How to simplify fractions

A fraction is considered to be “simplified” when it is expressed in the lowest term. That means the only common divisor between the numerator and denominator is 1, and no other.

METHODS IN SIMPLIFYING FRACTIONS

Method 1: Simplify Fractions by Repeated Division

  • Keep dividing the numerator and denominator by a common divisor until such time that the only remaining common divisor is 1.
  • Although there is no right way which common divisor to use in the beginning, I would suggest using the first five (5) prime numbers in order as possible common divisor:

2, 3, 5, 7, 11, …

Method 2: Simplify Fractions using the Greatest Common Factor

  • Find the greatest common factor (GCF) of the numerator and denominator.
  • Divide the top and bottom numbers of the fraction by the GCF to reduce to the lowest term.
  • You can find the GCF either by trial and error when the numbers are relatively small, or using Prime Factorization.

This is a simple illustration showing the fraction {8 over {12}} is being reduced to its simplest form. Can you see a pattern?

How to Simplify Fractions

Let’s go over a few more examples with detailed explanations.

Examples of How to Simplify Fractions

Example 1: Simplify the fraction below.

Simplify using Method 1: Repeated Division Method

It is obvious that 1 is not the only common divisor between the numerator and denominator. Since they are both even numbers, they must be divisible by 2.

  • Divide the top and bottom by 2. Here’s what we got after doing so.

How to Simplify Fractions

The output fraction after dividing the top and bottom by 2 is {2 over 4}. Can we stop here? Not yet! They can still be reduced by a second division of 2.

  • Divide again the top and bottom by 2. The answer is {1 over 2} (as the simplest form of {4 over 8} because the only divisor of its numerator and denominator is 1.

How to Simplify Fractions

Simplify using Method 2: Greatest Common Factor Method

In the above solution using repeated division, we have simplified {4 over 8} by dividing its numerator and denominator two times by the number 2. But wait! Is there a shortcut? Some of you may have observed that using a common divisor of 4 can directly simplify it with a single step!

In fact, the Greatest Common Factor (GCF) of this fraction is 4 because it is the LARGEST number that evenly divides the numerator and denominator. Because the numbers are small, the GCF can be determined by trial and error.

How to Simplify Fractions

Example 2: Simplify the fraction below.

Simplify using Method 1: Repeated Division Method

Start simplifying using the first few prime numbers (2, 3, 5, 7, 11, etc).

  • Divide the top and bottom numbers by the first prime number which is 2.

How to Simplify Fractions

  • We still have a common divisor! Divide the top and bottom by the next larger prime number which is 3. We should get the final answer after this step.

How to Simplify Fractions

Simplify using Method 2: Greatest Common Factor Method

To find the greatest common divisor, we are going to perform prime factorization on each number. Next, identify the common factors between them. Finally, multiply the common factors to get the required GCF that can simplify the fraction.

How to Simplify FractionsHow to Simplify Fractions

Since GCF = 6, use this number to divide the numerator and denominator to get the answer in a single step.

Example 3: Simplify the fraction below.

Simplify using Method 1: Repeated Division Method

We can start testing numbers 2, 3, 5, etc. to simplify this. But there is an obvious divisor that stands out! Since both numbers end with zero, they should be divisible by 10.

Now, 2 can’t divide both and so try 3.

Simplify using Method 2: Greatest Common Factor Method

Prime factorize each number and get the product of the common factors to obtain the needed GCF.

Simplify the given fraction in one-step using the divisor GCF = 30.

Example 4: Simplify the fraction below.

Solution:

Divide the numerator and denominator by a common divisor of 3.

Example 5: Simplify the fraction .

Solution:

Simplify using repeated division method.

  • Divide both numerator and denominator by 3, two times!

Example 6: Simplify the fraction below.

Solution:

Simplify this fraction by the greatest common factor method.

  • Find the GCF by prime factoring both the numerator and denominator. Identify the common factors. Multiply them together to get the required GCF.
  • After determining the GCF, divide the numerator and denominator to get the final answer.

Example 7: Simplify the fraction below.

