Operations with Fractions

We will expand or reduce the fractions to end up with two fractions with the same denominator.
A very common way to do this is by multiplying the denominators.

Only the numerators are added while the denominator remains unchanged.

$\frac{4}{5}+\frac{2}{3}=$
Solution:

We will multiply the numerators and obtain:
$\frac{12}{15}+\frac{10}{15}=$

We will obtain
$\frac{22}{15}=1\frac{7}{15}$

Click here for a deeper explanation on the addition of fractions with more exercises.

We will find the common denominator by expanding, simplifying, or multiplying the denominators.
We will end up with two fractions with the same denominator.

Only the numerators are subtracted while the denominator remains unchanged.

$\frac{5}{8}-\frac{1}{2}=$

Solution:
First step: Find the common denominator
We will multiply the denominators and obtain:
$\frac{10}{16}-\frac{8}{16}=$

Second step: Subtract the numerators and reduce the denominator
$\frac{2}{16}=\frac{1}{8}$

Click here for a more in-depth explanation on subtracting fractions with more exercises.

To multiply fractions, we will multiply numerator by numerator and denominator by denominator.

• In case there is a mixed number - we will convert it into a fraction and then multiply numerator by numerator and denominator by denominator.
• In case there is an integer - we will convert it into a fraction and then multiply numerator by numerator and denominator by denominator.
• The commutative property works - We can change the order of the fractions within the exercise without altering the result.

$3\frac{2}{4} \times \frac{2}{3}=$

Solution:
First, we will convert the mixed number to a fraction.

We will obtain:
$\frac{14}{4}=\frac{2}{3}$

Now, we will multiply numerator by numerator and denominator by denominator.
We will obtain:
$\frac{14 \times 2}{4 \times 3}=\frac{28}{12}=2\frac{4}{12}=2\frac{1}{3}$

Click here for a deeper explanation on fraction multiplication with more exercises.

• In case there is any mixed number - we will convert it into a fraction
• In case there is any whole number - we will convert it into a fraction

We will change the operation from divide to multiply and swap places between the numerator and the denominator in the fraction that is found after the divide sign.

$1\frac{4}{5}:\frac{2}{3}$

Solution:
First step: We will convert the mixed number to a fraction.
We will obtain:
$\frac{9}{5}:\frac{2}{3}=$

Second step: We will change the division operation to multiplication and swap places between the numerator and the denominator in the fraction that is after the division sign.
We will obtain:

$\frac{9}{5} \times \frac{3}{2}=$

Third step: We will multiply numerator by numerator and denominator by denominator.
We will obtain:
$\frac{9 \times 3}{5 \times 2}=$

$\frac{27}{10}=2\frac{7}{10}$

Click here for a more in-depth explanation on fraction division with more exercises.

When the numerators are equal and the denominators are different:
The larger fraction will be the one whose denominator is the smallest.
When the numerators are different and the denominators are equal:
The larger fraction will be the one whose numerator is the largest.
When both the numerators and the denominators are different:

We will find the common denominator by expanding, simplifying, or multiplying the denominators. (Let's remember to multiply both the numerator and the denominator)
In case there is any mixed number, we will convert it into a fraction and then, we will find the common denominator.

When obtaining two fractions with the same denominator, the larger fraction will be the one whose numerator is greater.

Place the corresponding sign $>,<,=$
$\frac{5}{10}$_____________________$\frac{5}{8}$

Solution:
The numerators are equal and the denominators are different, therefore, the larger fraction will be the one whose denominator is the smallest.

Place the corresponding sign $>,<,=$

$\frac{2}{5}$_____________________$\frac{4}{5}$

Solution:
The numerators are different and the denominators are the same, therefore, the larger fraction will be the one whose numerator is greater.

Place the corresponding sign $>,<,=$

$2\frac{4}{6}$_____________________$1\frac{4}{5}$

Solution:
We will convert the mixed numbers into fractions. We obtain:
$\frac{16}{6}$_____________________$\frac{9}{5}$
Now we will find the common denominator. We obtain:

$\frac{80}{30}$_____________________$\frac{54}{30}$

When the denominators are equal, the larger fraction will be the one whose numerator is greater.