# Simplifying Fractions

Consider the fractions, $\frac{952}{1904}$ and $\frac{1}{2}$. Both of them actually represent the exact same value. However, $\frac{1}{2}$ does appear significantly simpler than $\frac{952}{1904}$. In this case, $\frac{1}{2}$ is the fraction expressed in what is known as the simplest form or its lowest terms.

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## Simplifying fractions definition

Simplifying fractions is the way to put a fraction in its simplest form.

### Simplest form of a fraction definition

A fraction is in its simplest form if there are no more common factors between its numerator and denominator.

A fraction is in its simplest form if the greatest common divisor between its numerator and denominator is 1.

To see this, let's take the following example.

The fraction $\frac{8}{11}$ is in its simplest form.

In fact, the factors of 8 are 1, 2, 4 and 8, and the factors of 11 are 1 and 11.

We see that 1 is the highest (and only) factor of the numerator and denominator.

Hence$\frac{8}{11}$ is indeed in its simplest form.

However, the fraction $\frac{20}{88}$ is not in its simplest form.

In fact, the factors of 20 are 1,2,4,5,10, and 20, and the factors of 88 are 1,2,4,22,44,and 88. We notice that there are two common factors between 20 and 88 which are 2 and 4. Hence we deduce that our fraction is not in its simplest form and hence can be simplified furthermore.

We will see this in detail hereafter in the article.

## Simplifying fractions methods

There are two commonly used methods for simplifying fractions.

### Repeated division method for simplifying fractions

Repeatedly divide both the numerator and the denominator by the lowest prime number that is a common factor. Keep repeating this step until no other common prime factor remains.

### Using the greatest common divisor method for simplifying fractions

Divide both the numerator and the denominator by their Greatest Common Divisor. This would give the fraction in its simplest form.

In this article, we are not going to go through the process of finding the Greatest Common Divisor. To brush up on the topic, check out our article on Greatest Common Divisor.

### How to simplify mixed fractions?

We recall that a mixed fraction is a combination of a whole number and a proper fraction.

For example, $2\frac{1}{3}$, is the sum of 2 and $\frac{1}{3}$.

To simplify a mixed fraction we follow these steps,

• Convert it into an improper fraction,
• Continue with the standard simplification process by using either of the methods mentioned above.

## Simplifying fractions with exponents

For a fraction that contains exponents in the numerator and/or denominator, we use the Greatest Common Divisor method to simplify it.

Note that when there are exponents with a common base, in both the numerator and denominator, the common base with the lower exponent can be taken as part of the GCD.

For example, if the numerator contains 210 and the denominator contains 26, we include 26 as part of the GCD.

## Simplifying fractions with variables

For fractions with variables also known as algebraic fractions, we use the Greatest Common Divisor method for simplifying the numerator and the denominator in a way to put the fraction in its simplest form.

For finding the GCD of algebraic fractions, we treat exponents of variables the same way we treat numerical exponents - we take the lower exponent of the common variable as part of the GCD.

For example, if the numerator contains x10 and the denominator contains x6, we include x6 as part of the GCD.

## Simplifying fractions examples

In this section, we will look at multiple examples of simplifying fractions.

### Simplifying numerical fractions examples

Simplify $\frac{45}{144}$.

Solution

Method 1. Using the repeated division for simplifying fractions.

The factors of 45 are: 1,3,5,9, 15 and 45.

The factors of 144 are: 1,2,3,4,6,8,9,12,16,18,24,36,48,72 and 144.

We notice that the lowest prime number that is a common factor of the numerator and the denominator is 3. Thus, we divide the numerator and the denominator by 3 to get

$\frac{45}{144}=\frac{3×15}{3×48}=\frac{15}{48}$

15 and 48 are both dividable by 3, so by dividing by 3 we get,

$\frac{15}{48}=\frac{3×5}{3×16}=\frac{5}{16}$

There are no more common prime factors between the numerator 5 and the denominator 16.

Hence $\frac{5}{16}$ is the simplest form of the expression.

Method 2. Using the greatest common divisor of the numerator and the denominator.

The Greatest Common Divisor of the 45 and 144 is 9.

We divide both the numerator and denominator by 9 to get

$\frac{45}{144}=\frac{9×5}{9×16}=\frac{5}{16}$.

Simplify $\frac{48}{216}$

Solution

Method 1. Using the repeated division for simplifying fractions.

