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# Mixed Fractions – Definition & Operations (With Examples)

July 26, 2022

In general, a fraction is a part of a whole, which consist of a numerator and a denominator. Sometimes in a fraction, there is a whole part, for example, $5 \frac{2}{3}$, where $5$ is a whole number and $\frac {2}{3}$ is a fraction. Such a fraction is called a mixed fraction.

## What is a Mixed Fraction?

A fraction that is formed by combining a whole number and a fraction is called a mixed fraction. For example, if we combine $3$  (a whole number) and $\frac {2}{3}$ (a fraction), we get a mixed fraction.

Note:

• Since a mixed number consists of two different numbers – whole number and fraction, it is called a mixed number
• A mixed number is always greater than $1$ since it has a whole number. (The whole number is always greater than a proper fraction as a proper fraction is a part of a whole)
• A mixed fraction can always be represented as an improper fraction

## Relation Between a Mixed Fraction and Improper Fraction

An improper fraction is another way of representing a mixed fraction. The difference between an improper fraction and the proper fraction is

• In the case of a proper fraction, the numerator is less than the denominator
• In the case of an improper fraction, the numerator is greater than the denominator

## Converting An Improper Fraction to A Mixed Fraction

As seen above there is a relation between an improper fraction and a mixed fraction. Since the numerator in case of the improper fraction is greater than or equal to the denominator, we can divide the numerator by the denominator to get a quotient and a remainder.

For example, $\frac {19}{3}$ is an improper fraction, where $19$ (numerator) is greater than $3$ (denominator).

Dividing $19$ by $3$, we get $6$  as quotient and $1$ remainder.

The following steps are used to convert an improper fraction to a mixed fraction.

Step 1: Divide the numerator with the denominator.

Step 2: Find the remainder.

Step 3: Arrange the numbers in the following way, quotient followed by a fraction of remainder and divisor.

### Examples

Ex 1: $\frac{19}{3}$

Dividing $19$ by $3$, we get $6$  as quotient and $1$ as remainder.

So, $\frac {19}{3} = 6 \frac {1}{3}$

Ex 2: $\frac {57}{5}$

Dividing $57$ by $5$, we get the quotient as $11$ and the remainder as $2$.

Therefore, $\frac {57}{5} = 11 \frac {2}{5}$.

## Converting A Mixed Fraction to An Improper Fraction

As we can convert an improper fraction to a mixed fraction, we can also perform the reverse process. That is we can convert a mixed fraction to an improper fraction.

The steps involved in converting a mixed fraction to an improper fraction are

Step 1: Multiply the denominator of the mixed fraction with the whole number part.

Step 2: Add the numerator to the product obtained from step 1.

Step 3: Write the improper fraction with the sum obtained from step 2 in the numerator/denominator form.

### Examples

Ex 1: $8 \frac {2}{3}$

The whole part is $8$

The fraction part is $\frac {2}{3}$, where the numerator is $2$ and the denominator is $3$.

Multiplying the denominator with the whole part and adding the numerator: $3 \times 8 + 2 = 26$

Therefore, the improper fraction of $8 \frac {2}{3}$ is $\frac {26}{3}$

Ex 2: $11 \frac {5}{7}$

The whole part is $11$

The fraction part is $\frac {5}{7}$ where the numerator is $5$ and the denominator is $7$.

The improper fraction of $11 \frac{5}{7} = \frac{11 \times 7 + 5}{7} = \frac {82}{7}$

## Arithmetic Operations on Mixed Fractions

As you can perform any of the four basic arithmetic operations (addition, subtraction, multiplication, and division) on proper fractions, similarly, you can also perform the following operations on mixed fractions.

• Subtraction
• Multiplication
• Division

To understand the addition of mixed fractions, let’s understand these with the help of the following example.

Add $2 \frac {1}{4}$ and $3 \frac {1}{4}$

$2 \frac {1}{4}$ indicates that $2$ whole parts ($1$ + $1$) ($1$ = all $4$ parts) are shaded blue.

$3 \frac {1}{4}$ indicates that $3$ whole parts ($1$ + $1$ + $1$) ($1$ = all $4$ parts) are shaded green.

Now, adding these two fractions, we get

Therefore, $2 \frac {1}{4} + 3 \frac {1}{4} = 5 \frac {1}{2}$

Let us understand this with the help of an example.

Add $4 \frac {1}{4}$ and $5 \frac {1}{2}$

#### Steps To Perform Addition of Mixed Fractions

The following steps will be used to add these two fractions (or any two fractions in general).

