# Fractions: Improper Fractions & Mixed Numbers

Fractions are parts of a whole, for instance, \begin{align*} \frac{1} {2} \end{align*}. The three types of fractions are proper fractions, improper fractions and mixed numbers. A fraction is denoted as \begin{align*} \frac{x} {y} \end{align*}, where $x$ is the numerator and $y$ is the denominator.

1. Understanding improper fractions and mixed numbers
2. Express improper fractions as mixed numbers
3. Express mixed numbers as improper fractions

## 1. Understanding Improper Fractions And Mixed Numbers

• An improper fraction has a value equal to or greater than $1$
• The numerator is equal to or greater than the denominator.
• \begin{align*} \frac{4} {4} \end{align*}, \begin{align*} \frac{5} {4} \end{align*} and \begin{align*} \frac{10} {3} \end{align*} are examples of  improper fractions.
• A mixed number is made up of a whole number and a fraction.
• \begin{align*} 3\frac{1} {4} \end{align*}, \begin{align*} 1\frac{5} {7} \end{align*} and \begin{align*} 20\frac{2} {9} \end{align*} are examples of mixed numbers.
• An improper fraction can be expressed as a mixed number.

Example:

\begin{align*} \frac{23} {4} \end{align*} is an improper fraction.

\begin{align*} 5\frac{3} {4} \end{align*} is a mixed number.

Question 1:

Which of the following is an improper fraction?

1. \begin{align*} \frac{3} {4} \\ \end{align*}
2. \begin{align*} \frac{7} {8} \end{align*}
3. \begin{align*} \frac{5} {4} \end{align*}
4. \begin{align*} 5\frac{1} {5} \end{align*}

(3) \begin{align*} \frac{5} {4} \end{align*}

Question 2:

Which of the following is not an improper fraction?​​​

1. \begin{align*} \frac{8} {7} \end{align*}
2. \begin{align*} \frac{15} {3} \end{align*}
3. \begin{align*} \frac{9} {9} \end{align*}
4. \begin{align*} \frac{7} {10} \end{align*}

(4) \begin{align*} \frac{7} {10} \end{align*}

Question 3:

Which of the following is a mixed number?

1. \begin{align*} \frac{9} {7} \end{align*}
2. \begin{align*} 3\frac{6} {13} \end{align*}
3. \begin{align*} \frac{8} {8} \end{align*}
4. \begin{align*} \frac{1} {10} \end{align*}

(2) \begin{align*} 3\frac{6} {13} \end{align*}

Question 4:

What mixed number does the following represent?

1. \begin{align*} \frac{9} {4} \end{align*}
2. \begin{align*} \frac{63} {4} \end{align*}
3. \begin{align*} 6\frac{3} {4} \end{align*}
4. \begin{align*} 7\frac{3} {4} \end{align*}

Solution:

(3) \begin{align*} 6\frac{3} {4} \end{align*}

Question 5:

What improper fraction does the following represent?

Solution:

\begin{align*} \frac{8} {8}+\frac{8} {8}+\frac{5} {8} = \frac{21} {8} \end{align*}

\begin{align*} \frac{21} {8} \end{align*}

## 2. Express Improper Fractions As Mixed Numbers

There are 2 methods to express improper fractions as mixed numbers:

1. Expressing to the greatest possible number of wholes
2. Long Division

Question 1:

How many sevenths are there in 8 wholes?

Solution:

\begin{align*} ​1 \;\text{whole} &= \frac {7}{7} \\ \\ & = 7 \;\text{sevenths} \\ \\ \\ ​ 8 \;\text{wholes} &= \frac {56}{7} \\ \\ &= 56 \;\text{sevenths} ​ \end{align*}

$56$

Question 2:

How many fifths are there in 3 wholes?

Solution:

\begin{align*} ​1 \;\text{whole} &= \frac {5}{5} \\ \\ & = 5 \;\text{fifths} \\ \\ \\ ​ 3 \;\text{wholes} &= \frac {15}{5} \\ \\ &= 15 \;\text{fifths} ​ \end{align*}

$15$

Question 3:

Express \begin{align*} \frac{28} {5} \end{align*} as a mixed number.

Solution:

\begin{align*} \frac{28} {5} &=\frac{25} {5} +\frac{3} {5} \\ \\ &= 5+\frac{3} {5} \\ \\ &= 5\frac{3} {5} \end{align*}

or

\begin{align*} 5\frac{3} {5} \end{align*}

Question 4:

Express \begin{align*} \frac{26} {3} \end{align*} as a mixed number.

