##### Operations with Fractions

Doing operations with fractions means applying the four basic mathematical operations: addition, subtraction, multiplication and division.

Adding and subtracting fractions with the same denominator

Adding and subtracting fraction with the same denominator is easy. All you have to do is add or subtract the numerator while the denominator remains as it is. If the fraction that we get can be reduced to lowest terms then it’s better to do so.

Example 1: Add the fractions $\displaystyle \frac{1}{9}+\frac{2}{9}$

Since the denominator is the same we add the numerators and keep the same denominator

$\displaystyle \frac{1}{9}+\frac{2}{9}=\frac{{1+2}}{9}=\frac{3}{9}$

The we reduce the sum to the lowest term

$\displaystyle \frac{3}{9}=\frac{3}{{3\times 3}}=\frac{1}{3}$

So $\displaystyle \frac{1}{9}+\frac{2}{9}=\frac{3}{9}=\frac{1}{3}$

Example 2: Subtract the fractions $\displaystyle \frac{5}{7}-\frac{3}{7}$

Solution: Since the denominator is the same we substract the numerators and keep the same denominator

$\displaystyle \frac{5}{7}-\frac{3}{7}=\frac{{5-3}}{7}=\frac{2}{7}$

So, $\displaystyle \frac{5}{7}-\frac{3}{7}=\frac{2}{7}$

### Adding and subtracting fractions with different denominator

When adding and subtracting fraction with different denominators you have to turn the fractions into equivalent fractions by finding the least common denominator.Then reduce the sum into the lowest term if possible.

Example 3: Add the fractions $\displaystyle \frac{2}{5}+\frac{3}{{15}}$

Since the fractions have diffrent denominator we find the LCD

5 = 1 x 5

15 = 3 x 5

LCD(5, 15) = 3 x 5 = 15

Converting the fractions into equivalent fractions  with 15 as a denominator.

$\displaystyle \frac{2}{5}\times \frac{3}{3}=\frac{6}{{15}}$

$\displaystyle \frac{3}{{15}}$ is a fraction with the denominator 15 so no need to be converted.

Now since our fractions have similar denominators we add the numerators, the denominator stays the same and then we reduce the sum.

$\displaystyle \frac{6}{{15}}+\frac{3}{{15}}=\frac{9}{{15}}=\frac{3}{5}$

So, $\displaystyle \frac{2}{5}+\frac{3}{{15}}=\frac{3}{5}$

Example 4: Subtract the fraction $\displaystyle \frac{1}{{12}}-\frac{2}{9}$

We find the LCD since we have different denominators

12 = 2 x 2 x 3

9 = 3 x 3

LCD(9, 12) = 2 x 2 x 3 x 3 = 36

Converting the fractions into equivalent fractions with 36 as a denominator.

$\displaystyle \frac{1}{{12}}\times \frac{3}{3}=\frac{3}{{36}}$

$\displaystyle \frac{2}{9}\times \frac{4}{4}=\frac{8}{{36}}$

Now since the fractions have similar denominators we substract the numerators, the denominator stays the same.

$\displaystyle \frac{3}{{36}}-\frac{8}{{36}}=-\frac{5}{{36}}$.

So,$\displaystyle \frac{1}{{12}}-\frac{2}{9}=-\frac{5}{{36}}$

##### Adding and subtracting a fraction with a mixed one or two mixed fractions
• Convert the mixed fraction into a improper fraction.
• If the proper fractions have unlike denominators then find the LCD and change the fractions into equivalent one.
• Subtract or add the numerators and leave the dominator as it is.
• Reduce to lowest term if possible.

Example 5: Add the fractions $\displaystyle \frac{6}{7}+2\frac{3}{5}$

Firstly we convert the mixed fraction $\displaystyle 2\frac{3}{5}$ into a improper one.

The whole number is 2 and the denominator is 5, so 5 x 2 = 10

We add the product 10 to the numerator of our fraction: 3+10=13

Since the denominator stays the same our improper fraction is: $\displaystyle \frac{{13}}{5}$

The fractions $\displaystyle\frac{6}{7}$ and $\displaystyle \frac{{13}}{5}$ have diffrent denominators, so we find the LCD to turn them into equivalent one´s.

LCD(7, 5) = 35

$\displaystyle \frac{6}{7}\times \frac{5}{5}=\frac{{30}}{{35}}$

$\displaystyle \frac{{13}}{5}\times \frac{7}{7}=\frac{{91}}{{35}}$

Add the numerators and leave the denominator as it is :$\displaystyle \frac{{30}}{{35}}+\frac{{91}}{{35}}=\frac{{121}}{{35}}$

So, $\displaystyle \frac{6}{7}+2\frac{3}{5}=\frac{{121}}{{35}}$

Example 6: Subtract the mixed fractions $\displaystyle 3\frac{2}{7}-2\frac{3}{4}$

Firstly convert the mixed fractions into improper one.