Solution:

Find the greatest common factor between the numerator and denominator, and use this number to simplify the fraction.

  • Divide the numerator and denominator by GCF = 21.

Practice with Worksheets

You might also be interested in:

Adding and Subtracting Fractions with the Same Denominator Add and Subtract Fractions with Different Denominators Multiplying Fractions Dividing Fractions Equivalent Fractions Reciprocal of a Fraction

OPERATIONS WITH FRACTIONS

  • Enter an expression and click the Reduce fraction button.
  • The product of two fractions is defined as follows.
  • The product of two fractions is a fraction whose numerator is the product of the numerators and whose denominator is the product of the denominators of the given fractions.
  • In symbols,

Any common factor occurring in both a numerator and a denominator of either fraction can be divided out either before or after multiplying.

How to Simplify Fractions

The same procedures apply to fractions containing variables.

Solution First, we divide the numerator and denominator by the common factors to get

How to Simplify Fractions

Now, multiplying the remaining factors of the numerators and denominators yields

How to Simplify Fractions

If a negative sign is attached to any of the factors, it is advisable to proceed as if all the factors were positive and then attach the appropriate sign to the result. A positive sign is attached if there are no negative signs or an even number of negative signs on the factors; a negative sign is attached if there is an odd number of negative signs on the factors.

How to Simplify Fractions

When the fractions contain algebraic expressions, it is necessary to factor wherever possible and divide out common factors before multiplying.

How to Simplify Fractions

Solution First, we must factor the numerators and denominators to get

How to Simplify Fractions

Now, dividing out common factors yields

How to Simplify Fractions

We now multiply the remaining factors of the numerators and denominators to obtain

Note that when writing fractional answers, we will multiply out the numerator and leave the denominator in factored form. Very often, fractions are more useful in this form.

In algebra, we often rewrite an expression such as as an equivalent expression .Use whichever form is most convenient for a particular problem.

  1. Example 5
  2. Common Errors: Remember that we can only divide out common factors, never common terms! For example,
  3. because x is a term and cannot be divided out. Similarly,
  4. because 3 is not a factor of the entire numerator 3y + 2.

QUOTIENTS OF FRACTIONS

In dividing one fraction by another, we look for a number that, when multiplied by the divisor, yields the dividend. This is precisely the same notion as that of dividing one integer by another; a ÷ b is a number q, the quotient, such that bq = a.

To find , we look for a number q such that . To solve this equation for q, we multiply each member of the equation by . Thus,

In the above example, we call the number the reciprocal of the number . In general, the reciprocal of a fraction is the fraction . That is, we obtain the reciprocal of a fraction by “inverting” the fraction. In general,

  • The quotient of two fractions equals the product of the dividend and the reciprocal of the divisor.
  • That is, to divide one fraction by another, we invert the divisor and multiply. In symbols,
  • Example 1
  • As in multiplication, when fractions in a quotient have signs attached, it is advisable to proceed with the problem as if all the factors were positive and then attach the appropriate sign to the solution.
  • Example 2
  • Some quotients occur so frequently that it is helpful to recognize equivalent forms directly. One case is
  • In general,
  • Example 3
  • When the fractions in a quotient involve algebraic expressions, it is necessary to factor wherever possible and divide out common factors before multiplying.
  • Example 4
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How to Simplify Fractions

How to Simplify Fractions

  • Welcome to the Math Salamanders support page about How to Simplify Fractions.
  • Here you will find information and support about how to convert a fraction into its simplest form.

We have a simplify fraction calculator which will convert any fraction into its simplest form.

The calculator will also show you detailed working out to show how to get the answer.

How to Simplify Fractions

  • Simplify Fraction Calculator

This short video clip will show you how to reduce 4 different fractions into simplest form.

  1. Here you will find some simple information and advice about how to simplify a fraction.
  2. At the bottom of this page you will also find a printable resource sheet and some practice sheets which will help you understand and practice this math skill.
  3. Using these sheets will help your child to:
  • understand simplest form;
  • develop their knowledge of equivalent fractions.

