Operations with Fractions

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

Test yourself on operations with fractions!

$\frac{1}{3}+\frac{1}{4}=$

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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:

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

Question 2

$\frac{3}{8}+\frac{1}{4}=$

Question 3

$\frac{1}{4}+\frac{3}{4}=$

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

Solve the following:

$\frac{5}{9}:\frac{7}{18}=$

Question 2

Complete the following exercise:

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

Question 3

Complete the following exercise:

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

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:

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

Question 2

Complete the following exercise:

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

Question 3

Solve the following exercise:

$\frac{1}{4}+\frac{3}{6}=\text{?}$

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

Solve the following exercise:

$\frac{3}{4}+\frac{1}{6}=\text{?}$

Question 2

Solve the following exercise:

$\frac{2}{4}+\frac{2}{6}=\text{?}$

Question 3

$\frac{9}{10}+\frac{2}{5}=$

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

$\frac{3}{14}+\frac{3}{7}=$

Question 2

Solve the following exercise:

$\frac{1}{2}-\frac{2}{7}=\text{?}$

Question 3

$\frac{1}{3}+\frac{1}{4}=$

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:

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

Question 2

$\frac{3}{8}+\frac{1}{4}=$

Question 3

$\frac{1}{4}+\frac{3}{4}=$

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

Solve the following:

$\frac{5}{9}:\frac{7}{18}=$

Question 2

Complete the following exercise:

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

Question 3

Complete the following exercise:

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

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

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

Video Solution

Step-by-Step Solution

Let us solve the problem of multiplying the two fractions $\frac{2}{3}$ and $\frac{5}{7}$ .

Step 1: Identify the numerators and denominators. Here, the numerators are $2$ and $5$ , and the denominators are $3$ and $7$ .
Step 2: Multiply the numerators: $2 \times 5 = 10$ .
Step 3: Multiply the denominators: $3 \times 7 = 21$ .
Step 4: Put the results together in a new fraction: $\frac{10}{21}$ .
Step 5: Simplify the fraction if needed. In this case, $\frac{10}{21}$ is already in its simplest form as $10$ and $21$ have no common factors besides $1$ .

Therefore, the solution to the problem $\frac{2}{3} \times \frac{5}{7}$ is $\frac{10}{21}$ .

Answer

$\frac{10}{21}$

Exercise #2

Solve the following exercise:

$\frac{1}{3}+\frac{4}{9}=\text{?}$

Video Solution

Step-by-Step Solution

The problem involves adding the fractions $\frac{1}{3}$ and $\frac{4}{9}$ .

Step 1: Identify the Least Common Denominator (LCD).

The denominators are 3 and 9. The least common multiple of 3 and 9 is 9. Thus, the LCD is 9.

Step 2: Convert the fractions to have the common denominator.

The fraction $\frac{1}{3}$ must be converted to have the denominator of 9. Multiply both the numerator and denominator by 3:
$\frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9}$
The fraction $\frac{4}{9}$ already has the denominator of 9, so it remains unchanged.

Step 3: Add the equivalent fractions.

Add the numerators together, keeping the denominator:
$\frac{3}{9} + \frac{4}{9} = \frac{3+4}{9} = \frac{7}{9}$

Step 4: Simplify the result, if necessary.

The fraction $\frac{7}{9}$ is already in simplest form.

Therefore, the solution to the problem is $\frac{7}{9}$ .

Answer

$\frac{7}{9}$

Exercise #3

Complete the following exercise:

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

Video Solution

Step-by-Step Solution

To find the result of dividing $\frac{2}{4}$ by $\frac{4}{3}$ , follow these steps:

Step 1: Simplify the fraction $\frac{2}{4}$ . This becomes $\frac{1}{2}$ because both the numerator and the denominator can be divided by 2.
Step 2: Find the reciprocal of the fraction $\frac{4}{3}$ . The reciprocal is $\frac{3}{4}$ because it exchanges the numerator and the denominator.
Step 3: Multiply $\frac{1}{2}$ by $\frac{3}{4}$ . $\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}$
Step 4: The result is $\frac{3}{8}$ , which is already in its simplest form.

Therefore, the solution to the problem $\frac{2}{4}:\frac{4}{3}$ is $\frac{3}{8}$ .

Answer

$\frac{3}{8}$

Exercise #4

$\frac{1}{4}+\frac{7}{8}=$

Video Solution

Step-by-Step Solution

To find the sum $\frac{1}{4} + \frac{7}{8}$ , follow these steps:

Step 1: Identify the least common denominator (LCD) of the fractions. The denominators 4 and 8 have an LCD of 8.
Step 2: Convert $\frac{1}{4}$ to an equivalent fraction with a denominator of 8. Multiply both the numerator and the denominator by 2: $\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$ .
Step 3: The second fraction, $\frac{7}{8}$ , already has the correct denominator. Therefore, it remains $\frac{7}{8}$ .
Step 4: Add the numerators of the two fractions: $\frac{2}{8} + \frac{7}{8} = \frac{2+7}{8} = \frac{9}{8}$ .

Therefore, the sum of $\frac{1}{4}$ and $\frac{7}{8}$ is $\frac{9}{8}$ .

Answer

$\frac{9}{8}$

Exercise #5

$\frac{1}{4}\times\frac{4}{5}=$

Video Solution

Step-by-Step Solution

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

1*4 = 4

4*5 = 20

4/20

Note that we can simplify this fraction by 4

4/20 = 1/5

Answer

$\frac{1}{5}$

Do you think you will be able to solve it?

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

Solve the following exercise:

$\frac{1}{4}+\frac{3}{6}=\text{?}$

Start practice

Operations with Fractions