Informative line

Unit Proper And Improper Fractions

Proper Fraction

  • 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.

Illustration Questions

Which one of the following fractions represents a proper fraction?

A \(\dfrac {3}{2}\)

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

C \(\dfrac {1}{2}\)

D \(\dfrac {7}{2}\)

×

In a proper fraction, the numerator is smaller than the denominator.

Only in option (C), i.e., \(\dfrac {1}{2}\)

Numerator (1) is smaller than the denominator (2).

\(\therefore\,\dfrac {1}{2}\) is a proper fraction.

Hence, option (C) is correct.

Which one of the following fractions represents a proper fraction?

A

\(\dfrac {3}{2}\)

.

B

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

C

\(\dfrac {1}{2}\)

D

\(\dfrac {7}{2}\)

Option C is Correct

Unit 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.

Illustration Questions

Which one of the following options represents the unit fraction?

A \(\dfrac {3}{1}\)

B \(\dfrac {1}{3}\)

C \(\dfrac {4}{5}\)

D \(\dfrac {3}{2}\)

×

A unit fraction has the numerator 1.

Only in option (B), i.e., \(\dfrac {1}{3}\)

Numerator = 1

\(\therefore\,\dfrac {1}{3}\) is a unit fraction.

Hence, option (B) is correct.

Which one of the following options represents the unit fraction?

A

\(\dfrac {3}{1}\)

.

B

\(\dfrac {1}{3}\)

C

\(\dfrac {4}{5}\)

D

\(\dfrac {3}{2}\)

Option B is Correct

Proper Fractions Using Models

  • 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.

Illustration Questions

Consider the given figures. Some parts of the 2 big squares are shaded. Find the fraction for the shaded parts of the given squares altogether.

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

B \(\dfrac{1}{8}\)

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

D \(\dfrac{3}{8}\)

×

Given:

image

By observing the given squares, we can say that each big square has 4 equal parts.

\(\therefore\) One part of a big square represents one-fourth or \(\dfrac{1}{4}\).

Calculating fraction for the shaded parts 'altogether'

Each part represents one-fourth \(\left(\dfrac{1}{4}\right)\) and here, total 3 parts are shaded.

So, we have three-fourths \(=\dfrac{3}{4}\)

Hence, option (C) is correct.

Consider the given figures. Some parts of the 2 big squares are shaded. Find the fraction for the shaded parts of the given squares altogether.

image
A

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

.

B

\(\dfrac{1}{8}\)

C

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

D

\(\dfrac{3}{8}\)

Option C is Correct

Number Line Representation of Unit Fractions

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.

Illustration Questions

Which one of the following number lines represents the correct position of the fraction \(\dfrac {1}{6}\)?

A

B

C

D

×

Given fraction: \(\dfrac{1}{6}\)

 

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

image

Divide it into 6 equal parts.

image

Size of each part is \(\dfrac{1}{6}\).

image

\(\dfrac{1}{6}\) represents the length of the first segment.

image

Hence, option (D) is correct.

Which one of the following number lines represents the correct position of the fraction \(\dfrac {1}{6}\)?

A image
B image
C image
D image

Option D is Correct

Improper Fractions

  • 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. 

Illustration Questions

Which one of the following fractions represents an improper fraction?

A \(\dfrac{12}{5}\)

B \(\dfrac{6}{7}\)

C \(\dfrac{3}{5}\)

D \(\dfrac{13}{15}\)

×

In an improper fraction, the numerator is larger than the denominator.

Only in option (A), i.e., \(\dfrac{12}{5}\)

The numerator \((12)\) is larger than the denominator \((5)\) .

\(\therefore\; \dfrac{12}{5}\) is an improper fraction.

Hence, option (A) is correct.

Which one of the following fractions represents an improper fraction?

A

\(\dfrac{12}{5}\)

.

B

\(\dfrac{6}{7}\)

C

\(\dfrac{3}{5}\)

D

\(\dfrac{13}{15}\)

Option A is Correct

Improper Fractions using Models

  • 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.

Illustration Questions

Consider the given figures, some parts of these circles are shaded. Which one of the following options represents fraction for the shaded parts of the given figures altogether?

A \(\dfrac{16}{25}\)

B \(\dfrac{15}{8}\)

C \(\dfrac{4}{64}\)

D \(\dfrac{64}{15}\)

×

By observing the given figures, we can say that each circle has \(8\) equal parts.

\(\therefore\) One part of a circle represents one-eighth or \(\dfrac{1}{8}\).

image

Calculating fraction for shaded parts 'altogether':

Each part represents one-eighth \(\left(\dfrac{1}{8}\right)\).

Here, total 15 parts are shaded.

\(\therefore\) We have fifteen-eighths  \(=\dfrac{15}{8}\).

Hence, option (B) is correct.

Consider the given figures, some parts of these circles are shaded. Which one of the following options represents fraction for the shaded parts of the given figures altogether?

image
A

\(\dfrac{16}{25}\)

.

B

\(\dfrac{15}{8}\)

C

\(\dfrac{4}{64}\)

D

\(\dfrac{64}{15}\)

Option B is Correct

Number Line Representation of a Fraction

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:

Illustration Questions

Which one of the following number lines represents the correct position of the fraction \(\dfrac {5}{2}\)?

A

B

C

D

×

Given fraction: \(\dfrac {5}{2}\)

Define the interval from \(0\) to \(3\).

image

Divide each interval into \(2\) equal parts.

image

The size of each part is \(\dfrac {1}{2 }\) .

image

Thus, \(\dfrac {5}{2}\) represents the combined length of  \(5\)  parts.

Here,

\(0+\dfrac {1}{2}=\dfrac {1}{2}\)

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

\(\dfrac {1}{2}+\dfrac {1}{2}+\dfrac {1}{2}=\dfrac {3}{2}\)

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

\(\dfrac {1}{2}+\dfrac {1}{2}+\dfrac {1}{2}+\dfrac {1}{2}+\dfrac {1}{2} =\dfrac {5}{2}\)

\(\dfrac {1}{2}+\dfrac {1}{2}+\dfrac {1}{2}+\dfrac {1}{2}+\dfrac {1}{2}+\dfrac {1}{2} =\dfrac {6}{2}=3\)

image

The resulting number line representation is:

image

Hence, option (A) is correct.

Which one of the following number lines represents the correct position of the fraction \(\dfrac {5}{2}\)?

A image
B image
C image
D image

Option A is Correct

Practice Now