Tutoring: simplifying fractions - reducing them to the lowest terms
Steps to simplify a fraction, to reduce it to its lowest terms:
- A fraction fully simplified, a fraction reduced to its lowest terms is a fraction that can no longer be simplified, it has been reduced to its simplest equivalent fraction, the one having the smallest numerator and denominator possible - prime to each other.
- 1) Run the prime factorization of both the numerator and the denominator of the fraction.
- 2) Calculate the greatest common factor, GCF (or the greatest common divisor, GCD) of the fraction's numerator and denominator.
- 3) Divide both the numerator and the denominator of the fraction by their greatest common factor, GCF (GCD).
- The fraction thus obtained is called a reduced fraction or a fraction simplified to its lowest terms.
- A fraction reduced to its lowest terms may no longer be reduced and it is called an irreducible fraction.
Example: reduce the fraction 315/1,155 as much as possible, simplify it to the lowest terms.
1) Run the prime factorization of both the numerator and the denominator of the fraction.
- The numerator of the fraction is 315, its breaking down into prime factors is:
315 = 3 × 3 × 5 × 7 = 32 × 5 × 7 - The denominator of the fraction is 1,155, its breaking down into prime factors is:
1,155 = 3 × 5 × 7 × 11. 2) Calculate the greatest common factor, GCF (or the greatest common divisor, GCD) of the fraction's numerator and denominator.
- The greatest common factor, gcf (315; 1,155), is calculated by multiplying all the common prime factors of the numerator and the denominator, taken by their lowest powers (their lowest exponents):
- GCF (315; 1,155) = (32 × 5 × 7; 3 × 5 × 7 × 11) = 3 × 5 × 7 = 105
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3) Divide both the numerator and the denominator of the fraction by their greatest common factor, GCF (GCD).
- The numerator and denominator of the fraction are divided by their greatest common factor, GCF:
- 315/1,155 =
(32 × 5 × 7)/(3 × 5 × 7 × 11) =
((32 × 5 × 7) ÷ (3 × 5 × 7)) / ((3 × 5 × 7 × 11) ÷ (3 × 5 × 7)) =
3/11 - The fraction thus obtained is called a fraction reduced to the lowest terms.
Why reducing (simplifying) fractions to lower terms?
- When running operations with fractions we are often required to bring them to the same denominator, for example when adding, subtracting or comparing.
- Sometimes both the numerators and the denominators of those fractions are large numbers and doing calculations with such numbers could be difficult.
- By reducing (simplifying) a fraction, both its numerator and denominator are reduced to smaller values, much easier to work with, which will decrease the overall effort of working with that fraction.