Simplifying Mixed Fractions and Fractions

On this Page:
1) Common Factors
2) Common Factors & Simplifying
3) Lowest Common Denominator
4) Simplifying Mixed Fractions

On the fractions introduction page, it was explained how there can be equivalent fractions in Math. This is something that can help deal with with simplifying fractions examples.

Equivalent fractions are when fractions have with different numbers in them, but they both have the same overall value.

\bf{\frac{1}{2}}

\bf{\frac{2}{4}}

and

\bf{\frac{3}{6}}

are the same fraction,

but

\bf{\frac{1}{2}}

was the “simplest” form.

That’s because 1 and 2 are the smallest whole numbers for the numerator and denominator that could be used.

Common Factors

When aiming to simplify a fraction, we really want to find the largest common factor that the numerator and denominator in the fraction share.

A common factor of two whole numbers, is a lesser number that divides evenly into both the numbers, and any pair of numbers can possibly have more than one common factor.

For example with the fraction

\bf{\frac{6}{12}}

,

9 and 12 have a common factor of 3. =>

\bf{\frac{6}{3}}

= 2 ,

\bf{\frac{12}{3}}

= 4

So ^{$\frac{9}{12}$} can be simplified to ^{$\frac{3}{4}$}.

Simplifying Fractions Examples,
Greatest Common Factor

But there is a greater “common factor” that can simplify the fraction further.

The numerator 6 itself is greater than 3, and here is the greatest “common factor” that 6 and 12 share.

\bf{\frac{6}{6}}

= 1 ,

\bf{\frac{12}{6}}

= 2

The fraction ^{$\frac{6}{12}$} can be simplified further to ^{$\frac{1}{2}$},

in its simplest form.

Examples

1.1

\frac{6}{8}

can be simplified to

\frac{3}{4}

(

\bf{\frac{6}{2}}

= 3 ,

\bf{\frac{8}{2}}

= 4 )

1.2

\frac{8}{20}

can be simplified to

\frac{2}{5}

(

\bf{\frac{8}{4}}

= 2 ,

\bf{\frac{20}{4}}

= 5 )

1.3

\frac{18}{45}

can be simplified to

\frac{6}{15}

(

\bf{\frac{18}{3}}

= 6 ,

\bf{\frac{45}{3}}

= 15 )

Lowest Common Denominator

When it’s said that two or more fractions have a common denominator, it means that they simply both have the same denominator.

The fractions,

\bf{\frac{1}{4}}

\bf{\frac{3}{4}}

\bf{\frac{5}{4}}

all have a common denominator of 7.

There can also be times when fractions involved in simplifying fractions examples don’t share a common denominator to start with, but we may wish to re-write them so that there is a common denominator.

Find the Lowest Common Denominator

As they currently are, the fractions

\bf{\frac{1}{4}}

and

\bf{\frac{3}{7}}

do NOT share the same common denominator.

The lowest common denominator of these fractions, and indeed other fractions, happens to be lowest common multiple of the relevant denominators we’re looking at.
An approach to find the lowest common multiple of two or more numbers can be seen on the multiples of numbers page.

But for 4 and 7, the lowest common multiple is the number 28.
Thus for the fractions above, 28 is the lowest common denominator.

But along with changing the denominators to 28 in both the fractions.
The numerators also have to be correctly changed, to ensure that the overall value of each fraction is still the same as the original.

Rewriting the Fractions with
the Common Denominator:

For the fractions involved, we can just multiply each numerator by the same number that the denominator was multiplied by to make 28.

A division of 28 by the specific original denominator will give the number with which to multiply the numerator by.

1) 28 ÷ 4 = 7 =>

\bf{\frac{3\space\times\space7}{28}}

\bf{\frac{21}{28}}

2) 28 ÷ 7 = 4 =>

\bf{\frac{5\space\times\space4}{28}}

\bf{\frac{20}{28}}

The fractions $\bf{\frac{3}{4}}$ and $\bf{\frac{5}{7}}$ can be written as $\bf{\frac{21}{28}}$ and $\bf{\frac{20}{28}}$ so that they have a common denominator.

This approach can also be used when dealing with more than just two fractions.