Before looking at examples of simplifying fractions, it’s handy to have a recap of what equivalent fractions are, and what is the simplest form.
Equivalent Fractions
The understanding fractions page touched on the subject of equivalent fractions with a brief example of 3 different fractions.
The fractions \bf{\frac{1}{3}} , \bf{\frac{2}{6}} and \bf{\frac{3}{9}} are all the same fraction in terms of overall value.
But it’s the case that \bf{\frac{1}{3}} is the “simplest” form.
This is because 1 and 3 are the smallest whole numbers for the numerator on the top line and the denominator on the bottom line that could be used.
So it’s the case that the fraction \bf{\frac{3}{9}} can be simplified to \bf{\frac{1}{3}}.
This logic and approach will be shown further down on this page with more examples of simplifying fractions.
But before that, it’s also important to understand the concept of common factors in Math.
Examples of Simplifying Fractions,
Common Factors
When trying to simplify a fraction, the goal really is to find the largest “common factor” that both the numerator above and the denominator below share.
A common factor of two whole numbers, is a smaller number that can divide evenly into both of the whole numbers, and it’s the case that a pair of whole numbers can a number of common factors.
Factors and Common Factors:
If we consider the fraction \bf{\frac{8}{12}},It’s the case that 8 and 12 have a common factor of 2 => \bf{\frac{8}{2}} = 4 , \bf{\frac{12}{2}} = 6
The number 2 divides both numbers evenly, reducing each to a smaller whole number.
So \bf{\frac{8}{12}} can be simplified to \bf{\frac{4}{6}}.
However, there is a larger “common factor” that can be used to simplify the original fraction further.
The number 4 is larger than 2, and happens to be the largest “common factor” that 8 and 12 both share.
So the initial fraction \bf{\frac{8}{12}} can be simplified to \bf{\frac{2}{3}}, in its “simplest” form.
Examples
(1.1)
\bf{\frac{5}{10}} can be simplified to \bf{\frac{1}{2}}.
( {\frac{5}{5}} = 1 , {\frac{10}{5}} = 2 )
(1.2)
\bf{\frac{6}{14}} can be simplified to \bf{\frac{3}{7}}.
( {\frac{6}{2}} = 3 , {\frac{14}{2}} = 7 )
(1.3)
\bf{\frac{12}{27}} can be simplified to \bf{\frac{4}{9}}.
( {\frac{12}{3}} = 4 , {\frac{27}{3}} = 9 )
Lowest Common Denominator
When 2 or more fractions have a common denominator, they simply have the same denominator on the bottom line.
For example the fractions \bf{\frac{1}{7}} , \bf{\frac{4}{7}} , \bf{\frac{5}{7}}
all have a common denominator of 7.
But there can be times when fractions don’t have a common denominator initially, though you may wish to re-write them so that they do have a common denominator.
Finding Lowest Common Denominator:
As they sit, the fractions \bf{\frac{3}{4}} and \bf{\frac{5}{7}} do NOT share a common denominator.
The lowest common denominator of fractions turns out to be lowest common multiple of the relevant denominators.
Information on how to find the lowest common multiple of numbers can be viewed at basic-mathematics.com.
For 4 and 7, the lowest common multiple is 28.
So for both fractions, 28 is the lowest common denominator.
Changing the Fraction Denominators:
It’s the case that as well as changing the denominators to 28 in the fractions.The numerators are required to be changed also, to ensure that the value of each fraction is the same as before.
For our fractions, we just multiply the numerator by the same number that we multiplied the denominator by to make 28.
Division of 28 by the relevant original denominator gives the number to multiply the numerator by.
1) 28 ÷ 4 = 7 => \bf{\frac{3}{4}} = \bf{\frac{3\space\times\space7}{28}} = \bf{\frac{21}{28}}
2) 28 ÷ 7 = 4 => \bf{\frac{5}{7}} = \bf{\frac{5\space\times\space4}{28}} = \bf{\frac{20}{28}}
So the fractions \bf{\frac{3}{4}} and \bf{\frac{5}{7}} can be written as \bf{\frac{21}{28}} and \bf{\frac{20}{28}} if we want them to have a common denominator.
This process can also be used when dealing with more than only two fractions.
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