Tutoring: ordinary math fractions reducing (simplifying) to lower terms - how to reduce a fraction to its simplest form (irreducible)?
Tutoring: ordinary math fractions reducing to lower terms (simplifying)
Let's take an example, let's reduce (simplify) the fraction: 12/16
Numerator of the fraction: The number that is above the fraction line, 12, is called the numerator of the fraction;
Denominator of the fraction: The number that is under the fraction line, 16, is called the denominator of the fraction;
We notice that the two numbers, the numerator and the denominator, are dividing themselves without any remainder by 2: 12/16 = (12 ÷ 2)/(16 ÷ 2) = 6/8
Reduced (simplified) fraction, Equivalent fraction: The new fraction, 6/8, is called an equivalent fraction of the original one, 12/16; it represents the same value or proportion of the whole, and it comes from 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 (divizor): The number 2 that was used to divide the two numbers that make up the fraction is called a common factor or a divizor 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 further notice that the two new numbers, the new numerator and the new denominator, 6 and 8, are also dividing themselves without any rest by 2 (2 is a common factor or a common divizor of 6 and 8): 6/8 = (6 ÷ 2)/(8 ÷ 2) = 3/4
The new fraction, 3/4, is also a reduced fraction (a simplified fraction) and an equivalent of the fractions 12/16 și 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 its simplest form, the irreducible fraction?
Greatest common factor, GCF: To reduce a fraction to its simplest form, the irreducible fraction, we must divide both the numerator and the denominator of the fraction by its greatest common factor, gcf (12; 16).
Prime Factorization: One way to calculate the greatest common factor is to find all the prime factors of the two numbers and build their prime decomposition in exponential form, and then to multiply all the common prime factors by their lowest exponents, see below.
Fraction's numerator, 12 - prime decomposition in exponential form: 12 = 2 * 2 * 3 = 22 * 3.
Fraction's denominator, 16 - prime decomposition in exponential form: 16 = 2 * 2 * 2 * 2 = 24.
You can find the prime factors of numbers or build the prime decomposition in exponential form on this page at www.numere-prime.ro: numbers prime factorization.
The greatest common factor, GCF (12; 16), also called GCD, is calculated by multiplying all the common prime factors of both the numerator and the denominator, by their lowest exponents: GCF (12; 16) = (22 * 3; 24) = 22 = 4.
In the end, both fraction's numerator and denominator are divided 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.
Why reducing fractions to lower terms (simplifying)?
Operations with fractions often involve bringing them to the same denominator and sometimes both the numerators and the denominators are large numbers. Doing calculations with such large numbers could be difficult.
By simplifying, reducing a fraction to lower terms, both the numerator and denominator of the fraction are reduced to smaller values, much easier to work with, reducing the resulting computational effort.