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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.
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Simplification of Different Fractions

$6618 $

In order to simplify this fraction, there are three steps to follow.
1

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 $1.$
*simplified* or written *in its simplest form*.

$1866 =2⋅3⋅3=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 2

Find the Greatest Common Factor

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. 4

Reduce the Fraction

Finally, to reduce the fraction, divide its numerator and denominator by their GCF.

$6618 =66/618/6 =113 $

As a result, an equivalent fraction of $113 $ was obtained. It is the simplest form of the given fraction $6618 .$ The applet below illustrates how different fractions are simplified.

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.

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.

$ca +cb =ca+b ca −cb =ca−b $

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.

$ca +db =cdad+bc ca −db =cdad−bc $

$154 −127 $

The result can be found in four steps.
1

Find the Least Common Denominator

The least common denominator is the least common multiple of the numbers in the denominators.

The least common denominator is the product of the highest power of each prime factor.

$154 −127 $

The numbers need to be expressed as a product of their prime factors to find their LCM. Denominator | Prime Factorization |
---|---|

$15$ | $3⋅5$ |

$12$ | $2_{2}⋅3$ |

$LCD:2_{2}⋅3⋅5=60 $

2

Rewrite Each Fraction Using the LCD

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.

$154 −127 $

ExpandFrac

$ba =b⋅4a⋅4 $

$15⋅44⋅4 −127 $

Multiply

Multiply

$6016 −127 $

ExpandFrac

$ba =b⋅5a⋅5 $

$6016 −12⋅57⋅5 $

Multiply

Multiply

$6016 −6035 $

3

Subtract the Numerators

4

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.

$60-19 $

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.

$ba ⋅dc =b⋅da⋅c $

$65 ⋅43 $

The result of this multiplication can be found in three steps.
1

Multiply the Numerators

2

Multiply the Denominators

The product of the denominators is $6⋅4=24.$

3

Simplify if Possible

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 $65 $ and $43 $ simplified to its lowest terms is $85 .$

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

$ba ÷dc =ba ⋅cd $

$2512 ÷53 $

The quotient can be found in three steps.
1

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.

2

Multiply the Fractions

3

Simplfy if Possible

The resulting fraction can be simplified because $60$ and $75$ have a common factor.

$6075 =2_{2}⋅3⋅5=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 $54 .$