Mixed Fractions: Examples, Conversion Methods, and Mixed Equivalent Fractions

Mixed fraction is a type of fraction, represented by a combination of a whole number and a fraction.

• Fraction contains a numerator and a denominator.
• The numerator is above the line, while the denominator is below the line that separates these two.
• It is used to determine the parts of a whole object.
• Whole numbers are all positive integers including zero.

An example of a mixed fraction can be given as

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

In this example

• 2 is a whole number
• 3/4 is a fraction. 

Key Terms: Mixed fractions, Improper fractions, Proper fractions, Whole numbers, Addition and Subtraction of fractions, Integers, Types of fractions

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What is a Fraction?

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Fractions are defined in mathematics as parts of a whole.

• This whole can be a single thing or a collection of objects.
• When we cut a slice of cake from the whole, the portion is the fraction of the cake.
• The word fraction comes from the Latin word "Fractus" which means "broken".
• Examples of fractions are 1/2, 3/7, 3/5, 8/10, etc.
• The part above the line of a fraction is known as the numerator and the part below the line is known as the denominator.

There are three main types of fractions. They are

• Proper fractions: In this type of fraction, the numerator is less than the denominator.
• Improper fractions: In this type of fraction, the number is greater than the denominator.
• Mixed fractions: It is a combination of a whole number and a proper fraction.

Also check: 

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Mixed Fraction

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A mixed fraction is a type of fraction that consists of a whole number and a proper fraction.

Consider the example shown below

In this example

• 2 is a whole number
• 1/7 is a fraction, known as a proper fraction.

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Conversion of Mixed Fraction to Improper Fraction

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A fraction is said to be an improper fraction if its numerator is greater than the denominator.

Consider an example of a mixed fraction

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

The steps below show how to convert a mixed fraction to an improper fraction

Step 1:

From the given mixed fraction, determine the whole number and proper fraction. In this case

• 3 is the whole number
• 1/2 is a proper fraction.

Step 2:

Determine the numerator part and denominator part from the proper fraction.

• 1 is numerator
• 2 is denominator

Step 3:

Multiply the denominator of the fraction with the whole number.

i.e. 2 x 3 = 6

Step 4: 

Add the result obtained in Step 3 and the numerator of the fraction.

i.e. 6 + 1 = 7

This will be the numerator part of the improper fraction. The denominator part will be the same as for the given fixed fraction

Step 5:

The required improper fraction can be given as \(\frac {7}{2}\)

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Conversion of Improper Fraction to Mixed Fraction

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Consider an example of a mixed fraction

\(\frac {15}{7}\)

The steps below show how to convert an improper fraction to a mixed fraction

Step 1:

Divide the numerator part by the denominator part of the improper fraction.

i.e. 15 ÷ 7

Step 2:

The Quotient part of the answer obtained in Step 1 will be the integer part (whole number part) of the mixed fraction.

In this case, it is 2.

Step 3:

The remainder part of the answer obtained in Step 1 will be the numerator part of the mixed fraction.

In this case, it is 1.

Step 4:

The denominator part of the required mixed fraction will be the same as that of the numerator of the given improper fraction.

In this case, it is 7.

Step 5:

Therefore the required mixed fraction can be given as \(2 \frac {1}{7}\)

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Adding Mixed Fraction

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Adding mixed fractions can be done in two ways

1. If the denominator of the mixed fractions are same
2. If the denominator of the mixed fractions is different

Adding mixed fractions having the same denominator

Consider an example \(2\frac {2}{3}+2\frac {1}{3}\)

Step 1:

Convert the given mixed fractions to improper fractions

• \(2 \frac {2}{3}=\frac {8}{3}\)
• \(2 \frac {1}{3}=\frac {7}{3}\)

Step 2:

Add the numerators of both improper fractions.

i.e. 8 + 7 = 15

This will be the numerator part of the final result.

Step 3:

The denominator of the given fractions will be the denominator part of the final result i.e. 3

Step 4:

The final result is \(\frac {15}{3}=5\)

Adding mixed fractions having different denominators

Consider an example \(2\frac {1}{2}+2\frac {1}{4}\)

Step 1:

Convert the given mixed fractions to improper fractions

• \(2 \frac {1}{2}=\frac {5}{2}\)
• \(2 \frac {1}{4}=\frac {9}{4}\)

Step 2:

Find the LCM of the denominators.

