How to Simplify an Improper Fraction

Co-authored by David Jia

Last Updated: July 17, 2024 Fact Checked

Fractions are numbers that represent parts of whole numbers. If a fraction has a numerator greater than its denominator it is termed an “improper fraction” and can be simplified as a mixed number (a number that combines a whole number and a fraction). There is nothing wrong with an improper fraction, and in fact in mathematics it is often easier to work with than a mixed number; however, in our daily lives we use mixed numbers more than improper fractions, ^{[1]
X
Research source} so it is helpful to know how to create them.

Method 1

Method 1 of 2:

Using a Model

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1
Determine whether your fraction is improper. An improper fraction is a fraction that has a larger numerator than denominator. ^{[2]
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Research source}
- For example, ${\frac {10}{4}}$ is an improper fraction, because ten is greater than 4.
2
Interpret the denominator. The denominator is the number below the fraction bar. It tells you how many equal pieces a whole is divided into.
- For example, in the fraction ${\frac {10}{4}}$ , 4 is the denominator, and it tells you that a whole is divided into four equal parts, or quarters.
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3
Interpret the numerator. The numerator is the number above the fraction bar. It tells you how many pieces you have.
- For example, in the fraction ${\frac {10}{4}}$ , 10 is the numerator, and it tells you that you have 10 parts, or 10 quarters.
4
Draw circles to represent the whole. Divide each whole according to the denominator of your fraction.
- For example, if your denominator is 4, then divide each circle you draw into 4 equal pieces, or quarters.
5
Shade in pieces according to your numerator. The number in the numerator tells you how many pieces you should shade in.
- For example, if your fraction is ${\frac {10}{4}}$ , you will shade in 10 quarters.
6
Count how many whole circles you shaded in. To simplify an improper fraction, you must turn it into a mixed number, which includes a whole number and a fraction together. The number of whole circles you shaded in represents the whole number of your mixed fraction. Write this number down.
- For example, for the fraction ${\frac {10}{4}}$ , you should have shaded in 2 whole circles, so the whole number of your mixed number will be 2.
7
Count how many parts of a whole you shaded in. The leftover shaded parts will represent the fraction in your mixed number. Write this fraction next to your whole number, and you have your mixed number.
- For the fraction ${\frac {10}{4}}$ , you should have shaded in ${\frac {2}{4}}$ of a circle, so the fraction part of your mixed number will be ${\frac {2}{4}}$ . So ${\frac {10}{4}}$ is equal to $2{\frac {2}{4}}$ .
8
Simplify your answer, if necessary. Sometimes the fraction of your mixed number will need to be reduced before you reach your final, simplified answer. ^{[3]
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Research source}
- For example, if your mixed number is $2{\frac {2}{4}}$ , you could simplify it to $2{\frac {1}{2}}$

Method 2

Method 2 of 2:

Using Division

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1
Determine whether your fraction is improper. An improper fraction is a fraction that has a larger numerator than denominator. ^{[4]
X
Research source}
- For example, ${\frac {10}{4}}$ is an improper fraction, because $10>4$ .
2
Divide the numerator by the denominator. Remember that the fraction bar can be interpreted as a division symbol. ^{[5]
X
Research source} To simplify an improper fraction, you must turn it into a mixed number, which includes a whole number and a fraction together. The number of times you can divide the numerator evenly by the denominator will be the whole number of your mixed number. Write this number down, and note the remainder.
- The denominator will not divide evenly into the numerator. The remainder will be interpreted as the fraction part of your mixed number.
- For example, for the fraction ${\frac {10}{4}}$ complete the calculation $10\div 4=2R2$ . So the whole number of your fraction will be $2$ .
3
Turn the remainder into a fraction. To do this, take the remainder, and place it over the denominator of the original improper fraction. Combine this new fraction with the whole number, and you have your mixed number.
- For example, $10\div 4=2R2$ , so the fraction would be ${\frac {2}{4}}$ . So ${\frac {10}{4}}$ is equal to $2{\frac {2}{4}}$ .
4
Simplify your answer, if necessary. Sometimes the fraction of your mixed number will need to be reduced before you reach your final, simplified answer. ^{[6]
X
Research source} Divide the numerator and denominator of the fraction by their greatest common factor.
- For example, if your mixed number is $2{\frac {2}{4}}$ , you could simplify it to $2{\frac {1}{2}}$ .
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Joseph Meyer
Math Teacher
Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.

Joseph Meyer
Math Teacher

To simplify fractions, you can divide both the numerator and denominator by a common factor. This creates a new, easier-to-use fraction with smaller components, but it represents the same value. For instance, if you divide both the numerator and denominator of 6/12 by 2, you get 3/6, which is equal to 1/2.

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What if I have a big numerator, like 124?

Community Answer

As long as the denominator is less than 124 you have an improper fraction, and you can use the methods presented here to solve. The division method would be easiest, since drawing a model with 124 pieces would take some time.

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What if I want to simplify it without making it a mixed number? Like simplifying it into a regular fraction?

Donagan

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You cannot convert an improper fraction into a proper fraction.

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How do I reduce 42 by 18 5/9?

Donagan

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First change 42 to 41 9/9. Then subtract 18 from 41, and subtract 5/9 from 9/9. The difference is 23 4/9.

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To convert a mixed numbers back to improper fraction form, multiply the whole number by the denominator and add the product to the numerator.

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Keep your denominator. For example, $2{\frac {1}{2}}$ can be rewritten as ${\frac {5}{2}}$ , because $2\times 2+1=5$ .

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Some improper fractions can also represent whole numbers, such as ${\frac {24}{3}}$ .

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How to Simplify an Improper Fraction

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Using a Model

Using Division

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