All the Printable Fraction sheets in this section support Elementary Math Benchmarks for 4th Grade.

To simplify a fraction, you need to follow the 2 steps below:

How to Simplify Fractions

Step 1

Look for a common factor of both the numerator and denominator.

  • If there are no common factors (other than 1) then the fraction is already in its simplest form and you have finished.
  • Try to find the highest factor of both numbers (highest common factor) that you can!

Step 2

Reduce the numerator and denominator by dividing them both by the common factor. Now go back to Step 1.

Your fraction should now be in its simplest form.

Convert Fractions to Simplest Form printable sheet

  • Example 1) Convert into simplest form the fraction [ {35 over 45} ]
  • A common factor of both numbers is 5, so we need to divide the numerator and denominator by 5.
  • This gives us [ {35 over 45} ; = ; {35 ÷ 5 over 45 ÷ 5 } ; = ; {7 over 9} ]
  • There are no more common factors (except 1) and so this fraction is now in simplest form.
  • Final answer [ {35 over 45} ; = ; {7 over 9} ]
  • Example 2) Convert into simplest form the fraction [ {68 over 220} ]
  • A common factor of both numbers is 2, so we need to divide the numerator and denominator by 2.
  • This gives us [ {68 over 220} ; = ; {68 ÷ 2 over 220 ÷ 2 } ; = ; {34 over 110} ]
  • We are not finished yet, because 2 is still a common factor of both numbers.
  • Canceling the common factor of 2 gives us [ {34 over 110} ; = ; {34 ÷ 2 over 110 ÷ 2 } ; = ; {17 over 55} ]
  • There are no more common factors (except 1) and so this fraction is now in simplest form.
  • Final answer [ {68 over 220} ; = ; {17 over 55} ]
  • Note – a quicker way to simplify this fraction would be to have divided the numerator and denominator by 4 as this is a higher common factor than 2.
  • Example 3) Convert into simplest form the fraction [ {120 over 75} ]
  • A common factor of both numbers is 5, so we need to divide the numerator and denominator by 5.
  • This gives us [ {120 over 75} ; = ; {120 ÷ 5 over 75 ÷ 5 } ; = ; {24 over 15} ]
  • We are not finished yet, because 3 is a common factor of both numbers.
  • Canceling the common factor of 3 gives us [ {24 over 15} ; = ; {24 ÷ 3 over 15 ÷ 3 } ; = ; {8 over 5} ]
  • There are no more common factors (except 1) and so this fraction is now in simplest form.
  • Final answer [ {120 over 75} ; = ; {8 over 5} ]
  • Note – we could have simplified this fraction more quickly by seeing that the highest common factor was 15, and divided the numerator and denominator by this number first.
  1. Take a look at our NEW Simplifying Fractions Practice Zone for finding the simplest form for a range of fractions.
  2. You can choose from proper fractions, improper fractions or both.
  3. You can print out your results or benchmark your scores against future achievements.
  4. Great for using with a group of children as well as individually.

How to Simplify Fractions

  • Simplify Fractions Practice Zone
  • Simplifying Fractions Worksheet page

Greatest Common Factor Calculator

  • Our Greatest Common Factor calculator will tell you the highest common factor between 2 or more numbers.
  • If you type in the numerator and denominator of a fraction you want to simplify, the calculator will tell you which is the highest factor to divide them both by.
  • This will allow you to convert the fraction to simplest form in one quick go!
  • It will also list the factors of each of the numbers and tell you whether they are coprime or not.

How to Simplify Fractions

  • Greatest Common Factor Calculator

Here you will find the Math Salamanders free online Math help pages about Fractions.

There is a wide range of help pages including help with:

  • fraction definitions;
  • equivalent fractions;
  • converting improper fractions;
  • how to add and subtract fractions;
  • how to convert fractions to decimals and percentages;
  • how to simplify fractions.

How to Simplify Fractions

  • Learning Fractions Support

How to Print or Save these sheets

Need help with printing or saving? Follow these 3 easy steps to get your worksheets printed out perfectly!