$\displaystyle 2\frac{3}{4}=\frac{{11}}{4}$ and $\displaystyle 3\frac{2}{7}=\frac{{23}}{7}$

Since they have different denominator we change them into equaivalent fractions by finding the LCD.

LCD(4,7)=28

$\displaystyle \frac{{11}}{4}\times \frac{7}{7}=\frac{{77}}{{28}}$

$\displaystyle \frac{{23}}{7}\times \frac{4}{4}=\frac{{92}}{{28}}$

Subtract the numerators and leave the denominator as it is:

$\displaystyle \frac{{92}}{{28}}-\frac{{77}}{{28}}=\frac{{15}}{{28}}$

So, $\displaystyle 3\frac{2}{7}-2\frac{3}{4}=$

$\displaystyle \frac{{92}}{{28}}-\frac{{77}}{{28}}=\frac{{15}}{{28}}$

##### Adding and Subtracting Fractions by Using Formulas

There is an easy way to add and subtract fractions without having to find the LCD. This method consist in using formulas that involves cross multiplication of the fractions.

The formula for adding fractions is:

$\displaystyle \frac{a}{b}+\frac{c}{d}=\frac{{ad+bc}}{{bd}}$

Subtracting Fraction

The formula for subtracting fractions is:

$\displaystyle \frac{a}{b}-\frac{c}{d}=\frac{{ad-cd}}{{bd}}$

Example 7: Add $\displaystyle \frac{5}{9}+\frac{2}{5}$by using the formula above.

Solution: $\displaystyle \frac{5}{9}+\frac{2}{5}=\frac{{(5\times 5)+(9\times 2)}}{{9\times 5}}=\frac{{25+18}}{{45}}=\frac{{43}}{{45}}$

Example 8: Subtract$\displaystyle \frac{2}{3}-\frac{3}{9}$ by using the formula above.

Solution: $\displaystyle \frac{2}{3}-\frac{3}{9}=\frac{{(2\times 9)-(3\times 3)}}{{3\times 9}}=\frac{{18-9}}{{27}}=\frac{9}{{27}}=\frac{1}{3}$

### Multiplying and Dividing Fractions

Solution

$\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)$

$\displaystyle n(A\cap B)=n(A)+n(B)-n(A\cup B)$

$\displaystyle n(A\cap B)=12+10-18=4$

$\displaystyle n(A/B)=n(A)-n(A\cap B)$

$\displaystyle n(A/B)=12-4=8$

Multiplying Fractions
Multiplying fractions is even easier than adding and substracting them.All you have to do is multiply the numerators  and the denominators of the fraction then reduce the fraction if possible.If mixed fractions are involved don`t forget to turn them to improper fractions.

The formula for multiplying fractions is: $\displaystyle \frac{a}{b}\times \frac{c}{d}=\frac{{ac}}{{bd}}$

Example 9: Multiply $\displaystyle \frac{2}{9}\times \frac{3}{5}$.

Solution:  $\displaystyle \frac{2}{9}\times \frac{3}{5}=\frac{{(2\times 3)}}{{(9\times 5)}}=\frac{6}{{45}}=\frac{2}{{15}}$

Example 10: Multiply $\displaystyle 2\frac{1}{7}\times \frac{3}{8}$.

Solution: Firstly we have to turn the mixed fraction into a improper one.

$\displaystyle 2\frac{1}{7}=\frac{{15}}{7}$

Then we multiply by using the formula above.

$\displaystyle 2\frac{1}{7}\times \frac{3}{8}=\frac{{15}}{7}\times \frac{3}{8}=\frac{{45}}{{56}}$

### Dividing Fractions

Dividing two fractions is the same as multiplying the first fraction with the reciprocal of the second fraction. So all you have to do is keep the first fraction, change the division sign to multiplication and then flip the second fraction. Then we use the multiplication formula. Don´t forget to reduce the fraction if possible. If it´s easier for you, you can apply directly the formula for dividing fractions.

The formula for dividing fractions is: $\displaystyle \frac{a}{b}\div \frac{c}{d}=\frac{{ad}}{{bc}}$

Example 11: Divide $\displaystyle \frac{2}{4}\div \frac{5}{9}$.

Three simply steps:

Keep the first fraction, flip the second one and then multiply them.

$\displaystyle \frac{2}{4}\div \frac{5}{9}=\frac{2}{4}\times \frac{9}{5}=\frac{{18}}{{20}}=\frac{9}{{10}}$

Using the formula

$\displaystyle \frac{2}{4}\div \frac{5}{9}=\frac{{(2\times 9)}}{{(4\times 5)}}=\frac{{18}}{{20}}=\frac{9}{{10}}$