Operations with Fractions

Operations with Fractions

In this article, we will learn how to perform mathematical calculations with fractions.

More reading material:

  • Addition of fractions
  • Subtraction of fractions
  • Multiplication of fractions
  • Division of fractions
  • Comparison of fractions

Sum of Fractions

First step: Find the common denominator

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.


Second step: Addition of the numerators

Only the numerators are added while the denominator remains unchanged.

Let's look at an example

45+23=\frac{4}{5}+\frac{2}{3}=
Solution:

First step: Obtain the common denominator

We will multiply the numerators and obtain:
1215+1015=\frac{12}{15}+\frac{10}{15}=

Second step: Add the numerators

We will obtain
2215=1715\frac{22}{15}=1\frac{7}{15}

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


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Subtraction of Fractions

First step: Find the common denominator

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


Second step: Subtraction of numerators

Only the numerators are subtracted while the denominator remains unchanged.

Let's look at an example

5812=\frac{5}{8}-\frac{1}{2}=

Solution:
First step: Find the common denominator
We will multiply the denominators and obtain:
1016816=\frac{10}{16}-\frac{8}{16}=

Second step: Subtract the numerators and reduce the denominator
216=18\frac{2}{16}=\frac{1}{8}

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


Multiplication of Fractions

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.

Example

324×23=3\frac{2}{4} \times \frac{2}{3}=

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

We will obtain:
144=23\frac{14}{4}=\frac{2}{3}

Now, we will multiply numerator by numerator and denominator by denominator.
We will obtain:
14×24×3=2812=2412=213\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.


Division of Fractions

First step: Convert all the numbers in the exercise to fractions.

  • 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

Second step: Change the division operation to multiplication and swap the places of the numerator and denominator in the second 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.


Third step: Multiply numerator by numerator and denominator by denominator

Let's look at an example

145:231\frac{4}{5}:\frac{2}{3}

Solution:
First step: We will convert the mixed number to a fraction.
We will obtain:
95:23=\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:

95×32=\frac{9}{5} \times \frac{3}{2}=

Third step: We will multiply numerator by numerator and denominator by denominator.
We will obtain:
9×35×2=\frac{9 \times 3}{5 \times 2}=

2710=2710\frac{27}{10}=2\frac{7}{10}

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


Comparison of Fractions

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:


First step

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.


Second step

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


Let's look at some examples

Example 1

Place the corresponding sign  >,<,= >,<,=
510\frac{5}{10}_____________________58\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.


Example 2

Place the corresponding sign  >,<,= >,<,=

25\frac{2}{5}_____________________45\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.


Example 3

Place the corresponding sign  >,<,= >,<,=

2462\frac{4}{6}_____________________1451\frac{4}{5}

Solution:
We will convert the mixed numbers into fractions. We obtain:
166\frac{16}{6}_____________________95\frac{9}{5}
Now we will find the common denominator. We obtain:

8030\frac{80}{30}_____________________5430\frac{54}{30}

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