 3 and 4, the numerator and the denominator of the fraction, are coprime numbers (prime to each other), not having any common factors other than 1.
How to reduce the fraction 12/16 to the lowest terms? Irreducible fraction
Greatest common factor, GCF. To reduce a fraction to the lowest terms (or, in another words, to its simplest form, irreducible), we must divide both the numerator and the denominator of the fraction by their greatest common factor, gcf (12; 16).
Prime Factors. One way to calculate the greatest common factor, GCF, is to break down each of the two numbers to prime factors and then write them as products of those primes, in exponential form. After that, multiply all the common primes by their lowest exponents, see it below.
The prime factorization of the numerator and the denominator:
To reduce the fraction to its lowest terms, divide the numerator and the denominator by their greatest common factor, GCF:
12/16 = (12 ÷ 4) / (16 ÷ 4) = 3/4
Irreducible fraction. The end fraction 3/4 is called a reduced fraction, simplified. Since the numerator and the denominator are coprime numbers (prime to each other), their greatest common factor is 1, so this fraction is in its simplest form (it cannot be reduced anymore). This fraction is called an irreducible fraction.
The fraction 3/4 is also an equivalent of the original fraction 12/16, representing the same value or proportion of the whole. As we saw above:
3/4 = 6/8 = 12/18 - all these are equivalent fractions, calculated by reducing the original one.
Equivalent fractions can be calculated not only by reducing but also by expanding a fraction, that is, by multiplying the numerator and denominator by the same number other than zero, that is, the inverse of the process of simplification, but this is another discussion.
Why reducing fractions to lower terms (simplifying)?
In order to do calculations with fractions, sometimes we are also required to build them up to the same denominator or the same numerator. Sometimes both the numerators and the denominators are large numbers. Doing calculations with such large numbers could be time and resource consuming.
By simplifying, by reducing a fraction to lower terms, both the numerator and denominator of the fraction are reduced to smaller values which are easier to work with.