We notice first both the numerator 48 and the denominator 216 are even numbers, so they are both dividable by 2,

$\frac{48}{216}=\frac{2×24}{2×108}$

We divide by 2 to get

$\frac{48}{216}=\frac{\overline{)2}×24}{\overline{)2}×108}=\frac{24}{108}$

The same goes for 24 and 108, both numbers are even, so they are dividable by 2,

$\frac{24}{108}=\frac{2×12}{2×54}$

We divide by 2 to get,

$\frac{24}{108}=\frac{\overline{)2}×12}{\overline{)2}×54}=\frac{12}{54}$

12 and 54 are both even numbers so they are dividable by 2 too, so we have

$\frac{12}{54}=\frac{2×6}{2×27}$

We divide by 2 to get,

$\frac{12}{54}=\frac{\overline{)2}×6}{\overline{)2}×27}=\frac{6}{27}$

Now 6 and 27 have 3 as their lowest common prime factor. Dividing by 3, we get

$\frac{6}{27}=\frac{2×3}{3×9}$

Dividing by 3, we get,

$\frac{6}{27}=\frac{\overline{)3}×2}{\overline{)3}×9}=\frac{2}{9}$

There are no more common prime factors between the numerator 2 and the denominator 9.

So $\frac{2}{9}$ is the fraction expressed in its simplest form.

Method 2. Using the greatest common divisor of the numerator and the denominator.

The factors of 48 are: 1,2,4,6,24, and 48.

The factors of 216 are: 1,2,3,4,6,8,9, 12, 18, 24,27,36,54, 72, 108 and 216.

Thus, the Greatest Common Divisor of the 48 and 216 is 24.

In fact, by dividing both the numerator and denominator by 24, we get

$\frac{48}{216}=\frac{2×24}{9×24}=\frac{2}{9}$

Simplify $\frac{240}{90}$

Solution

Method 1. Using the repeated division for simplifying fractions.

We first notice that both 240 and 90 are divisible by 10, hence dividing by 10 we get,

$\frac{240}{90}=\frac{24×10}{9×10}=\frac{24}{9}$

Now 24 and 9 are both dividable by 3, so dividing by 3 we get,

$\frac{24}{9}=\frac{3×8}{3×3}=\frac{8}{3}$

Next, 8 and 3 have no ore common factors, thus $\frac{8}{3}$is the simplest form of the given fraction.

Method 2. Using the greatest common divisor of the numerator and the denominator.

The factors of 240 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, and 240.

The factors of 90 are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.

We notice that the greatest common divisor of 240 and 90 is 30.

Dividing both the numerator and the denominator by 30, we get

$\frac{240}{90}=\frac{8×30}{3×30}=\frac{8}{3}$.

### Simplifying mixed fractions examples

Simplify $3\frac{10}{15}$

Solution

Firstly, we must turn $3\frac{10}{15}$ into an improper fraction. We can do this by writing the integer part of the mixed fraction as a fraction with the same denominator as the fractional part.

$3\frac{10}{15}=\frac{3×15}{15}+\frac{10}{15}=\frac{45}{15}+\frac{10}{15}=\frac{55}{15}$

The final step is to simplify the improper fraction using either the repeated division method or the greatest common divisor method. Using either of these methods, we find that the simplified fraction is $\frac{11}{3}.$

Therefore

$3\frac{10}{15}=\frac{11}{3}$

Simplify $4\frac{32}{60}$

Solution

Firstly, we must turn $4\frac{32}{60}$ into an improper fraction. To do this, we can again express the integer part of the mixed fraction as a fraction with the same denominator as the fractional part.

$4\frac{32}{60}=\frac{4×60}{60}+\frac{32}{60}=\frac{240}{60}+\frac{32}{60}=\frac{272}{60}$

Again, the final step is to simplify the improper fraction using either the repeated division method or the greatest common divisor method. Using either of these methods, we find that the simplified fraction is $\frac{68}{15}$.

Therefore,

$4\frac{32}{60}=\frac{68}{15}$

Simplify $12\frac{12}{30}$

Firstly, we turn $12\frac{12}{30}$ into an improper fraction. We do this by expressing the integer part of the mixed fraction as a fraction with the same denominator as the fraction part.

$12\frac{12}{30}=\frac{12×30}{30}+\frac{12}{30}=\frac{360}{30}+\frac{12}{30}=\frac{372}{30}$

Finally, we simplify the improper fraction with either the repeated divisor method or the greatest common divisor method. Using either of these methods we find that the simplified fraction is $\frac{62}{5}$.

Therefore,

$12\frac{12}{30}=\frac{62}{5}$

### Simplifying fractions with exponents examples

Simplify $\frac{{2}^{8}{3}^{2}{7}^{5}{11}^{2}}{{2}^{5}{3}^{2}{5}^{5}{11}^{3}}$.

