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.

In this article, the lesson objectives are: 

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

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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. \(\begin{align*} 5\frac{1} {5} \end{align*}\)

Answer:

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

Answer:

(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. \(\begin{align*} \frac{1} {10} \end{align*}\)

Answer:

(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:

Answer:

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

Question 5:

What improper fraction does the following represent? 

Solution: 

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

Answer: 

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

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:

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

Answer:

\(56\)

Question 2: 

How many fifths are there in 3 wholes? 

Solution:

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

Answer:

\(15\)

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

Answer:

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

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

Answer:

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

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

Answer:

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

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

Answer:

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

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.

Answer:

(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.

Answer:

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

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.

Answer:

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

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.

Answer:

(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.

Answer:

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

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.

Answer:

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