Operations with Fractions

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

Test yourself on operations with fractions!

Complete the following exercise:

$\frac{1}{2}:\frac{3}{5}=\text{?}$

Practice more now

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

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

First step: Obtain the common denominator

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

Second step: Add the numerators

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

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

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Test your knowledge

Question 1

Complete the following exercise:

$\frac{1}{5}:\frac{1}{10}=\text{?}$

Question 2

Complete the following exercise:

$\frac{1}{2}:\frac{1}{4}=\text{?}$

Question 3

Complete the following exercise:

$\frac{1}{2}:\frac{2}{3}=\text{?}$

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

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

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.

Do you know what the answer is?

Question 1

Complete the following exercise:

$\frac{1}{2}:\frac{5}{3}=\text{?}$

Question 2

Complete the following exercise:

$\frac{2}{4}:\frac{4}{3}=\text{?}$

Question 3

Complete the following exercise:

$\frac{3}{4}:\frac{5}{6}=\text{?}$

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

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

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.

Check your understanding

Question 1

Complete the following exercise:

$\frac{1}{3}:\frac{1}{4}=\text{?}$

Question 2

Complete the following exercise:

$\frac{5}{2}:\frac{1}{4}=\text{?}$

Question 3

$\frac{1}{6}\times\frac{2}{3}=$

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.

Do you think you will be able to solve it?

Question 1

$\frac{2}{5}\times\frac{1}{2}=$

Question 2

$\frac{3}{4}\times\frac{1}{2}=$

Question 3

$\frac{2}{3}\times\frac{1}{4}=$

Third step: Multiply numerator by numerator and denominator by denominator

Let's look at an example

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

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.

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:

Test your knowledge

Question 1

$\frac{1}{4}\times\frac{3}{2}=$

Question 2

$\frac{2}{3}\times\frac{5}{7}=$

Question 3

Complete the following exercise:

$\frac{1}{2}:\frac{3}{5}=\text{?}$

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.

Do you know what the answer is?

Question 1

Complete the following exercise:

$\frac{1}{5}:\frac{1}{10}=\text{?}$

Question 2

Complete the following exercise:

$\frac{1}{2}:\frac{1}{4}=\text{?}$

Question 3

Complete the following exercise:

$\frac{1}{2}:\frac{2}{3}=\text{?}$

Let's look at some examples

Example 1

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.

Example 2

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.

Check your understanding

Question 1

Complete the following exercise:

$\frac{1}{2}:\frac{5}{3}=\text{?}$

Question 2

Complete the following exercise:

$\frac{2}{4}:\frac{4}{3}=\text{?}$

Question 3

Complete the following exercise:

$\frac{3}{4}:\frac{5}{6}=\text{?}$

Example 3

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.

Examples and exercises with solutions for operations with fractions

Exercise #1

Complete the following exercise:

$\frac{1}{2}:\frac{3}{5}=\text{?}$

Video Solution

Step-by-Step Solution

To solve the division $\frac{1}{2} \div \frac{3}{5}$ , we will follow the multiplication by the reciprocal method. Here are the steps:

Step 1: Find the reciprocal of the divisor $\frac{3}{5}$ , which is $\frac{5}{3}$ .
Step 2: Multiply the dividend $\frac{1}{2}$ by the reciprocal found in Step 1: $\frac{1}{2} \times \frac{5}{3}$ .
Step 3: Multiply the numerators: $1 \times 5 = 5$ .
Step 4: Multiply the denominators: $2 \times 3 = 6$ .
Step 5: Combine the results to form the fraction $\frac{5}{6}$ .

The simplified result of $\frac{1}{2} \div \frac{3}{5}$ is $\frac{5}{6}$ .

Answer

$\frac{5}{6}$

Exercise #2

Complete the following exercise:

$\frac{1}{5}:\frac{1}{10}=\text{?}$

Video Solution

Step-by-Step Solution

To solve this problem, we need to divide the fraction $\frac{1}{5}$ by the fraction $\frac{1}{10}$ . When dividing fractions, the procedure involves multiplying by the reciprocal of the divisor (the second fraction).

