# Operations on Fractions

Pronunciation: /ˌɒ.pəˈreɪ.ʃənz ɒn ˈfræk.ʃənz/ Explain

An operation is something like addition and multiplication. An operation on a fraction applies one of these operations to fractions.

### How to Add and Subtract Fractions

To add or subtract fractions, one must first find a common denominator. The least common denominator is used since it simplifies the calculations.

StepExpressionsDescription
1 These are the fractions to add.
2 Since the denominators are already equal, there is no need to find a least common denominator.
3 Write a new fraction with the addition on top and the common denominator on the bottom.
4 Add the numerator. Since the numerator 3 and the denominator 4 are relatively prime, the fraction can not be simplified further. The problem is done.
Table 1: Addition of fractions example 1.

StepExpressionsDescription
1 These are the fractions to add.
2 The first step in finding the least common denominator is finding the prime factorization of each of the denominators.
3 Combine the prime factors to get the least common denominator. 6 is the least common denominator.
4 Find the number to multiply by the first fraction. Since the least common denominator is 6, and the denominator of the first fraction is 3, what times 3 equals 6?
5 Find the number to multiply by the second fraction. Since the least common denominator is 6, and the denominator of the second fraction is 2, what times 2 equals 6?
6 We will use the property of multiplying by 1 to transform the fractions.
7 Apply the property of multiplying by 1.
8 Multiply the fractions.
9 Since the denominators are equal, add the numerators.
10 Add the numerator. Since the numerator 5 and the denominator 6 are relatively prime, the fraction can not be simplified further. The problem is complete.
Table 2: Addition of fractions example 2.

Subtraction Example 1
StepExpressionsDescription
1 These are the fractions to subtract.
2 The first step in finding the least common denominator is finding the prime factorization of each of the denominators.
3 Combine the prime factors to get the least common denominator. 12 is the least common denominator.
4 Find the number to multiply by the first fraction. Since the least common denominator is 12, and the denominator of the first fraction is 12, what times 12 equals 12? 1 times 12 equals 12. This means that the first fraction already has the least common denominator.
5 Find the number to multiply by the second fraction. Since the least common denominator is 12, and the denominator of the second fraction is 6, what times 6 equals 12?
6 We will use the property of multiplying by 1 to transform the fractions.
7 Apply the property of multiplying by 1.
8 Multiply the fraction.
9 Since the denominators are equal, subtract the numerators.
10 The numerator and the denominator are not relatively prime. Find and cancel the common factor.
Table 3: Addition of fractions example 3.

### General Case of Addition

To derive the general case, start with addition of two arbitrary fractions: .
The denominators are not equal, so the fractions can not yet be added. Since the denominators are b and d, b · d will always be a common denominator. However, b · d may not be the least common denominator.

First use the property of multiplying by 1. Apply the property of multiplying by 1 to the fraction to get .
But, since and ,
substitute b / b and d / d into the expression, getting .
Multiply the fractions to get .
Since the denominators are now common, add the numerators: .
We can now conclude .

### How to Multiply Fractions

To multiply fractions, multiply the numerators by each other and the denominators by each other: .

Multiplication Example 1
StepExpressionsDescription
1 These are the fractions to multiply.
2 Multiply the numerators and the denominators.
3 Expand the numerator and denominator into prime factors.
4 Cancel the common factors, then multiply out the numerator and denominator. Since 1 is prime relative to 6, this fraction can not be reduced any further. The problem is done.
Table 4: Multiplication of fractions example 1.

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Multiplication Example 2
StepExpressionsDescription
1 These are the fractions to multiply.
2 Multiply the numerators and the denominators.
3 Expand the numerator and denominator into factors.
4 Cancel the common factors, then multiply out the numerator and denominator. Since 1 is prime relative to 8, this fraction can not be reduced any further. The problem is done.
Table 5: Multiplication of fractions example 2.

### How to Divide Fractions

When dividing fractions, use the property of fractions that dividing by a fraction is the same as multiplying by its reciprocal: .

Division Example 1
StepExpressionsDescription
1 These are the fractions to divide.
2 Change the division problem into a multiplication problem by flipping the divisor.
3 Cancel the common factors and simplify the fraction.
Table 6: Division of fractions example 1.

Division Example 2
StepExpressionsDescription
1 These are the fractions to divide.
2 Change the division problem into a multiplication problem by flipping the divisor.
3 Find the common factors.
4 Cancel the common factors and simplify the fraction.
Table 7: Division of fractions example 2.

Derivation of the Division Rule for Fractions
StepExpressionsDescription
1 Start with a general case of fractions to divide. a, b, c and d represent arbitrary values.
2 The definition of division is . Use the definition of division to transform the divisor.
3 This step will use the property of multiplying by 1: . Use to property of multiplying by 1 to multiplying the right side by 1.
4 This step will use the property of dividing a value by itself: . Use the substitution property of equality to replace the 1 with .
5 Since , replace the d in the denominator of the last fraction with .
6 Combine the two fractions on the right.
7 Cancel the d's.
8 Simplify the denominator on the right.
9 Use the fact that to replace with .
10 We have shown that Table 8: Derivation of the division rule for fractions.

### How to Raise a Fraction to a Power

If a fraction is raised to a power, the numerator and denominator can each be raised to the same power: Exponentiation of Fractions Example 1
StepExpressionsDescription
1 This is the fraction to raise to a power.
2 Apply the power rule for fractions.
3 Simplify the numerator and denominator.
Table 6: Exponentiation of fractions example 1.

McAdams, David E. Operations on Fractions. 4/27/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/o/operationsonfractions.html.

### Revision History

4/27/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
9/5/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
1/15/2009: Initial version. (McAdams, David E.)

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