Step 1: Add the whole parts $4 + 5 = 9$

Step 2: Add the fractions $\frac{1}{4} + \frac{1}{2} = \frac{3}{4}$

Step 3: Write the whole part and fraction together

Step 4: $4 \frac {1}{4} + 5 \frac {1}{2} = 9 \frac {3}{4}$

#### Examples

Ex 1: Add $2 \frac {4}{7} + 5 \frac {2}{7}$

The whole part of $2 \frac {4}{7}$ is $2$

The whole part of $5 \frac {2}{7}$ is $5$

The fraction part of $2 \frac {4}{7}$ is $\frac {4}{7}$

The fraction part of $2 \frac {2}{7}$ is $\frac {2}{7}$

Adding whole parts: $2 + 5 = 7$

Adding fraction parts: $\frac {4}{7} + \frac {2}{7} = \frac {4 + 2}{7} = \frac {6}{7}$

Therefore, $2 \frac {4}{7} + 5 \frac {2}{7} = 7 \frac {6}{7}$

Ex 2: Add $8 \frac {5}{13} + 9 \frac {3}{13}$

The whole part of $8 \frac {5}{13}$ is $8$

The whole part of $9 \frac {3}{13}$ is $9$

The fraction part of $8 \frac {5}{13}$ is $\frac {5}{13}$

The fraction part of $9 \frac {3}{13}$ is  $\frac {3}{13}$

Adding whole parts: $8 + 9 = 17$

Adding fraction parts: $\frac {5}{13} + \frac {3}{13} = \frac {5 + 3}{13} = \frac {8}{13}$

Therefore, $8 \frac {5}{13} + 9 \frac {3}{13} = 17 \frac {8}{13}$

### Subtraction of Mixed Fractions

The process of subtraction is similar to that addition of mixed fractions. In subtraction also, the whole parts and fraction parts are subtracted separately, and then the resultant whole part and fraction part are written together.

#### Steps To Perform Subtraction of Mixed Fractions

The following steps will be used to add these two fractions (or any two fractions in general).

Step 1: Subtract the whole parts $9 – 3 = 6$

Step 2: Subtract the fractions $\frac {1}{2} – \frac {1}{4} = \frac {1}{4}$

Step 3: Write the whole part and fraction together

Step 4: $9 \frac {1}{2} – 3 \frac {1}{4} = 6 \frac {1}{4}$

#### Examples

Ex 1: $6 \frac {5}{7} – 2 \frac {2}{7}$

The whole part of $6 \frac {5}{7}$ is $6$

The whole part of $2 \frac {2}{7}$ is $2$

The fraction part of $6 \frac {5}{7}$ is $\frac {5}{7}$

The fraction part of $2 \frac {2}{7}$ is $\frac {2}{7}$

Subtracting the whole parts $6 – 2 = 4$

Subtracting the fraction parts $\frac {5}{7} – \frac {2}{7} = \frac {5 – 2}{7} = \frac {3}{7}$

Therefore, $6 \frac {5}{7} – 2 \frac {2}{7} = 4 \frac {3}{7}$

Ex 2: $16 \frac {11}{19} – 12 \frac {6}{19}$

The whole part of $16 \frac {11}{19}$ is $16$

The whole part of $12 \frac {6}{19}$ is $12$

The fraction part of $16 \frac {11}{19}$ is $\frac {11}{19}$

The fraction part of $12 \frac {6}{19}$ is $\frac {6}{19}$

Subtracting the whole parts $16 – 12 = 4$

Subtracting the fraction parts $\frac {11}{19} – \frac {6}{19} = \frac {11 – 6}{19} = \frac {5}{19}$

Therefore, $16 \frac {11}{19} – 12 \frac {6}{19} = 4 \frac {5}{19}$

### Multiplication of Mixed Fractions

The first step in multiplying mixed fractions is to change them into improper fractions. Next, follow the steps of multiplication of fractions to multiply them.

#### Steps for Multiplication of Mixed Fractions

Step 1: Change mixed fractions to improper fractions

Step 2: Multiply the numerators.

Step 3: Multiply the denominators.

Step 4: Reduce (if necessary) the resultant fraction to its lowest terms or simplest form.

#### Examples

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

$1 \frac {4}{5} = \frac {1\times 5 + 4}{5} = \frac {9}{5}$

$2 \frac {1}{3} = \frac {2\times 3 + 1}{3} = \frac {7}{3}$

$\frac {4}{5} \times 2 \frac {1}{3} = \frac {9}{5} \times \frac {7}{3} = \frac {9 \times 7}{5 \times 3} = \frac {63}{15} = 4 \frac {3}{15} = 4 \frac {1}{5}$

