Simplifying Fractions

Consider the fractions, 9521904 and 12. Both of them actually represent the exact same value. However, 12 does appear significantly simpler than 9521904. In this case, 12 is the fraction expressed in what is known as the simplest form or its lowest terms.

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    In this article, we will learn more about different methods of simplifying fractions.

    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 811 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.

    Hence811 is indeed in its simplest form.

    However, the fraction 2088 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, 213, is the sum of 2 and 13.

    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 45144.

    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

    45144=3×153×48=1548

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

    1548=3×53×16=516

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

    Hence 516 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

    45144=9×59×16=516.

    Simplify 48216

    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,

    48216=2×242×108

    We divide by 2 to get

    48216=2×242×108=24108

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

    24108=2×122×54

    We divide by 2 to get,

    24108=2×122×54=1254

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

    1254=2×62×27

    We divide by 2 to get,

    1254=2×62×27=627

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

    627=2×33×9

    Dividing by 3, we get,

    627=3×23×9=29

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

    So 29 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

    48216=2×249×24=29

    Simplify 24090

    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,

    24090=24×109×10=249

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

    249=3×83×3=83

    Next, 8 and 3 have no ore common factors, thus 83is 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

    24090=8×303×30=83.

    Simplifying mixed fractions examples

    Simplify 31015

    Solution

    Firstly, we must turn 31015 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.

    31015 = 3×1515 + 1015=4515 + 1015=5515

    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 113.

    Therefore

    31015 = 113

    Simplify 43260

    Solution

    Firstly, we must turn 43260 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.

    43260 = 4×6060 + 3260= 24060 + 3260= 27260

    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 6815.

    Therefore,

    43260 = 6815

    Simplify 121230

    Firstly, we turn 121230 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.

    121230 = 12 ×3030 + 1230= 36030 + 1230= 37230

    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 625.

    Therefore,

    121230 = 625

    Simplifying fractions with exponents examples

    Simplify 283275112253255113.

    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 283275112 and 253255113 is 2532112.

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

    283275112253255113=28327511225321122532551132532112=237555111=23 755511

    Thus, the simplest form of the given fraction is

    23 755511.

    Simplify 3452849832825

    Solution

    The greatest common divisor of the numerator and denominator is 32825. Dividing both the numerator and denominator by the greatest common divisor gives

    3282985

    Simplify 41534103544123210293

    Solution

    The greatest common divisor of the numerator and denominator in this case is 41232102. Dividing both the numerator and denominator by the greatest common divisor gives

    4332541093

    Simplifying fractions with variables examples

    Simplify 12b5c230ab3c

    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 12b5c2 and 30ab3c is 6b3c.

    Next, by dividing both the numerator and denominator by 6b3c, we get

    12b5c230 ab3c=12b5c26b3c30 ab3c6b3c=2b2c5a

    Thus, the simplest form of the given fraction is

    2b2c5a

    Simplify 14a3b21 ac

    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 of14a3band 21 ac is 7a.

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

    14a3b21ac=14a3b7a21ac7a=2a2b3c

    Hence, the simplest form of the given fraction is,

    2a2b3c

    Simplify 49x2y335y2z2

    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 49x2y3and 35y2z2 is 7y2.

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

    49x2y335y2z2=49 x2 y37 y235 y2z27y2=7x2y5z2

    Hence, the simplest form of the given fraction is,

    7x2y5z2

    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.
    Frequently Asked Questions about Simplifying Fractions

    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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