Informative line

Formation Of Fractions

Fraction and Types of Fraction

  • A fraction consists of two numbers and both the numbers are separated by a line known as fraction bar. 

For example: \(\dfrac{2}{3},\dfrac{2}{7},\dfrac{4}{9}\) etc. are fractions.

  • A fraction is a part of whole.
  • A fraction has two parts:
  1. Numerator 
  2. Denominator
  • Numerator: The number written above the fraction bar is known as numerator.

For example: \(\dfrac{4}{5}\)

\(4\) is the numerator of the given fraction.

  • Denominator: The number written below the fraction bar is known as denominator.

For example: \(\dfrac{5}{7}\)

\(7\) is the denominator of the given fraction.

  • There are four types of fractions:
  1. Proper fractions
  2. Improper fractions
  3. Mixed numbers
  4. Unit fractions

1. Proper fraction: 

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

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

\(2\) is smaller than \(3\).

\(\therefore\;\;\dfrac{2}{3}\) is a proper fraction.

2. Improper fraction: 

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

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

\(5\) is larger than \(2\).

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

3. Mixed number:

  • A mixed number is a number which has both a whole number and a proper fraction.

For example: \(2\dfrac{1}{3}\)

In \(2\dfrac{1}{3}\), whole number \(=2\)

Proper fraction \(=\dfrac{1}{3}\)

\(\therefore\) \(2\dfrac{1}{3}\) is a mixed number.

4. 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 is a unit fraction?

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

B \(\dfrac{7}{4}\)

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

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

×

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

In option (C), the numerator is \(1\).

So, it is a unit fraction.

Hence, option (C) is correct.

Which one of the following is a unit fraction?

A

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

.

B

\(\dfrac{7}{4}\)

C

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

D

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

Option C is Correct

Formation of Fractions through Figures

  • To form the fractions through a figure, follow the steps given below:

Step 1  Count the number of total parts in the figure and take this as the denominator of the fraction.

Step 2  Count the number of parts for which fraction is to be calculated in the figure and take this as the numerator.
Example : If fraction is asked for shaded portion then count the number of shaded parts for numerator.

In the example of given circle :

Shaded parts \(=3\) 

Unshaded parts \(=5\)

Total number of parts = 8

Fraction for shaded parts:

Denominator = Total number of parts = 8

Numerator = Number of shaded parts = 3

\(\therefore\) The fraction representing shaded parts \(=\dfrac{3}{8}\) 

Fraction for unshaded parts:

Denominator = Total number of parts = 8

Numerator = Number of unshaded parts = 5

\(\therefore\) The fraction representing the unshaded parts\(=\dfrac{5}{8}\)

NOTE : If the figure is not divided into equal parts, then first we need to divide it into equal parts.

For example:

The given figure is not divided into equal parts, so first we make all parts equal.

Now, the figure is divided into 8 equal parts.

Number of shaded parts \(=3\)

Total number of parts are taken as denominator and number of shaded parts are taken as numerator.

So, the fraction representing the shaded parts \(=\dfrac{3}{8}\)

Illustration Questions

Which fraction is representing the shaded parts?

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

B \(\dfrac{5}{6}\)

C \(\dfrac{5}{8}\)

D \(\dfrac{3}{9}\)

×

In the given figure, all the parts are not equal.

So, first we make all parts equal.

image

Number of total parts are \(8\), so we take \(8\) as the denominator.

image

Number of shaded parts are \(5\), so we take \(5\) as the numerator.

image

So, the fraction representing the shaded parts \(=\dfrac{5}{8}\)

Hence, option (C) is correct.

Which fraction is representing the shaded parts?

image
A

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

.

B

\(\dfrac{5}{6}\)

C

\(\dfrac{5}{8}\)

D

\(\dfrac{3}{9}\)

Option C is Correct

Equivalent Fractions

  • Equivalent fractions are fractions which have the same value, even though they look different.

For example:

\(\dfrac {1}{2}=\dfrac {2}{4}=\dfrac {4}{8}\) are equivalent fractions.

  • The fundamental fact about equivalent fractions is that a fraction does not change when its numerator and denominator are multiplied or divided by a same non-zero whole number.
  • To understand it easily, let us consider the following example for the fraction \(\dfrac {1}{3}\):

Multiply both numerator and denominator by the same non-zero whole number, like 2 in this case.