Solution:

\begin{align*} \frac{26} {3} &=\frac{24} {3} +\frac{2} {3} \\ \\ &= 8+\frac{2} {3} \\ \\ &= 8\frac{2} {3} \end{align*}

or

\begin{align*} 8\frac{2} {3} \end{align*}

Question 5:

Convert \begin{align*} \frac{19} {3} \end{align*} to a mixed number.

Solution:

\begin{align*} \frac{19} {3} &=\frac{18} {3} +\frac{1} {3} \\ \\ &= 6+\frac{1} {3} \\ \\ &= 6\frac{1} {3} \end{align*}

or

\begin{align*} 6\frac{1} {3} \end{align*}

Question 6:

Convert \begin{align*} \frac{26} {5} \end{align*} to a mixed number.

Solution:

\begin{align*} \frac{26} {5} &=\frac{25} {5} +\frac{1} {5} \\ \\ &= 5+\frac{1} {5} \\ \\ &= 5\frac{1} {5} \end{align*}

or

\begin{align*} 5\frac{1} {5} \end{align*}

## 3. Express Mixed Numbers As Improper Fractions

We can express mixed numbers as improper fractions by expressing the wholes as improper fractions and then adding the remaining fraction.

Example:

\begin{align*} 3\frac{2} {5} &=3 +\frac{2} {5} \\ \\ &= \frac{15} {5}+\frac{2} {5} \\ \\ &= \frac{17} {5} \end{align*}

The short-cut method is as shown.

Question 1:

Convert \begin{align*} 8\frac{3} {8} \end{align*} to an improper fraction.

1. \begin{align*} \frac{8} {67} \end{align*}
2. \begin{align*} \frac{24} {8} \end{align*}
3. \begin{align*} \frac{67} {8} \end{align*}
4. \begin{align*} \frac{83} {8} \end{align*}

Solution:

\begin{align*} 8\frac{3} {8} &=8 +\frac{3} {8} \\ \\ &= \frac{64} {8}+\frac{3} {8} \\ \\ &= \frac{67} {8} \end{align*}

The short-cut method is as shown.

(3) \begin{align*} \frac{67} {8} \end{align*}

Question 2:

Convert \begin{align*} 6\frac{4} {7} \end{align*} to an improper fraction.

Solution:

\begin{align*} 6\frac{4} {7} &=6 +\frac{4} {7} \\ \\ &= \frac{42} {7}+\frac{4} {7} \\ \\ &= \frac{46} {7} \end{align*}

The short-cut method is as shown.

\begin{align*} \frac{46} {7} \end{align*}

Question 3:

Convert \begin{align*} 13\frac{1} {6} \end{align*} to an improper fraction.

Solution:

\begin{align*} 13\frac{1} {6} &=13 +\frac{1} {6} \\ \\ &= \frac{79} {6} \end{align*}

The short-cut method is as shown.

\begin{align*} \frac{79} {6} \end{align*}

Question 4:

How many thirds are there in \begin{align*} 2\frac{1} {3} \end{align*}?

1. \begin{align*} \frac{21} {3} \end{align*}
2. \begin{align*} \frac{7} {3} \end{align*}
3. \begin{align*} 21 \end{align*}
4. \begin{align*} 7 \end{align*}

Solution:

Express the mixed number as an improper fraction with denominator 3.

\begin{align*} 2\frac{1} {3} &=2 +\frac{1} {3} \\ \\ &= \frac{6} {3}+\frac{1} {3} \\ \\ &= \frac{7} {3} \\ \\ &= 7 \;\text{third} \end{align*}

The short-cut method is as shown.

(4) $7$

Question 5:

How many halves are there in \begin{align*} 5\frac{1} {2} \end{align*}?

Solution:

Express the mixed number as an improper fraction with denominator $2$.

\begin{align*} 5\frac{1} {2} &=5 +\frac{1} {2} \\ \\ &= \frac{10} {2}+\frac{1} {2} \\ \\ &= \frac{11} {2} \\ \\ &= 11 \;\text{halves} \end{align*}

The short-cut method is as shown.

\begin{align*} 11 \end{align*}

Question 6:

How many eighths are there in \begin{align*} 6\frac{3} {4} \end{align*}?

Solution:

Express the mixed number to an improper fraction with denominator \begin{align*} 8 \end{align*}.

\begin{align*} 6\frac{3} {4} &= \frac{27} {4} \\ \\ &= \frac{54} {8} \\ \\ &= 54 \; \text{eights} \end{align*}

The short-cut method is as shown.

\begin{align*} 54 \end{align*}

Continue Learning
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Multiplication And Division Decimals
Model Drawing Strategy Division
Fractions Factors And Multiples
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Conversion Of Time
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