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Operations With Fractions


Simplifying a Fraction

Fractions of the same value can be written using different pairs of numerators and denominators. This is why some fractions can be simplified to equivalent fractions with a smaller numerator and denominator. Consider the following example.
In order to simplify this fraction, there are three steps to follow.
Determine Whether the Fraction Can Be Simplified
To determine whether the given fraction can be simplified, split the numerator and denominator into prime factors and see if there are any common factors other than
The numerator and denominator share factors and Therefore, the fraction can be simplified. If the numerator and denominator did not have common factors other than the fraction would be said to be simplified or written in its simplest form.
Find the Greatest Common Factor
The greatest common factor (GCF) of the numbers and can be found by multiplying all their common factors.
If the numerator and denominator share only one common factor, then that factor is their GCF.
Reduce the Fraction
Finally, to reduce the fraction, divide its numerator and denominator by their GCF.
As a result, an equivalent fraction of was obtained. It is the simplest form of the given fraction


Simplification of Different Fractions
The applet below illustrates how different fractions are simplified.
Applet that shows how to simplify different fractions


Adding and Subtracting Fractions

The first step in adding and subtracting fractions is to check if they share the same denominator. Here, the methods of performing these operations on fractions and how to convert unlike fractions to like fractions will be discussed with examples.

Adding and Subtracting Like Fractions

The numerators of like fractions are added or subtracted when finding the sum or difference of the like fractions, respectively. The denominator remains the same in these situations.

Adding and Subtracting Unlike Fractions

Unlike fractions must first be converted to like fractions when the operation deals with the sum or difference. One way to convert them is to multiply the numerator and denominator of each fraction by the denominator of the other. Then, the given operation can be performed.

Another way is to find the least common denominator (LCD) of the fractions. Consider the example of subtracting from
The result can be found in four steps.
Find the Least Common Denominator
The least common denominator is the least common multiple of the numbers in the denominators.
The numbers need to be expressed as a product of their prime factors to find their LCM.
Denominator Prime Factorization
The least common denominator is the product of the highest power of each prime factor.
Rewrite Each Fraction Using the LCD
The LCD was found to be Now the fractions will be multiplied by the appropriate factors to make the denominators equal to The first fraction must be multiplied by and the second fraction by to get the LCD.
Subtract the Numerators
The numbers in the numerators can now be subtracted because both fractions have the same denominator.
Simplify if Possible
Check if the resulting fraction can be simplified or not. It cannot be simplified since the numerator and denominator do not have any common factors.
The process for adding unlike fractions is similar to the just performed example of subtracting unlike fractions.


Multiplying Fractions

The product of two fractions is equal to the product of the numerators divided by the product of the denominators. The resulting fraction is then simplified to its lowest terms, if possible.

Here, and are not When multiplying fractions, it makes no difference whether they are like or unlike fractions. Consider multiplying by
The result of this multiplication can be found in three steps.
Multiply the Numerators
The numerator of the first fraction is and the numerator of the second is The product of the numerators is then
Multiply the Denominators
The product of the denominators is
Simplify if Possible
Note that is the greatest common factor of and Divide both the numerator and the denominator by to simplify the fraction.
Therefore, the product of and simplified to its lowest terms is


Dividing Fractions

Dividing a fraction by another fraction is the same as multiplying the first fraction by the reciprocal of the second fraction.

Here, and are not The division of two fractions can then be considered as a multiplication of two fractions. Consider the following division of two fractions.
The quotient can be found in three steps.
Multiply by the Reciprocal of the Divisor
The division of fractions begins by keeping the first fraction as is. Then change the division sign with the multiplication sign and write the reciprocal of the second fraction. Note that the reciprocal of a fraction is found by switching the numerator and denominator of the fraction.
Multiply the Fractions
The result is now a multiplication of two fractions. The product of the fractions is equal to the product of the numerators divided by the product of the denominators.
Simplfy if Possible
The resulting fraction can be simplified because and have a common factor.
The greatest common factor of the numbers is Simplify the fraction by
The divison expression is equal to
The same steps above are also used when dividing a fraction by a whole number. This is because every whole number can be thought of as a fraction with a denominator of