• A proper fraction is a fraction in which the numerator is smaller than the denominator.

For example: 

\(\dfrac{2}{3}\begin{matrix} \longrightarrow\text{Numerator}\\ \;\;\,\longrightarrow\text{Denominator} \end{matrix}\)

Since \(2\) is smaller than \(3\), therefore \(\dfrac {2}{3}\) is a proper fraction.

• A unit fraction is a fraction in which the numerator is 1.

For example:

\(\dfrac {1}{2},\,\dfrac {1}{3},\,\dfrac {1}{8},\,\dfrac {1}{11}\) are unit fractions.

• A proper fraction is a fraction in which the numerator is smaller than the denominator.

For example: 

\(\dfrac{2}{3}\begin{matrix} \longrightarrow\text{Numerator}\\ \;\;\,\longrightarrow\text{Denominator} \end{matrix}\)

Since \(2\) is smaller than \(3\), therefore \(\dfrac {2}{3}\) is a proper fraction.

• To understand the concept of proper fractions using fraction for shaded or unshaded parts of a figure, consider this example:
• Ronnie and Jacob together buy 3 pizzas, as shown in figure b₁.

• Each pizza has 4 equal slices.
• If they eat 9 slices in total and leave the remaining pizza for Ronnie's sister, then

'how many pizza slices are left for Ronnie's sister?'

Consider figure b₂ for the remaining pizza slices.

• We can observe that each slice represents \(\dfrac{1}{4}\) or one-quarter and only 3 slices are left.

\(\therefore\) In fraction, we can represent it as-

\(\dfrac{3}{4}\)

Now, in \(\dfrac{3}{4}\), the numerator 3 is less than the denominator 4.

Thus, \(\dfrac{3}{4}\) is a proper fraction.

A unit fraction is a fraction in which the numerator is \(1\).

For example: \(\dfrac {1}{2},\dfrac {1}{3},\dfrac {1}{8},\dfrac {1}{11}\) are unit fractions.

Let us consider \(\dfrac {1}{b}\) as a unit fraction.

To represent \(\dfrac {1}{b}\) on a number line:

• Define interval from \(0\) to \(1\).
• Divide it into 'b' equal parts.
• The size of each part is \(\dfrac {1}{b}\).
• The length from point \(0\) to \(\dfrac {1}{b}\) represents the unit fraction.

For example: Represent \(\dfrac {1}{5}\) on a number line.

Step 1 : Define the interval from \(0\) to \(1\).

Step 2 : Divide it into 5 equal parts.

Step 3 : Size of each part is \(\dfrac {1}{5}\).

Step 4 : By observing the number line, we can say that the distance from \(0\) to point \(\dfrac {1}{5}\) represents the unit fraction.

• An improper fraction is a fraction in which the numerator is larger than the denominator.

For example:

\(\dfrac{5}{2}\begin{matrix} \longrightarrow\text{Numerator}\\ \;\;\,\longrightarrow\text{Denominator} \end{matrix}\)  

• Since \(5\) is larger than \(2\), therefore \(\dfrac{5}{2}\) is an improper fraction. 

• Since \(5\) is larger than \(2\)

\(\therefore\;\dfrac{5}{2}\) is an improper fraction. 

• An improper fraction is a fraction in which the numerator is larger than the denominator.

For example: \(\dfrac{5}{2}\)

\(5\) is larger than \(2.\)

\(\therefore\;\dfrac{5}{2}\) is an improper fraction.

• To understand improper fractions by using fraction for shaded or unshaded parts of a figure, consider this example:
• Mr. James buys two pizzas, as shown in figure \(a_1\).
• Each pizza has \(4\) equal slices.

• If he eats \(3\) slices out of \(4\) of one pizza then "how many pizza slices are left altogether?"
• Consider figure \(a_2\) for the remaining pizza slices.
• We can observe that each slice represents \(\dfrac{1}{4}\) or one-quarter.

• \(\therefore\) We have five quarters, i.e., \(\dfrac{5}{4}\)

• Now, in \(\dfrac{5}{4}\), the denominator \(5\) is greater than the numerator \(4.\)

• Thus, it is an improper fraction.

• We can also write \(\dfrac{5}{4}\) as:
• \(\dfrac{5}{4}=\dfrac{4}{4}+\dfrac{1}{4}\)\(=1+\dfrac{1}{4}\)
• Here, \(1\) represents one whole pizza and \(\dfrac{1}{4}\) represents one-fourth part(one slice of second pizza) that is left.

• We can observe that each slice represents \(\dfrac{1}{4}\) or one-quarter.

\(\therefore\) We have five quarters, i.e., \(\dfrac{5}{4}\)

• Now, in \(\dfrac{5}{4}\), the denominator \(5\) is greater than the numerator \(4.\)

Thus, it is an improper fraction.

• We can also write \(\dfrac{5}{4}\) as:

\(\dfrac{5}{4}=\dfrac{4}{4}+\dfrac{1}{4}\)

\(1+\dfrac{1}{4}\)

• Here, \(1\) represents one whole pizza and \(\dfrac{1}{4}\) represents one-fourth part that is left.

To represent the fraction \(\dfrac {a}{b}\) on a number line:

• Define the interval starting from zero.
• Divide each interval into 'b' equal parts.
• The size of each part is \(\dfrac {1}{b}\).
• The fraction \(\dfrac {a}{b}\) represents the combined length of 'a' parts of size \(\dfrac {1}{b}\).

For example: Represent \(\dfrac {3}{4}\)  on a number line.

Step 1: Define the interval from 0 to 1.

Step 2: Divide the interval into 4 equal parts.

Step 3 : The size of each part is \(\dfrac {1}{4}\).

Step 4: Thus, \(\dfrac {3}{4}\) represents the combined length of \(3\) parts.

Here,

\(\dfrac {0}{4}+\dfrac {1}{4} =\dfrac {1}{4}\)

\(\dfrac {1}{4}+\dfrac {1}{4} =\dfrac {2}{4}\)

\(\dfrac {1}{4}+\dfrac {1}{4}+\dfrac {1}{4} =\dfrac {3}{4}\)

\(\dfrac {1}{4}+\dfrac {1}{4}+\dfrac {1}{4}+\dfrac {1}{4} =\dfrac {4}{4}=1\)

Step 5: The resulting number line representation is: