Operations on Fractions

Pronunciation: /ˌɒ pəˈreɪ ʃənz ɒn ˈfræk ʃənz/ ?

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

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Addition and Subtraction of Fractions

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

Addition Example 1
StepExpressionsDescription
1(1/4)+(2/4)These are the fractions to add.
2(1/4)+(2/4) with the 4s highlighted.Since the denominators are already equal, there is no need to find a least common denominator.
3(1/4)+(2/4)=(1+2)/4Write a new fraction with the addition on top and the common denominator on the bottom.
43/4Add 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.

Addition Example 2
StepExpressionsDescription
1(1/3)+(1/2)These are the fractions to add.
23=1*3, 2=1*2.The first step in finding the least common denominator is finding the prime factorization of each of the denominators.
31*2*3=6Combine the prime factors to get the least common denominator. 6 is the least common denominator.
43*2=6Find 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?
52*3=6Find 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?
61=(2/2) and 1=(3/3)We will use the property of multiplying by 1 to transform the fractions.
7(2/2)*(1/3)+(3/3)*(1/2)Apply the property of multiplying by 1.
8(2/2)*(1/3)+(3/3)*(1/2)=(2*1)/(2*3)+(3*1)/(3*2)=(2/6)+(3/6)Multiply the fractions.
9(2/6)+(3/6)=(2+3)/6Since the denominators are equal, add the numerators.
10(2+3)/6=5/6Add the numerator. Since the numerator 5 and the denominator 6 are relatively prime, the fraction can not be simplified further. The problem is done.
Table 2: Addition of fractions example 2.

Addition Example 3
StepExpressionsDescription
1(5/12)-(1/6)These are the fractions to subtract.
212=1*2^2*3, 6=1*2*3.The first step in finding the least common denominator is finding the prime factorization of each of the denominators.
31*2^2*3=12Combine the prime factors to get the least common denominator. 12 is the least common denominator.
412*1=12Find 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.
52*6=12Find 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?
61=(2/2)We will use the property of multiplying by 1 to transform the fractions.
7(5/12)-(2/2)*(1/6)Apply the property of multiplying by 1.
8(5/12)-(2/2)*(1/6)=(5/12)-(2*1)(2*6)=(5/12)-(2/12)Multiply the fraction.
9(5/12)-(2/12)=(3/12)Since the denominators are equal, subtract the numerators.
10(3/12)=(1*3)/(4*3)=(1/4)*1=(1/4)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:

(a/b)+(c/d).
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
(a/b)+(c/d)=1*(a/b)+1*(c/d).
But, since
(b/b)=1,b!=0 and (d/d)=1, d!=0,
substitute b/b and d/d into the expression, getting
(a/b)+(c/d)=(d/d)*(a/b)+(b/b)*(c/d).
Multiply the fractions to get
(d/d)*(a/b)+(b/b)*(c/d)=(ad)/(bd)+(bc)/(bd).
Since the denominators are now common, add the numerators:
(ad)/(bd)+(bc)/(bd)=(ad+bc)/(bd).
We can now conclude
a/b+c/d=(ad+bc)/(bd) and a/b-c/d=(ad-bc)/(bd).

Multiplication of Fractions

To multiply fractions, multiply the numerators by each other and the denominators by each other:
(a/b)*(c/d)=(ac/bd).

Multiplication Example 1
StepExpressionsDescription
1(1/4)*(2/3)These are the fractions to multiply.
2(1/4)*(2/3) = (1*2)/(3*4) = 2/(3*4)Multiply the numerators and the denominators.
34=2*2 implies that 2/(3*4) = (2)/(3*2*2)Expand the numerator and denominator into factors.
4(2)/(3*2*2) cancel one 2, = 1/(3*2) = 1/6Cancel 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.

Multiplication Example 2
StepExpressionsDescription
1(5/12)*(3/10)These are the fractions to multiply.
2(5/12)*(3/10) = (5*3)/(12*10)Multiply the numerators and the denominators.
312=2*2*3, 10=2*5 implies that (5*3)/(12*10) = (5*3)/(2*2*3*2*5)Expand the numerator and denominator into factors.
4(5*3)/(2*2*3*2*5) cancel one 3 and one 5 = 1/(2*2*2) = 1/8Cancel 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.

Division of Fractions

When dividing fractions, use the property of fractions that dividing by a fraction is the same as multiplying by its reciprocal:
(a/b)/(c/d)=(a/b)*(d/c).

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

Division Example 2
StepExpressionsDescription
1(1/12)/(7/3)These are the fractions to divide.
2(1/12)/(7/3) = (1/12)*(3/7)Change the division problem into a multiplication problem by flipping the divisor.
3(1/12)*(3/7) = (1/3*4)*(3/7)Find the common factors.
4(1/3*4)*(3/7) cancel the e = (1/5)*(1/7) = (1/28)Cancel the common factors and simplify the fraction.
Table 7: Division of fractions example 2.

Derivation of the Division Rule for Fractions
StepExpressionsDescription
1(a/b)/(c/d)Start with a general case of fractions to divide. a, b, c and d represent arbitrary values.
2(a/b)*(1/(c/d))The definition of division is m/n=m*(1\n). Use the definition of division to transform the divisor.
3(a/b)*(1/(c/d))=(a/b)*(1/(c/d))*1This step will use the property of multiplying by 1: m=1*m. Use to property of multiplying by 1 to multiplying the right side by 1.
4(a/b)*(1/(c/d))*1=(a/b)*(1/(c/d))*(d/d)This step will use the property of dividing a value by itself: d/d=1, d!=0. Use the substitution property of equality to replace the 1 with d/d.
5(a/b)*(1/(c/d))*(d/1)Since m=m/1, replace the d in the denominator of the last fraction with d/1.
6(a/b)*((1*d)/((c/d)*(d/1)))Combine the two fractions on the right.
7(a/b)*((1*d)/((c/d)*(d/1))) with the rightmost d's crossed out.Cancel the d's.
8(a/b)*(d/(c/1))Simplify the denominator on the right.
9(a/b)*(d/c)Use the fact that m=m/1 to replace c/1 with c.
10(a/b)/(c/d)=(a/b)*(d/c)We have shown that (a/b)/(c/d)=(a/b)*(d/c)
Table 8: Derivation of the division rule for fractions.

Raising Fractions to a Power

If a fraction is raised to a power, the numerator and denominator can each be raised to the same power:
(a/b)^m=(a^m)/(b^m)

Exponentiation of Fractions Example 1
StepExpressionsDescription
1(3/4)^2This is the fraction to raise to a power.
2(3/4)^2=(3^2)/(4^2)Apply the power rule for fractions.
3(3^2)/(4^2)=9/16Simplify the numerator and denominator.
Table 6: Exponentiation of fractions example 1.

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Operations on Fractions. 2009-01-15. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/o/operationsonfractions.html.

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2009-01-15: Initial version (McAdams, David.)

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