Ex 2: $3 \frac {4}{5} \times 2 \frac {5}{7}$

$3 \frac {4}{5} = \frac {3\times5 + 4}{5} = \frac {19}{5}$

$2 \frac {5}{7} = \frac {2\times 7 + 5}{7} = \frac {19}{7}$

$3 \frac {4}{5} \times 2 \frac {5}{7} = \frac {19}{5} \times \frac {19}{7} = \frac {19 \times 19}{5 \times 7} = \frac {361}{35} = 10 \frac {11}{35}$

Ex 3: $5 \frac {2}{9} \times 3 \frac {1}{6}$

$5 \frac {2}{9} = \frac {5\times 9 + 2}{9} = \frac {47}{9}$

$3 \frac {1}{6} = \frac {3\times 6 + 1}{6} = \frac {19}{6}$

$5 \frac {2}{9} \times 3 \frac {1}{6} = \frac {47}{9} \times \frac {19}{6} = \frac {47 \times 19}{9 \times 6} = \frac {893}{54} = 16 \frac {29}{54}$

### Division of Mixed Fractions

The first step in the division of a fraction with a mixed number is to change the mixed number to an improper fraction. Next, follow the steps of the division of fractions to divide the numbers.

#### Steps For Division Of Mixed Fractions

Step 1: Convert mixed fractions to improper fractions

Step 2: Write the reciprocal of the second fraction, i.e., reciprocal of $\frac {c}{d}$ which is $\frac {d}{c}$

Step 3: Multiply $\frac {a}{b}$ by $\frac {d}{c}$ using the steps mentioned below (These are the same as that of multiplying fractions)

Step 4: Multiply the numerators.

Step 5: Multiply the denominators.

Step 6: Reduce (if necessary) the resultant fraction to its lowest terms or simplest form.

#### Examples

Ex 1: Divide $5 \frac {2}{3}$ by  $2 \frac {6}{7}$ i.e., $5 \frac {2}{3} \div 2 \frac {6}{7}$

$5 \frac {2}{3} = \frac {5 \times 3 + 2}{3} = \frac {17}{3}$

$2 \frac {6}{7} = \frac {2 \times 7 + 6}{7} = \frac {20}{7}$

$5 \frac {2}{3} \div 2 \frac {6}{7} = \frac {17}{3} \div \frac {20}{7} = \frac {17}{3} \times \frac {7}{20}$

$= \frac {17 \times 7}{3 \times 20} = \frac {119}{60} = 1 \frac {59}{60}$

$5 \frac {2}{3} \div 2 \frac {6}{7} = 1 \frac {59}{60}$

Ex 2: Divide $8 \frac {4}{5}$ by $2 \frac {5}{6}$ i.e., $8 \frac {4}{3} \div 2 \frac {5}{6}$

$8 \frac {4}{5} = \frac {8 \times 5 + 4}{5} = \frac {44}{5}$

$2 \frac {5}{6} = \frac {2 \times 6 + 5}{6} = \frac {17}{6}$

$8 \frac {4}{5} \div 2 \frac {5}{6} = \frac {44}{5} \div \frac {17}{6} = \frac {44}{5} \times \frac {6}{17}$

$= \frac {44 \times 6}{5 \times 17} = \frac {264}{85} = 3 \frac {9}{85}$

Therefore, $8 \frac {4}{3} \div 2 \frac {5}{6} = 3 \frac {9}{85}$

## Conclusion

Mixed fractions are another way of representing improper fractions which consist of two parts – a whole part and a fraction part. You can perform any of the arithmetic operations on mixed fractions by converting them into improper fractions.

## Practice Problems

1. Convert the following improper fractions to mixed fractions
1. $\frac {22}{5}$
2. $\frac {69}{7}$
3. $\frac {38}{17}$
2. Convert the following mixed fractions to improper fractions
1. $5 \frac {3}{13}$
2. $7 \frac {2}{9}$
3. $14 \frac {6}{11}$
3. Add the following mixed fractions
1. $15 \frac {2}{19} + 7 \frac {4}{19}$
2. $2 \frac {2}{3} + 3 \frac {3}{5}$
3. $6 \frac {9}{15} + 2 \frac {3}{4}$
4. Subtract the following mixed fractions
1. $15 \frac {6}{7} – 12 \frac {2}{7}$
2. $11 \frac {8}{9} – 5 \frac {2}{9}$
3. $64 \frac {2}{3} – 35 \frac {1}{2}$
5. Multiply the following mixed fractions
1. $2 \frac {1}{2} \times 5 \frac {3}{4}$
2. $11 \frac {3}{4} \times 3 \frac {4}{5}$
3. $6 \frac {9}{11} \times 5 \frac {2}{3}$
6. Divide the following mixed fractions
1. $5 \frac {3}{4} \div 3 \frac {2}{3}$
2. $16 \frac {1}{2} \div 2 \frac {1}{3}$
3. $18 \frac {4}{5} \div 7 \frac {2}{9}$