How to Print or Save these sheets

Need help with printing or saving? Follow these 3 easy steps to get your worksheets printed out perfectly!

  1. The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

Simplification of Fractions | How to Simplify Fraction? | reducing fractions

In simplification of fractions parenthesis can also be used. The three parenthesis (1st), {2nd}, [3rd] are used commonly.

  • Examples on simplification of fractions:
  • 1. 3 1/3 ÷ 5/3 – 1/10 of 2 ½ + 7/4
  • Solution:
  • = 10/3 ÷ 5/3 – 1/10 of 5/2 + 7/4

3 1/3 ÷ 5/3 – 1/10 of 2 ½ + 7/4= (3 × 3 + 1)/3 ÷ 5/3 – 1/10 of (2 × 2 + 1)/2 + 7/4
How to Simplify Fractions

                      [‘of’ simplified]

= 10/3 × 3/5 – ½ × ½ + 7/4                  [‘÷’ simplified]

How to Simplify Fractions

= 2/1 – ¼ + 7/4                   [‘×’ simplified]= (2 × 4)/1 × 4) – (1 × 1)/4 × 1) + (7 × 1)/4 × 1)= 8/4 – ¼ + 7/4[Now the denominators are same of all the fractions]= (8 – 1 + 7)/4                  [‘+’ and ‘-‘ simplified]= 14/4= 7/2

  1. 2. 45 of 3/5 ÷ 1 2/3 + 3 of 1/3 – 10
  2. Solution:
  3. = 45 × 3/5 ÷ 5/3 + 3 × 1/3 – 10                [‘of’ simplified]
See also:  The mighty onion

45 of 3/5 ÷ 1 2/3 + 3 of 1/3 – 10= 45 of 3/5 ÷ (1 × 3 + 2)/3 + 3 of 1/3 – 10= 45 of 3/5 ÷ 5/3 + 3 of 1/3 – 10
How to Simplify Fractions

  • = 9 × 3 × 3/5 + 3 × 1/3 – 10             [‘÷’ simplified],  [‘×’ simplified]
  • = (27 × 3)/5 + 1 – 10
  • = 81/5 + 1 – 10= (81 × 1)/(5 × 1) + (1 × 5)/(1 × 5) – (10 × 5)/(1 × 5)= 81/5 + 5/5 – 50/5[Now the denominators are same of all the fractions]= (81 + 5 – 50)/5                     [‘+’ and ‘-‘ simplified]
  • = 36/5

= 7 1/5

3.

  1. 43 of 1/86 ÷ 1/14 × 2/7 + 9/4 – ¼
  2. Solution:
  3. = 43 × 1/86 ÷ 1/14 × 2/7 + 9/4 – ¼

43 of 1/86 ÷ 1/14 × 2/7 + 9/4 – ¼
How to Simplify Fractions

  • = 2/1 + 9/4 – ¼
  • = (2 × 4)/1 × 4) + (9 × 1)/4 × 1) – (1 × 1)/4 × 1)= 8/4 + 9/4 – 1/4[Now the denominators are same of all the fractions]= (8 + 9 – 1)/4= 16/4 = 4
  • 4. 9/10 ÷ (3/5 + 2 1/10)
  • Solution:
  • = 9/10 × 10/27

9/10 ÷ (3/5 + 2 1/10)= 9/10 ÷ (3/5 + 21/10)= 9/10 ÷ ((6 +21)/10)[Solve within brackets]= 9/10 ÷ 27/10
How to Simplify Fractions

  1. = 1/3
  2. 5. (7 ¼ – 6 1/4) of (2/5 + 3/15)
  3. Solution:
  4. = 4/4 × 9/15

(7 ¼ – 6 1/4) of (2/5 + 3/15)= (29/4 – 25/4) of (2/5 + 3/15)= ((29 – 25)/4) × ((6 + 3)/15)[Solve within brackets]
How to Simplify Fractions

  •           [Reduce to lowest term]
  • = 1 × 3/5
  • = 3/5
  • 6. {18 + (2 ½ + 4/5)} of 1/1000
  • Solution:
  • These are the examples of simplification of fractions.