\(\Rightarrow\dfrac {1×2}{3×2}=\dfrac {2}{6}\)

The Greatest Common Factor of 2 and 6 is 2, so we divide both the numerator and the denominator by 2.

We get \(\dfrac{1}{3}\) which is same as the original fraction.

The new fraction \(\left(\dfrac{2}{6}\right)\)  is the equivalent fraction of the original fraction.

Equivalent fraction of \(\dfrac {1}{3}=\dfrac {2}{6}\)

Though, \(\dfrac {1}{3}\) and \(\dfrac {2}{6}\) look different but they have the same value.

Illustration Questions

Which one of the following is an equivalent fraction of \(\dfrac {3}{2}\)?

A \(\dfrac {9}{6}\)

B \(\dfrac {6}{9}\)

C \(\dfrac {12}{4}\)

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

×

Given fraction: \(\dfrac {3}{2}\)

Option (A): \(\dfrac {9}{6}\)

To get 9 from 3, we need to multiply the numerator (3) by 3.

\(\therefore\) The denominator (2) will also get multiplied by 3.

Thus, the equivalent fraction of \(\dfrac {3}{2}=\dfrac {3×3}{2×3}=\dfrac {9}{6}\)

Hence, option (A) is correct.

Option (B): \(\dfrac {6}{9}\)

To get 6 from 3, we need to multiply the numerator (3) by 2.

\(\therefore\) The denominator (2) will also get multiplied by 2.

Thus, the equivalent fraction of \(\dfrac{3}{2}=\dfrac {3×2}{2×2}=\dfrac {6}{4}\)

Hence, option (B) is incorrect.

Option (C): \(\dfrac {12}{4}\)

To get 12 from 3, we need to multiply the numerator (3) by 4.

\(\therefore\) The denominator (2) will also get multiplied by 4.

Thus, the equivalent fraction of \(\dfrac {3}{2}\) =  \(\dfrac {3×4}{2×4}=\dfrac {12}{8}\)

Hence, option (C) is incorrect.

Option (D): \(\dfrac {15}{8}\)

To get 15 from 3, we need to multiply the numerator (3) by 5.

\(\therefore\) The denominator (2) will also get multiplied by 5.

Thus, the equivalent fraction of \(\dfrac {3}{2}\) =  \(\dfrac {3×5}{2×5}=\dfrac {15}{10}\)

Hence, option (D) is incorrect.

Which one of the following is an equivalent fraction of \(\dfrac {3}{2}\)?

A

\(\dfrac {9}{6}\)

.

B

\(\dfrac {6}{9}\)

C

\(\dfrac {12}{4}\)

D

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

Option A is Correct

Number Line Representation of Unit and Simple Fractions

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.

Number Line Representation of Simple Fractions

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 \(\dfrac {1}{6}\)?

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

Pictorial Representation of Fractions [Figure is Not Divided into Parts]

  • Pictorial representation of fractions simply means visual representation of fractions by using figures.

We can understand the concept with the following example:

For example: Shade \(\dfrac{5}{6}\) part of the given rectangle.

First, we divide the given rectangle into \(6\) equal parts because the denominator is \(6\) and denominator represents the total parts.

Now, we shade \(5\) parts of the rectangle.

Shaded part represents \(\dfrac{5}{6}\) of the figure.

Shaded part represents \(\dfrac{5}{6}\) of the figure.

Illustration Questions

Which one of the following fractions represents the shaded part for the given figure?

A \(\dfrac{2}{5}\)

B \(\dfrac{9}{4}\)

C \(\dfrac{4}{9}\)

D \(\dfrac{5}{9}\)

×

Divide the figure into equal parts as the denominator represents the total parts.

image

The figure is divided into \(9\) equal parts and there are \(4\) shaded parts.

The denominator represents the total parts of the figure, so the denominator is \(9\).

The numerator represents the number of shaded parts of the figure, so the numerator is \(4\).

Fraction representing the shaded parts  

\(=\dfrac{4}{9}\)

Hence, option (C) is correct.

Which one of the following fractions represents the shaded part for the given figure?

image
A

\(\dfrac{2}{5}\)

.

B

\(\dfrac{9}{4}\)

C

\(\dfrac{4}{9}\)

D

\(\dfrac{5}{9}\)

Option C is Correct

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