A fraction represents a part of a whole. Whole numbers are complete numbers. However, fractions are numbers that are less than whole numbers or are numbers that lie between whole numbers. In a fraction, there are two numbers: the upper number is called the numerator, and the lower number is called the denominator. Based on the numerator and denominator that constitute a fraction, there may be different types of fractions such as proper fraction, improper fraction, unit fraction, equivalent fraction, like fraction, unlike fraction. A mixed fraction is a combination of a whole number and a fraction. Fractions help us show parts of things in everyday life.

In this article, we will learn about the concept of mixed fractions, in brief, steps to convert improper fractions to mixed fractions and vice versa, different mathematical operations on mixed fractions, and related solved examples.

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What is Mixed Fractions?

Mixed fractions are those fractions that are formed by combining a whole number with a fraction.

For example,

\(8\frac{1}{2}\) is a mixed fraction,

Upon further simplification, we get,

\(8+\frac{1}{2}\)

\(=\frac{17}{2}\) = 8.5

Formulas for Mixed Fractions

Mixed fractions and improper fractions are two ways to represent the same value. Knowing how to convert between them is very important. Here are the standard methods in simple terms:

1. Converting an Improper Fraction to a Mixed Fraction
   An improper fraction has a numerator larger than or equal to the denominator. To convert it into a mixed fraction:

• Divide the numerator by the denominator.
• The quotient becomes the whole number.
• The remainder becomes the numerator of the fraction, keeping the same denominator.

Formula: Improper Fraction = Numerator ÷ Denominator = Whole Number + Remainder / Denominator

Example:
7/3 = 2 1/3
(7 ÷ 3 = 2 remainder 1 → 2 1/3)

2. Converting a Mixed Fraction to an Improper Fraction
   To convert a mixed fraction into an improper fraction:

• Multiply the whole number by the denominator.
• Add the numerator of the fraction.
• Place the result over the original denominator.

Formula: Improper Fraction = (Whole Number × Denominator + Numerator) / Denominator

Example:
2 1/3 = (2 × 3 + 1) / 3 = 7/3

Steps to convert Improper fraction to Mixed fraction

An improper fraction is a type of fraction in which the numerator is greater than the denominator.

For example,

\(=\frac{6}{4}\), where the numerator “6” is greater than the denominator “4”, making it an improper fraction.

Let’s learn to convert this improper fraction to a mixed fraction.

Step 1) To divide the numerator with the denominator.

Step 2) Find the remainder from the given division.

Step 3) arrange the numbers in this way : \(\text quotient \frac{remainder}{divisor}\).

For example,

Steps to convert Mixed fraction to Improper Fraction

Just like how we could convert an improper fraction to a mixed fraction, now we will understand the steps to convert a mixed fraction to an improper fraction.

Step 1) Multiple the whole number of the mixed fraction to the denominator.

Step 2) Add the numerator to the product obtained in step 1, which becomes the numerator of the improper fraction.

Step 3) write the obtained result from step 2 as a fraction.

Let’s understand the concept in brief by taking an example,

Consider the fraction,

\(7\frac{3}{4}\)

Where “7” is the whole number and \(\frac{3}{4}\) is a fraction.

According to step 1,

We multiply the whole number with the denominator,

\(7\times 4\) = 28

Step 2) Add “28” to the numerator of the fraction = 28 + 3 = 31.

Hence, the obtained result in the form of an improper fraction is \(\frac{31}{4}\).

Learn about Difference Between Percentage and Percentile

Operations on Mixed Fractions

In this article, we will learn about the four basic operations on mixed fractions

1. Addition of Mixed Fractions.
2. Subtraction of Mixed fractions.
3. Multiplication of Mixed fractions.
4. Division of Mixed fractions.

Addition of Mixed Fractions

Learn the addition of mixed fractions by the following steps.

Step 1) Convert the given mixed fraction into an improper fraction.

Step 2) If the denominators of the given fractions are the same, then we add the numerators directly and obtain the result.

Step 3) If the denominators are different, then we find their LCM to make the denominators the same and then add the numerators to obtain the result.

For example,

\(2\frac{3}{4}and\ 3\frac{1}{2}\)

\(2\frac{3}{4}+\ 3\frac{1}{2}\)

\(2\frac{3}{4}=\frac{8+3}{4}=\frac{11}{4}\)

and

\(3\frac{1}{2}=\frac{6+1}{2}=\frac{7}{2}\)

hence ,

\(2\frac{3}{4}+\ 3\frac{1}{2}=\frac{11}{4}+\frac{7}{2}\).

For the denominators “4” and “2”, the LCM is 4,

Hence we multiply the fraction \(\frac{7}{2}\) by 2, to make the denominators equal,

\(\frac{11}{4}+\frac{7\times 2}{2\times 2}=\frac{11}{4}+\frac{14}{4}\)

\(=\frac{11+14}{4}\)

\(=\frac{25}{4}\)which is the required result after addition of two mixed fractions.

Learn about Like Fractions and Unlike Fractions

Subtraction of Mixed Fractions

The subtraction of mixed fractions is very similar to the concept of addition of mixed fractions.

Let’s understand this concept by the following steps,

Step 1) Convert the given mixed fraction into an improper fraction.

Step 2) If the denominators of the given fractions are the same, then we subtract the numerators directly and obtain the result.

Step 3) If the denominators are different, then we find their LCM to make the denominators the same and then subtract the numerators to obtain the result.

For example,

Consider two fractions,

\(5\frac{1}{3}and\ 3\frac{1}{2}\)

We can clearly see that the denominators are not equal.