{18 + (2 ½ + 4/5)} of 1/1000= {18 + (5/2 + 4/5)} of 1/1000 = {18 + ((25 + 8)/10)} of 1/1000 = {18 + 33/10} of 1/1000= {(180 + 33)/10} of 1/1000= 213/10 of 1/1000= 213/10 × 1/1000= (213 × 1)/(10 × 1000)= 213/10000 = 0.0213


  1. Multiplication
    is Repeated Addition.
  2. ● Multiplication
    of Fractional Number by a Whole Number.
  3. ● Multiplication
    of a Fraction by Fraction.
  4. ● Properties
    of Multiplication of Fractional Numbers.
  5. ● Multiplicative
    Inverse.
  6. ● Worksheet
    on Multiplication on Fraction.
  7. ● Division
    of a Fraction by a Whole Number.
  8. ● Division
    of a Fractional Number.
  9. ● Division
    of a Whole Number by a Fraction.
  10. ● Properties
    of Fractional Division.
  11. ● Worksheet
    on Division of Fractions.
  12. ● Simplification
    of Fractions.
  13. ● Worksheet
    on Simplification of Fractions.
  14. ● Word
    Problems on Fraction.
  15. ● Worksheet
    on Word Problems on Fractions.
  16. 5th Grade Numbers Page
  17. 5th Grade Math Problems
  18. From Simplification of Fractions to HOME PAGE

How to Simplify Fractions: Lowest Terms Fractions

In this post, we are going to learn how we can calculate the lowest terms fractions by reducing them.
Before we start, let’s see what lowest terms fractions are: lowest terms fractions are fractions that cannot be simplified further.
How do we find the lowest terms of a fraction? There are two methods:

  • Method 1: Divide the numerator and denominator by a common factor between them until there are no more common factors. Let’s look at an example:

How to Simplify Fractions

Let’s reduce the fraction 28/42. Both the numerator and denominator can be divided by 2. This would leave us at 14/21 and since 14 and 21 are multiples of 7, we can divide by 7. In dividing 14 and 21 by 7, we end up with 2/3, which is in the lowest terms since there is no longer a common factor between numerator and denominator.

  • Method 2: Divide the numerator and denominator by the greatest common factor (GCF). Let’s see how we would reduce 90/120 with this method:

How to Simplify Fractions

We calculate the greatest common factor between 90 and 120. As shown above, we take the common prime factors of 90 and 120 (including repeats), which are 2, 3 and 5, and multiply them: 2 x 3 x 5 = 30.

How to Simplify Fractions

Let’s look at a trick to put fractions in lowest terms.

As shown in the image below, we will break down both the numerator (90) and the denominator (120) into prime factors and write the fraction with all of the factors.

Cross out the factors found both in the numerator and denominator, and multiply the factors that are not crossed out. In the end, we have 3/4, which is the same result obtained using method 2.

How to Simplify Fractions

If you want to learn more about fractions or elementary math, you can try Smartick for free.

Learn More:

  • Learn and Practice How to Subtract or Add Fractions
  • Review Factoring with Examples
  • Greatest Common Factor (GCF)
  • Two Ways of Dividing Fractions and Some Examples
  • Practice Adding Fractions with Examples

Simplifying Fractions

How to Simplify FractionsProblem: Josephine ate four-eighths of a pie and Penelope ate six-twelfths of a pie. If both pies are the same size, then which girl ate more pie?

  • Analysis:
  • ?
  • We need to simplify these fractions in order to compare them more easily.

The numerator and denominator of a fraction are called its terms. If we simplify a fraction, then we are reducing it to lowest terms. Reducing a fraction to lowest terms will not change its value; the reduced fraction will be an equivalent fraction. All we need to do is divide the numerator and the denominator by the same nonzero whole number. This is shown below.

Solution: Since four-eighths and six-twelfths can both be reduced to one-half, these fractions are equivalent. Therefore, Josephine and Penelope both ate the same amount of pie (one-half).

