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Operations With Fractions
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Reference
Operations With Fractions
Method
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. 18/66 In order to simplify this fraction, there are three steps to follow.
1
Determine Whether the Fraction Can Be Simplified
expand_more 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 1.
18 &= 2* 3* 3 66 &= 2* 3* 11
The numerator and denominator share factors 2 and 3. Therefore, the fraction can be simplified. If the numerator and denominator did not have common factors other than 1, the fraction would be said to be simplified or written in its simplest form.
2
Find the Greatest Common Factor
expand_more The greatest common factor (GCF) of the numbers 18 and 66 can be found by multiplying all their common factors. GCF(18,66) = 2* 3= 6
If the numerator and denominator share only one common factor, then that factor is their GCF.
3
Reduce the Fraction
expand_more Finally, to reduce the fraction, divide its numerator and denominator by their GCF.
18/66=18/ 6/66/ 6=3/11
As a result, an equivalent fraction of 311 was obtained. It is the simplest form of the given fraction 1866.
Extra
Simplification of Different FractionsThe applet below illustrates how different fractions are simplified. 
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Method
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.
a/c + b/c=a + b/c [0.8em] a/c - b/c=a - b/c
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.
a/c + b/d=ad + bc/cd [0.8em] a/c - b/d=ad - bc/cd
Another way is to find the least common denominator (LCD) of the fractions. Consider the example of subtracting 712 from 415. 4/15 - 7/12 The result can be found in four steps.
1
Find the Least Common Denominator
expand_more The least common denominator is the least common multiple of the numbers in the denominators.
4/15 - 7/12
The numbers need to be expressed as a product of their prime factors to find their LCM.
| Denominator | Prime Factorization |
|---|---|
| 15 | 3 * 5 |
| 12 | 2^2 * 3 |
The least common denominator is the product of the highest power of each prime factor. LCD: 2^2 * 3 * 5 = 60
2
Rewrite Each Fraction Using the LCD
expand_more The LCD was found to be 60. Now the fractions will be multiplied by the appropriate factors to make the denominators equal to 60. The first fraction must be multiplied by 4 and the second fraction by 5 to get the LCD.
4/15 - 7/12
ExpandFrac
a/b=a * 4/b * 4
4 * 4/15 * 4- 7/12
Multiply
Multiply
16/60 - 7/12
ExpandFrac
a/b=a * 5/b * 5
16/60 - 7 * 5/12 * 5
Multiply
Multiply
16/60 - 35/60
3
Subtract the Numerators
expand_more 4
Simplify if Possible
expand_more 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.
- 19/60
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Method
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.
a/b * c/d = a * c/b * d
Here, b and d are not 0. When multiplying fractions, it makes no difference whether they are like or unlike fractions. Consider multiplying 56 by 34. 5/6 * 3/4 The result of this multiplication can be found in three steps.
1
Multiply the Numerators
expand_more 2
Multiply the Denominators
expand_more The product of the denominators is 6 * 4 = 24.
3
Simplify if Possible
expand_more Note that 3 is the greatest common factor of 15 and 24. Divide both the numerator and the denominator by 3 to simplify the fraction.
Therefore, the product of 56 and 34 simplified to its lowest terms is 58.
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Method
Dividing Fractions
Dividing a fraction by another fraction is the same as multiplying the first fraction by the reciprocal of the second fraction.
a/b ÷ c/d = a/b * d/c
Here, b, c, and d are not 0. The division of two fractions can then be considered as a multiplication of two fractions. Consider the following division of two fractions. 12/25 ÷ 3/5 The quotient can be found in three steps.
1
Multiply by the Reciprocal of the Divisor
expand_more 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. 
2
Multiply the Fractions
expand_more 3
Simplfy if Possible
expand_more The resulting fraction can be simplified because 60 and 75 have a common factor. 60 & = 2^2 * 3 * 5 75 & = 3 * 5^2
The greatest common factor of the numbers is 3 *5 =15. Simplify the fraction by 15.
The divison expression is equal to 45.
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