The LCM of 3 and 2 is 6,

\(5\frac{1}{3}=\frac{16}{3}\)

And

\(3\frac{1}{2}=\frac{7}{2}\)

Multiplying \(\frac{16}{3}\) by 2 and \(\frac{7}{2}\) by 3, we get,

\(\frac{32}{6}-\frac{21}{6}\)

\(=\frac{11}{6}\) is the required result upon subtraction of two mixed fractions.

Multiplication of Two Mixed Fractions

Understand the following steps for multiplication of Two Mixed Fractions

Step 1) to convert the given mixed fractions to improper fractions.

Step 2) multiply the numerator with the numerator of the other fraction.

Step 3) multiply the denominator with the denominator of the other fraction.

Let’s understand the concept with the help of an example,

Multiply : \(3\frac{2}{4}and\ 5\frac{1}{2}\)

Let us simplify each of the given mixed fractions,

\(3\frac{2}{4}=\frac{12+2}{4}=\frac{14}{4}\)

and

\(5\frac{1}{2}=\frac{11}{4}\)

\(\frac{14}{4}\times \frac{11}{4}\)

\(\frac{154}{16}\)

\(\frac{77}{8}\)

Division of Mixed Fractions

Let’s understand the division of mixed fractions from the following steps,

Step 1) Convert the given mixed fraction into an improper fraction.

Step 2) To multiply the first fraction with the multiplicative inverse of the second fraction and obtain the result in its simplified form.

For example,

Consider the two fractions \(6\frac{1}{2}and\ 7\frac{1}{4}\)

\(6\frac{1}{2}=\frac{13}{2}\)

And \(7\frac{1}{4}=\frac{29}{4}\)

\(\frac{\frac{13}{2}}{\frac{29}{4}}\)

\(\frac{13}{2}\times \frac{4}{29}\)

\(\frac{26}{29}\) is the result obtained upon division of two improper fractions.

Types of Fractions

Fractions are numbers that represent a part of a whole, and they can be divided into three main types. Here is a detailed explanation of each type in simple terms:

1. Proper Fraction
   A proper fraction is a fraction where the numerator (the top number) is smaller than the denominator (the bottom number).
   Example: 3/4, 2/5, 7/8
2. Improper Fraction
   An improper fraction is a fraction where the numerator is greater than or equal to the denominator. This means the fraction is equal to or more than one whole.
   Example: 7/4, 9/3, 5/2
3. Mixed Fraction
   A mixed fraction is a combination of a whole number and a proper fraction. It is another way to write an improper fraction.
   Example: 1 3/4, 2 1/2, 3 5/6

Mixed Equivalent Fractions

Equivalent fractions are those fractions, where despite the values of numerator and denominator being different they have the same value as the result.

\(\frac{1}{2}and\frac{4}{8}\)Upon solving them we get the final answer as 1/2, which is an example of an equivalent fraction.

Similarly, consider two mixed fractions,

\(3\frac{2}{4}and\ 3\frac{1}{2}\)

Lets simplify each of them,

\(3\frac{2}{4}=\frac{14}{4}=\frac{7}{2}\)

\(3\frac{1}{2}=\frac{7}{2}\),

Hence

\(3\frac{2}{4}and\ 3\frac{1}{2}\) are an example of mixed equivalent fractions.

Properties of Mixed Fractions

Mixed fractions have some important properties that make them easy to use in calculations. Here they are in simple terms:

1. Combination of Whole Number and Fraction
   A mixed fraction is made up of a whole number and a proper fraction. For example, 2 1/3 has 2 as the whole number and 1/3 as the fraction.
2. Greater than 1
   Every mixed fraction is greater than 1 because it contains a whole number.
3. Can be Converted
   A mixed fraction can be easily converted into an improper fraction, and vice versa. This is useful for doing calculations like addition, subtraction, multiplication, and division.
4. Represents a Part and a Whole
   Mixed fractions show a quantity that includes a complete part (whole number) and a partial part (fraction).
5. Used in Real Life
   Mixed fractions are often used in measurements, cooking, and time, because they are easier to understand than improper fractions.

Mixed Fractions Solved Examples

Example 1: Add the following mixed fractions,
\(3\frac{1}{2}and\ 4\frac{3}{4}\).

Solution:

Lets simplify each of the given mixed fractions,

\(3\frac{1}{2}=\frac{7}{2}\)

And \(4\frac{3}{4}=\frac{19}{4}\).

Since the denominators are not equal,

The LCM of 2 and 4 is 4,

Hence

\(\frac{7}{2}=\frac{14}{4}\)

\(\frac{14}{4} +\frac{19}{4}\)

\(=\frac{33}{4}\).

Example 2: Convert the fraction \(\frac{9}{4}\) into a mixed fraction.

Solution: 

\(9\div 4\)

where ,

Quotient = 2

Remainder = 1

hence ,

We can write the given fraction in the form of mixed fraction as,

\(\text quotient \frac{remainder}{divisor}\).

\(2\frac{1}{4}\).

Example 3: Divide the two mixed fractions 2 1/4 and 6 1/2

Solution:

1. Convert the mixed fractions into improper fractions:
   • 2 1/4 = 9/4
   • 6 1/2 = 13/2
2. To divide fractions, multiply the first fraction by the reciprocal of the second fraction:
   9/4 ÷ 13/2 = 9/4 × 2/13
3. Multiply the numerators and denominators:
   (9 × 2) / (4 × 13) = 18/52
4. Simplify the fraction: 18/52 = 9/26

Answer: 9/26

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