 = 

Dividing the numerator and the denominator of a fraction by the same nonzero whole number does not change the value of a fraction. This is because dividing by one does not change the value of a number.

Definition: A fraction is in lowest terms when the greatest common factor (GCF) of the numerator and denominator is 1.

To simplify a fraction (reduce it to lowest terms), the numerator and the denominator must be divided by the same nonzero whole number. Let's look at some examples of reducing a fraction to lowest terms.

Example 1:

How to Simplify Fractions

Method 1:

How to Simplify Fractions

Answer:

How to Simplify Fractions

With method 1, it can take several steps to reduce a fraction to lowest terms. Let's see what happens when we start by dividing the numerator and the denominator by their greatest common factor, instead of by their lowest common factor.

  1. Example 1:
  2. Method 2:
  3. Answer:

With method 2, we found the greatest common factor of the numerator and the denominator. We then divided the numerator and denominator by their GCF. With this method it takes only two steps to reduce a fraction to lowest terms. So, method 2 is more efficient for reducing a fraction to lowest terms. The two methods for simplifying fractions are summarized below.

Reducing a Fraction to Lowest Terms (Simplest Form)

Method 1:

  1. Find the common factors of the numerator and the denominator.
  2. Divide the numerator and the denominator by their lowest common factor.
  3. Find the common factors of the numerator and denominator in the resulting fraction.
  4. Keep dividing by a common factor until there are no common factors other than 1

Method 2:

  1. Find the GCF of the numerator and the denominator.
  2. Divide the numerator and the denominator by their greatest common factor (GCF).

For the remainder of this lesson, we will use method 2 to reduce a fraction to lowest terms. Let's look at some more examples.

  • Example 2:
  • Solution:
  • Answer:
  • Example 3:
  • Solution:
  • Answer:
  • Example 4:
  • Solution:
  • Answer:
  • Example 5:
  • Solution:
  • Answer:
See also:  How 8 fonts got their names

Summary: To simplify a fraction (reduce it to lowest terms), the numerator and the denominator must be divided by the same nonzero whole number. A fraction is in lowest terms when the greatest common factor (GCF) of its numerator and denominator is one.

Exercises

In Exercises 1 through 5, click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Note: To write the fraction two-thirds, enter 2/3 into the form.

1.
2.
3.
4.
5.

How to Reduce a Fraction to Its Lowest Terms

Even if fractions look different, they can actually represent the same amount; in other words, one of the fractions will have reduced terms compared to the other. You may need to reduce the terms of fractions to work with them in an equation.

Reducing fractions to their lowest terms involves division. But because you can’t always divide, reducing takes some finesse.

Here you will learn the formal way to reduce fractions, which works in all cases. Then you will learn a more informal way that you can use after you’re more comfortable.

Method 1: Reduce fractions the formal way

Reducing fractions the formal way relies on an understanding of how to break down a number into its prime factors.

Here’s how to reduce a fraction:

    • Break down both the numerator (top number) and denominator (bottom number) into their prime factors.
    • For example, suppose you want to reduce the fraction 12/30. Break down both 12 and 30 into their prime factors:
    1. Cross out any common factors.
    2. In this example, you cross out a 2 and a 3, because they’re common factors — that is, they appear in both the numerator and denominator:
  1. Multiply the remaining numbers to get the reduced numerator and denominator.

    This shows you that the fraction12/30 reduces to 2/5:

    How to Simplify Fractions

As another example, here’s how you reduce the fraction 32/100:

How to Simplify Fractions

This time, cross out two 2s from both the top and the bottom as common factors. The remaining 2s on top, and the 5s on the bottom, aren’t common factors. So the fraction 32/100 reduces to 8/25.