Solution

As stated earlier in the article, when simplifying fractions with exponents in the numerator and denominator, we use the Greatest Common Divisor method.

When having the same base, the lowest exponent is the common factor to be considered.

Hence, the Greatest Common Divisor of ${2}^{8}{3}^{2}{7}^{5}{11}^{2}$ and ${2}^{5}{3}^{2}{5}^{5}{11}^{3}$ is ${2}^{5}{3}^{2}{11}^{2}$.

Next, dividing both the numerator and denominator by the GCD, we get

Thus, the simplest form of the given fraction is

$\frac{{2}^{3}{7}^{5}}{{5}^{5}11}$.

Simplify $\frac{{3}^{4}{5}^{2}{8}^{4}{9}^{8}}{{3}^{2}{8}^{2}5}$

Solution

The greatest common divisor of the numerator and denominator is ${3}^{2}{8}^{2}5$. Dividing both the numerator and denominator by the greatest common divisor gives

${3}^{2}{8}^{2}{9}^{8}5$

Simplify $\frac{{4}^{15}{3}^{4}{10}^{3}{5}^{4}}{{4}^{12}{3}^{2}{10}^{2}{9}^{3}}$

Solution

The greatest common divisor of the numerator and denominator in this case is ${4}^{12}{3}^{2}{10}^{2}$. Dividing both the numerator and denominator by the greatest common divisor gives

$\frac{{4}^{3}{3}^{2}{5}^{4}10}{{9}^{3}}$

### Simplifying fractions with variables examples

Simplify $\frac{12{b}^{5}{c}^{2}}{30a{b}^{3}c}$

Solution

As stated earlier in the article, when simplifying fractions with variables in the numerator and denominator, we use the Greatest Common Divisor method.

When having the same base, the lowest exponent is the common factor to be considered.

Hence, the Greatest Common Divisor of $12{\mathrm{b}}^{5}{\mathrm{c}}^{2}$ and $30{\mathrm{ab}}^{3}\mathrm{c}$ is $6{\mathrm{b}}^{3}\mathrm{c}$.

Next, by dividing both the numerator and denominator by $6{\mathrm{b}}^{3}\mathrm{c}$, we get

Thus, the simplest form of the given fraction is

$\frac{2{b}^{2}c}{5a}$

Simplify $\frac{14{a}^{3}b}{21ac}$

Solution

As stated earlier in the article, when simplifying fractions with variables in the numerator and denominator, we use the Greatest Common Divisor method.

When having the same base, the lowest exponent is the common factor to be considered.

Hence, the GCD of$14{\mathrm{a}}^{3}\mathrm{b}$and $21\mathrm{ac}$ is $7a$.

Next, by dividing both the numerator and denominator by $7a$ we get,

Hence, the simplest form of the given fraction is,

$\frac{2{a}^{2}b}{3c}$

Simplify $\frac{49{x}^{2}{y}^{3}}{35{y}^{2}{z}^{2}}$

Solution

As stated earlier in the article, when simplifying fractions with variables in the numerator and denominator, we use the Greatest Common Divisor method.

When having the same base, the lowest exponent is the common factor to be considered.

Hence, the Greatest Common Divisor of $49{\mathrm{x}}^{2}{\mathrm{y}}^{3}$and $35{\mathrm{y}}^{2}{\mathrm{z}}^{2}$is $7{\mathrm{y}}^{2}.$

By dividing both the numerator and denominator by the GCD, we get

Hence, the simplest form of the given fraction is,

$\frac{7{x}^{2}y}{5{z}^{2}}$

## Simplifying Fractions - Key takeaways

• A fraction is in its simplest form if there are no more common factors between its numerator and denominator.
• We can reduce a fraction to its simplest form by repeatedly dividing both the numerator and the denominator by the lowest prime common factor until no such factor remains.
• We can reduce a fraction to its simplest form by dividing both the numerator and the denominator by their Greatest Common Divisor.

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How to simplify fractions?

To simplify fractions, you can divide both the numerator and denominator by their Greatest Common Divisor

How do you simplify fractions with mixed numbers?

To simplify fractions with mixed numbers, convert the mixed fraction into a pure fraction and then divide the numerator and denominator by their GCD.

How to simplify fractions with division?

To simplify fractions using division, repeatedly divide both the numerator and the denominator by the lowest prime number that is a common factor. Keep repeating this step until no other common prime factor remains.

How to simplify improper fractions?

To simplify improper fractions, you can divide both the numerator and denominator by their Greatest Common Divisor.

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