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

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

Video Solution

Step-by-Step Solution

To solve this problem, we'll begin by finding a common denominator for the fractions $\frac{1}{3}$ and $\frac{1}{4}$ .
Step 1: Identify the denominators, which are 3 and 4. Multiply these to get a common denominator: $3 \times 4 = 12$ .

Step 2: Convert each fraction to an equivalent fraction with the common denominator of 12.

To convert $\frac{1}{3}$ to a denominator of 12, multiply both the numerator and the denominator by 4: $\frac{1 \times 4}{3 \times 4} = \frac{4}{12}$ .
To convert $\frac{1}{4}$ to a denominator of 12, multiply both the numerator and the denominator by 3: $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$ .

Step 3: Add the resulting fractions: $\frac{4}{12} + \frac{3}{12} = \frac{4 + 3}{12} = \frac{7}{12}$ .

Thus, the sum of $\frac{1}{3}$ and $\frac{1}{4}$ is $\frac{7}{12}$ .

Answer

$\frac{7}{12}$

Exercise #2

Complete the following exercise:

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

Video Solution

Step-by-Step Solution

To solve the problem of dividing two fractions, we apply the method of multiplication by the reciprocal:

Step 1: The given problem is $\frac{3}{4} \div \frac{5}{6}$ .
Step 2: Find the reciprocal of the divisor $\frac{5}{6}$ , which is $\frac{6}{5}$ .
Step 3: Multiply the dividend $\frac{3}{4}$ by the reciprocal of the divisor: $\frac{3}{4} \times \frac{6}{5}$
Step 4: Perform the multiplication: $\frac{3 \times 6}{4 \times 5} = \frac{18}{20}$
Step 5: Simplify the resulting fraction $\frac{18}{20}$ : $\frac{18 \div 2}{20 \div 2} = \frac{9}{10}$

Therefore, the correct result of the division $\frac{3}{4} \div \frac{5}{6}$ is $\frac{9}{10}$ .

Answer

$\frac{9}{10}$

Exercise #3

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

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Identify the least common multiple (LCM) of the denominators 8 and 4.
Step 2: Convert $\frac{1}{4}$ to a fraction with the denominator of 8.
Step 3: Add the fractions $\frac{3}{8}$ and $\frac{2}{8}$ .

Now, let's work through each step:
Step 1: The LCM of 8 and 4 is 8, so this will be the common denominator.
Step 2: Transform $\frac{1}{4}$ into a fraction with the denominator of 8. Multiply the numerator and the denominator by 2 to get $\frac{2}{8}$ .
Step 3: Now, add the fractions with the same denominator: $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$ .

Therefore, the sum of $\frac{3}{8}$ and $\frac{1}{4}$ is $\frac{5}{8}$ , which matches choice 4.

Answer

$\frac{5}{8}$

Exercise #4

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

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Identify the denominators of the fractions.
Step 2: Because the denominators are the same, add the numerators.
Step 3: Simplify the resulting fraction if necessary.

Now, let's work through each step:
Step 1: Both fractions, $\frac{1}{4}$ and $\frac{3}{4}$ , have the same denominator, 4.
Step 2: Since the denominators are the same, we can add the numerators: $1 + 3 = 4$ .
Step 3: The resulting fraction is $\frac{4}{4}$ , which simplifies to $1$ .

Therefore, the solution to the problem is $1$ .

Answer

$1$

Exercise #5

Solve the following:

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

Step-by-Step Solution

To solve $\frac{5}{9} \div \frac{7}{18}$ , we will proceed with the following steps:

Step 1: Identify the dividend and divisor: $\frac{5}{9}$ and $\frac{7}{18}$ .
Step 2: Find the reciprocal of the divisor. The reciprocal of $\frac{7}{18}$ is $\frac{18}{7}$ .
Step 3: Multiply the dividend by the reciprocal of the divisor: $\frac{5}{9} \times \frac{18}{7}.$
Step 4: Multiply the numerators: $5 \times 18 = 90$ .
Step 5: Multiply the denominators: $9 \times 7 = 63$ .
Step 6: The resulting fraction from the multiplication is $\frac{90}{63}.$
Step 7: Simplify $\frac{90}{63}$ . The greatest common divisor (GCD) of 90 and 63 is 9. Divide both the numerator and the denominator by 9: $\frac{90 \div 9}{63 \div 9} = \frac{10}{7}.$
Step 8: Convert $\frac{10}{7}$ into a mixed number. Since $10$ divided by $7$ is $1$ with a remainder of $3$ , this is $1\frac{3}{7}.$

So, the solution to the problem is $1\frac{3}{7}$ .

Answer

$1\frac{3}{7}$

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