The LCM of 2 and 4 is 4.

Step 3:

Multiply both Denominators and Numerators of both fractions with a number such that they have the LCM as their new Denominator.

Multiply the numerator and Denominator of  5/2 with 2 and 9/4 with 1.

Step 4:

Add the Numerator and keep the Denominators the same.

10/4 + 9/4 = 19/4

Step 5:

If the answer obtained in Step 4 is an improper fraction, then convert it into a mixed fraction i.e. \(4 \frac {3}{4}\)

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Subtracting Mixed Fraction

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Subtracting mixed fractions can be done in two ways

1. If the denominator of the mixed fractions are same
2. If the denominator of the mixed fractions is different

Subtracting mixed fractions having the same denominator

Consider an example \(2\frac {2}{3}-2\frac {1}{3}\)

Step 1:

Convert the given mixed fractions to improper fractions

• \(2 \frac {2}{3}=\frac {8}{3}\)
• \(2 \frac {1}{3}=\frac {7}{3}\)

Step 2:

Subtract the numerators of both improper fractions.

i.e. 8 - 7 = 1

This will be the numerator part of the final result.

Step 3:

The denominator of the given fractions will be the denominator part of the final result i.e. 3

Step 4:

The final result is \(\frac {1}{3}\)

Subtracting mixed fractions having different denominators

Consider an example \(2\frac {1}{2}-2\frac {1}{4}\)

Step 1:

Convert the given mixed fractions to improper fractions

• \(2 \frac {1}{2}=\frac {5}{2}\)
• \(2 \frac {1}{4}=\frac {9}{4}\)

Step 2:

Find the LCM of the denominators.

The LCM of 2 and 4 is 4.

Step 3:

Multiply both Denominators and Numerators of both fractions with a number such that they have the LCM as their new Denominator.

Multiply the numerator and Denominator of  5/2 with 2 and 9/4 with 1.

Step 4:

Subtract the Numerator and keep the Denominators the same.

10/4 - 9/4 = 1/4

Step 5:

Therefore the final answer is \(\frac {1}{4}\)

If the answer obtained in Step 4 will be an improper fraction, then convert it into a mixed fraction.

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Multiplying Mixed Fraction

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Multiplying mixed fractions refers to the procedure of multiplying any two mixed fractions.

Consider an example \(2\frac {1}{2}\times 2\frac {1}{4}\)

Step 1:

Convert the given mixed fractions to improper fractions

• \(2 \frac {1}{2}=\frac {5}{2}\)
• \(2 \frac {1}{4}=\frac {9}{4}\)

Step 2:

Multiply the numerators of both fractions separately and the denominators of both fractions separately i.e.

\(\frac {5 \times 9}{2 \times 4} = \frac {45}{8}\)

This can be the required result.

Step 3:

If the result is an improper fraction, then convert it into a mixed fraction.

In this case, \(\frac {45}{8}=5 \frac {5}{8}\) 

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Mixed Equivalent Fractions

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When the values of two fractions are similar after simplification, they are said to be equivalent fractions.

For example: \(\frac {1}{2}\) and \(\frac{2}{4}\) are two equivalent fractions since \(\frac{2}{4} = \frac {1}{2}\)

In nature, two mixed fractions that are equal to each other are equivalent. As a result, if we convert any two equal fractions into mixed fractions, the quotient remaining after dividing the numerator by the denominator must be the same.

For example: \(\frac{5}{2}\) and \(\frac{10}{4}\) are two equivalent fractions.

• When we divide 5 by 2 we get a quotient equal to 2 and a remainder equal to 1.
• So \(\frac{5}{2}\) could be written in the form of a mixed fraction as \(2\frac{1}{2}\).
• Similarly, when we divide 10 by 4 we get a quotient equal to 2 and the remainder equal to 2.
• Therefore, \(\frac {10}{4}=2\frac{2}{4}\).

Hence, for both mixed fractions \(2\frac{1}{2}\) and \(2\frac{2}{4}\), the quotient value is equal to 2.

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Things to remember

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• The ratio of a fraction can only contain whole numbers or positive integers.
• The numerator indicates how many components of the total are represented by the fraction.
• The Denominator indicates how many equal portions are separated into the entire.
• Fraction division and multiplication do not require a common denominator.
• A common denominator is required for fraction addition and subtraction.

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