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.

\frac{1 8}{6 6}

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

1 .

1866 = 2 \cdot 3 \cdot 3 = 2 \cdot 3 \cdot 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.

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 \cdot 3 = 6

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.

\frac{1 8}{6 6} = \frac{1 8 / 6}{6 6 / 6} = \frac{3}{1 1}

As a result, an equivalent fraction of

\frac{3}{1 1}

was obtained. It is the simplest form of the given fraction

\frac{1 8}{6 6} .

Extra

Simplification of Different Fractions

The applet below illustrates how different fractions are simplified.

$Applet that shows how to simplify different fractions$

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.

\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \frac{a}{c} - \frac{b}{c} = \frac{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.

\frac{a}{c} + \frac{b}{d} = \frac{a d + b c}{c d} \frac{a}{c} - \frac{b}{d} = \frac{a d - b c}{c d}

Another way is to find the least common denominator (LCD) of the fractions. Consider the example of subtracting

\frac{7}{1 2}

from

\frac{4}{1 5} .

\frac{4}{1 5} - \frac{7}{1 2}

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.

\frac{4}{1 5} - \frac{7}{1 2}

The numbers need to be expressed as a product of their prime factors to find their LCM.

Denominator	Prime Factorization
$15$	$3 \cdot 5$
$12$	$2^{2} \cdot 3$

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

LCD : 2^{2} \cdot 3 \cdot 5 = 60

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.

\frac{4}{1 5} - \frac{7}{1 2}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 4}{b \cdot 4}$

\frac{4 \cdot 4}{1 5 \cdot 4} - \frac{7}{1 2}

Multiply

\frac{1 6}{6 0} - \frac{7}{1 2}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 5}{b \cdot 5}$

\frac{1 6}{6 0} - \frac{7 \cdot 5}{1 2 \cdot 5}

Multiply

\frac{1 6}{6 0} - \frac{3 5}{6 0}

Subtract the Numerators

The numbers in the numerators can now be subtracted because both fractions have the same denominator.

\frac{1 6}{6 0} - \frac{3 5}{6 0}

SubFrac

Subtract fractions

\frac{1 6 - 3 5}{6 0}

SubTerm

Subtract term

\frac{- 1 9}{6 0}

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.

\frac{- 1 9}{6 0}

The process for adding unlike fractions is similar to the just performed example of subtracting unlike fractions.

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.

$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$

Here,

b

and

d

are not

0 .

When multiplying fractions, it makes no difference whether they are like or unlike fractions. Consider multiplying

\frac{5}{6}

\frac{3}{4} .

\frac{5}{6} \cdot \frac{3}{4}

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

Multiply the Numerators

The numerator of the first fraction is

5

and the numerator of the second is

3 .

The product of the numerators is then

15 .

\frac{5}{6} \cdot \frac{3}{4}

MultFrac

Multiply fractions

\frac{5 \cdot 3}{6 \cdot 4}

Multiply

\frac{1 5}{6 \cdot 4}

Multiply the Denominators

The product of the denominators is

6 \cdot 4 = 24 .

\frac{1 5}{6 \cdot 4}

Multiply

\frac{1 5}{2 4}

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.

\frac{1 5}{2 4}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

\frac{1 5 / 3}{2 4 / 3}

SimpQuot

Simplify quotient

\frac{5}{8}

Therefore, the product of

\frac{5}{6}

and

\frac{3}{4}

simplified to its lowest terms is

\frac{5}{8} .

Method

Dividing Fractions

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

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

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.

\frac{1 2}{2 5} \div \frac{3}{5}

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.

\frac{1 2}{2 5} \cdot \frac{5}{3}

MultFrac

Multiply fractions

\frac{1 2 \cdot 5}{2 5 \cdot 3}

Multiply

\frac{6 0}{7 5}

Simplfy if Possible

The resulting fraction can be simplified because

60

and

75

have a common factor.

6075 = 2^{2} \cdot 3 \cdot 5 = 3 \cdot 5^{2}

The greatest common factor of the numbers is

3 \cdot 5 = 15 .

Simplify the fraction by

15 .

\frac{6 0}{7 5}

ReduceFrac

$\frac{a}{b} = \frac{a / 1 5}{b / 1 5}$

\frac{6 0 / 1 5}{7 5 / 1 5}

SimpQuot

Simplify quotient

\frac{4}{5}

The divison expression is equal to

\frac{4}{5} .

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

1 .

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Extra

Adding and Subtracting Like Fractions

Adding and Subtracting Unlike Fractions