Method 2: Reduce fractions the informal way

Here’s an easier way to reduce fractions after you get comfortable with the concept:

    • If the numerator (top number) and denominator (bottom number) are both divisible by 2 — that is, if they’re both even — divide both by 2.
    • For example, suppose you want to reduce the fraction 24/60. The numerator and the denominator are both even, so divide them both by 2:
    1. Repeat Step 1 until the numerator or denominator (or both) is no longer divisible by 2.
    2. In the resulting fraction, both numbers are still even, so repeat the first step again:
    • Repeat Step 1 using the number 3, and then 5, and then 7, continuing testing prime numbers until you’re sure that the numerator and denominator have no common factors.
    • Now, the numerator and the denominator are both divisible by 3, so divide both by 3:

    Neither the numerator nor the denominator is divisible by 3, so this step is complete. At this point, you can move on to test for divisibility by 5, 7, and so on, but you really don’t need to. The numerator is 2, and it obviously isn’t divisible by any larger number, so you know that the fraction 24/60 reduces to 2/5.

Simplify Fractions

Use this page to reduce a fraction to it's lowest terms. Enter the numerator and denominator as integer numbers. Then press the “Simplify Fraction” button, to calculate and display the simplified fraction.

You may enter positive or negative numbers for both numerator and denominator as long as their value is between -2147483648 and 2147483647.

This calculator may generate improper fractions, that is, fractions that have the numerator larger than their denominator.

How are fractions reduced to their lowest terms

Fractions can be simplified by dividing the numerator and denominator by their greatest common factor. This is how this calculator does it.

When reducing fractions by hand it may be easier to repeatedly divide numerator and denominator by factors that are common to both of them. The process is complete when the numerator and the denominator have no more factors in common.

Reduced fractions are fractions that have been reduced to their simplest form.
When the GCF of the numerator and denominator is equal to 1, the fraction cannot be reduced any further.

Simplifying Fractions Calculator

Convert improper fractions to mixed numbers in simplest form. This calculator also simplifies proper fractions by reducing to lowest terms and showing the work involved.

In order to simplify a fraction there must be:

  1. A number that will divide evenly into both the numerator and denominator so it can be reduced, or
  2. The numerator must be greater than the denominator, (an improper fraction), so it can be converted to a mixed number.

What is an Improper Fraction?

An improper fraction is any fraction where the numerator is greater than the denominator. Examples of improper fractions are 16/3, 81/9, 525/71.

How to Convert an Improper Fraction to a Mixed Number

  1. Divide the numerator by the denominator
  2. Write down the whole number result
  3. Use the remainder as the new numerator over the denominator. This is the fraction part of the mixed number.

Example: Convert the improper fraction 16/3 to a mixed number.

  1. Divide 16 by 3: 16 ÷ 3 = 5 with remainder of 1
  2. The whole number result is 5
  3. The remainder is 1. With 1 as the numerator and 3 as the denominator, the fraction part of the mixed number is 1/3.
  4. The mixed number is 5 1/3. So 16/3 = 5 1/3.

When possible this calculator first reduces an improper fraction to lowest terms before finding the mixed number form.

Example: Convert the improper fraction 45/10 to a mixed number.

  • This calculator reduces the improper fraction to lowest terms by dividing numerator and denominator by the greatest common factor (GCF). The GCF of 45 and 10 is 5.
  • ( dfrac{45div5}{10div5} = dfrac{9}{2})
  • Use this reduced improper fraction and divide 9 by 2: 9 ÷ 2 = 4 with remainder of 2
  • The whole number result is 4
  • The remainder is 1. With 1 as the numerator and 2 as the reduced denominator, the fraction part of the mixed number is 1/2.
  • The mixed number 4 1/2. So 45/10 = 4 1/2.

Related Calculators

For additional explanation of factoring numbers to find the greatest common factor (GCF) see the Greatest Common Factor Calculator.

If your improper fraction numbers are large you can use the Long Division with Remainders Calculator to find whole number and remainder values when simplifying fractions by hand.

To perform math operations on fractions before you simplify them try our Fractions Calculator. This calculator will also simplify improper fractions into mixed numbers.

Notes

This calculator performs the reducing calculation faster than other calculators you might find. The primary reason is that it utilizes Euclid's Algorithm for reducing fractions, as shown on The Math Forum.

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