How to Tell if a Proper Fraction Is Simplified
Look for unit fractions., Determine whether the denominator is a multiple of the numerator., Determine whether the denominator is a prime number., Find the difference between the numerator and the denominator., Reduce Fractions that aren’t already...
Step-by-Step Guide
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Step 1: Look for unit fractions.
A unit fraction is one that has 1 as the numerator.
Unit fractions cannot be simplified any further.For example, 14{\displaystyle {\frac {1}{4}}}, 12{\displaystyle {\frac {1}{2}}}, 1100{\displaystyle {\frac {1}{100}}}, and 167{\displaystyle {\frac {1}{67}}} are all simplified, because they have 1 as the numerator. -
Step 2: Determine whether the denominator is a multiple of the numerator.
One way to reduce a fraction is to divide by a greatest common factor.If the numerator is a multiple of the denominator, that means each can be divided by a greatest common factor (the numerator).
These types of fractions can be reduced to a unit fraction.
For example, 26{\displaystyle {\frac {2}{6}}} is not simplified, because 6 is a multiple of
2.
The numerator and denominator can still be divided by a common factor of 2, simplifying the fraction to 13{\displaystyle {\frac {1}{3}}}.
The fraction 25{\displaystyle {\frac {2}{5}}} is simplified, because 5 is not a multiple of
2. , A prime number is a number that is only divisible by itself and
1.If the denominator is prime, the fraction cannot be simplified any further.This is because the denominator can only be divided by itself, so whatever number appears in the numerator will not have a common factor.
For more information about prime numbers, you can read Check if a Number Is Prime.
For example, 1523{\displaystyle {\frac {15}{23}}} is simplified, because 23 is a prime number.
The only factors of 23 are 23 and 1, so it is impossible to find a greatest common factor you can use to simplify the numerator and denominator. (If the numerator were 1, it would be a unit fraction and thus already simplified.
If the numerator were 23, the fraction would equal
1.) , If the difference is 1, then the fraction is simplified.
For example, you know that 78{\displaystyle {\frac {7}{8}}} is simplified, because 7−8=1{\displaystyle 7-8=1} , A fraction can be simplified by looking for the largest number that can be evenly divided into the numerator and denominator, and then dividing each by that number.
For example, if you are reducing 515{\displaystyle {\frac {5}{15}}}, divide the numerator and denominator by
5.
This will give you 13{\displaystyle {\frac {1}{3}}}, which you know cannot be reduced further because it is a unit fraction. , Do not make any calculations: 15{\displaystyle {\frac {1}{5}}}, 110{\displaystyle {\frac {1}{10}}}, 510{\displaystyle {\frac {5}{10}}}.
The fractions 15{\displaystyle {\frac {1}{5}}} and 110{\displaystyle {\frac {1}{10}}} are in their reduced, or simplified, form, because each is a unit fraction, with 1 as the numerator.
You should know that 510{\displaystyle {\frac {5}{10}}} is not simplified, because 10 is a multiple of
5. , Franny says that 12109{\displaystyle {\frac {12}{109}}} is a simplified fraction.
Without making any calculations, how do you know she is correct? Since 109 is a prime number, you can tell that the fraction is simplified. 109 is only divisible by 109 and 1, so it shares no common factors with
12. , Do not make any calculations: 45{\displaystyle {\frac {4}{5}}}, 47{\displaystyle {\frac {4}{7}}}, 48{\displaystyle {\frac {4}{8}}} Since 5−4=1{\displaystyle 5-4=1}, you know that 45{\displaystyle {\frac {4}{5}}} is simplified.
Since 7 is a prime number, you know that 47{\displaystyle {\frac {4}{7}}} is simplified.
Since 8 is a multiple of 4, you know that 48{\displaystyle {\frac {4}{8}}} is NOT simplified. -
Step 3: Determine whether the denominator is a prime number.
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Step 4: Find the difference between the numerator and the denominator.
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Step 5: Reduce Fractions that aren’t already simplified.
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Step 6: Determine which of the following fractions are in their reduced form.
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Step 7: Consider the following problem.
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Step 8: Determine which fraction is NOT simplified.
Detailed Guide
A unit fraction is one that has 1 as the numerator.
Unit fractions cannot be simplified any further.For example, 14{\displaystyle {\frac {1}{4}}}, 12{\displaystyle {\frac {1}{2}}}, 1100{\displaystyle {\frac {1}{100}}}, and 167{\displaystyle {\frac {1}{67}}} are all simplified, because they have 1 as the numerator.
One way to reduce a fraction is to divide by a greatest common factor.If the numerator is a multiple of the denominator, that means each can be divided by a greatest common factor (the numerator).
These types of fractions can be reduced to a unit fraction.
For example, 26{\displaystyle {\frac {2}{6}}} is not simplified, because 6 is a multiple of
2.
The numerator and denominator can still be divided by a common factor of 2, simplifying the fraction to 13{\displaystyle {\frac {1}{3}}}.
The fraction 25{\displaystyle {\frac {2}{5}}} is simplified, because 5 is not a multiple of
2. , A prime number is a number that is only divisible by itself and
1.If the denominator is prime, the fraction cannot be simplified any further.This is because the denominator can only be divided by itself, so whatever number appears in the numerator will not have a common factor.
For more information about prime numbers, you can read Check if a Number Is Prime.
For example, 1523{\displaystyle {\frac {15}{23}}} is simplified, because 23 is a prime number.
The only factors of 23 are 23 and 1, so it is impossible to find a greatest common factor you can use to simplify the numerator and denominator. (If the numerator were 1, it would be a unit fraction and thus already simplified.
If the numerator were 23, the fraction would equal
1.) , If the difference is 1, then the fraction is simplified.
For example, you know that 78{\displaystyle {\frac {7}{8}}} is simplified, because 7−8=1{\displaystyle 7-8=1} , A fraction can be simplified by looking for the largest number that can be evenly divided into the numerator and denominator, and then dividing each by that number.
For example, if you are reducing 515{\displaystyle {\frac {5}{15}}}, divide the numerator and denominator by
5.
This will give you 13{\displaystyle {\frac {1}{3}}}, which you know cannot be reduced further because it is a unit fraction. , Do not make any calculations: 15{\displaystyle {\frac {1}{5}}}, 110{\displaystyle {\frac {1}{10}}}, 510{\displaystyle {\frac {5}{10}}}.
The fractions 15{\displaystyle {\frac {1}{5}}} and 110{\displaystyle {\frac {1}{10}}} are in their reduced, or simplified, form, because each is a unit fraction, with 1 as the numerator.
You should know that 510{\displaystyle {\frac {5}{10}}} is not simplified, because 10 is a multiple of
5. , Franny says that 12109{\displaystyle {\frac {12}{109}}} is a simplified fraction.
Without making any calculations, how do you know she is correct? Since 109 is a prime number, you can tell that the fraction is simplified. 109 is only divisible by 109 and 1, so it shares no common factors with
12. , Do not make any calculations: 45{\displaystyle {\frac {4}{5}}}, 47{\displaystyle {\frac {4}{7}}}, 48{\displaystyle {\frac {4}{8}}} Since 5−4=1{\displaystyle 5-4=1}, you know that 45{\displaystyle {\frac {4}{5}}} is simplified.
Since 7 is a prime number, you know that 47{\displaystyle {\frac {4}{7}}} is simplified.
Since 8 is a multiple of 4, you know that 48{\displaystyle {\frac {4}{8}}} is NOT simplified.
About the Author
Charlotte Fox
Writer and educator with a focus on practical creative arts knowledge.
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