Let's start with the solution:

First, determine the reciprocal of $\frac{1}{10}$ . The reciprocal of a fraction is obtained by swapping its numerator and denominator. Thus, the reciprocal of $\frac{1}{10}$ is $\frac{10}{1}$ .
Now multiply $\frac{1}{5}$ by the reciprocal of $\frac{1}{10}$ , which is $\frac{10}{1}$ :

$\frac{1}{5} \times \frac{10}{1} = \frac{1 \times 10}{5 \times 1} = \frac{10}{5}$

Simplify the fraction $\frac{10}{5}$ :

$\frac{10}{5} = 2$

Therefore, the result of $\frac{1}{5} : \frac{1}{10}$ is $2$ .

Answer

$2$

Exercise #3

Complete the following exercise:

$\frac{1}{2}:\frac{1}{4}=\text{?}$

Video Solution

Step-by-Step Solution

To solve the division of two fractions $\frac{1}{2} : \frac{1}{4}$ , follow these steps:

Step 1: Identify the operation: The problem involves dividing $\frac{1}{2}$ by $\frac{1}{4}$ .
Step 2: Use the reciprocal: In fraction division, multiply by the reciprocal of the second fraction. Thus, $\frac{1}{2} : \frac{1}{4} = \frac{1}{2} \times \frac{4}{1}$ .
Step 3: Perform the multiplication: Now compute the multiplication by multiplying the numerators and the denominators: $\frac{1 \times 4}{2 \times 1} = \frac{4}{2}$ .
Step 4: Simplify the fraction: The fraction $\frac{4}{2}$ simplifies to $2$ .

Thus, the solution to the division $\frac{1}{2} : \frac{1}{4}$ is $2$ . Therefore, the correct answer choice is $2$ (Choice 1).

Answer

$2$

Exercise #4

Complete the following exercise:

$\frac{1}{2}:\frac{2}{3}=\text{?}$

Video Solution

Step-by-Step Solution

To solve the division of fractions problem $\frac{1}{2} \div \frac{2}{3}$ , we follow these steps:

Step 1: Rewrite the division problem as multiplication by the reciprocal: $\frac{1}{2} \div \frac{2}{3}$ becomes $\frac{1}{2} \times \frac{3}{2}$ .
Step 2: Multiply the numerators together: $1 \times 3 = 3$ .
Step 3: Multiply the denominators together: $2 \times 2 = 4$ .
Step 4: Form the new fraction from the resulting numerator and denominator: $\frac{3}{4}$ .

Thus, the result of dividing $\frac{1}{2}$ by $\frac{2}{3}$ is $\frac{3}{4}$ .

The correct answer is $\frac{3}{4}$ .

Answer

$\frac{3}{4}$

Exercise #5

Complete the following exercise:

$\frac{1}{2}:\frac{5}{3}=\text{?}$

Video Solution

Step-by-Step Solution

To solve the problem of dividing the fractions $\frac{1}{2}$ by $\frac{5}{3}$ , we proceed as follows:

We can simplify a division of fractions by multiplying the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator.

First, we find the reciprocal of $\frac{5}{3}$ , which is $\frac{3}{5}$ .

Next, we multiply the fractions $\frac{1}{2}$ and $\frac{3}{5}$ :

$\frac{1}{2} \times \frac{3}{5} = \frac{1 \times 3}{2 \times 5}.$

This results in

$\frac{3}{10}.$

Thus, the solution to $\frac{1}{2}:\frac{5}{3}$ is $\frac{3}{10}$ .

Answer

$\frac{3}{10}$

Do you think you will be able to solve it?

Question 1

Complete the following exercise:

$\frac{1}{3}:\frac{1}{4}=\text{?}$

Question 2

Complete the following exercise:

$\frac{5}{2}:\frac{1}{4}=\text{?}$

Question 3

$\frac{1}{6}\times\frac{2}{3}=$

Start practice

Operations with Fractions