Informative line

# 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.

#### 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

Denominator = Total number of parts = 8

Numerator = Number of shaded parts = 3

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

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 $$=\dfrac{3}{8}$$

#### 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.

Number of total parts are $$8$$, so we take $$8$$ as the denominator.

Number of shaded parts are $$5$$, so we take $$5$$ as the numerator.

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

Hence, option (C) is correct.

### Which fraction is representing the shaded parts?

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.

#### 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:

#### 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 $$1$$ .

Divide it into 6 equal parts.

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

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

Hence, option (D) is correct.

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

A
B
C
D

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.

#### 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.

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 $$4$$.

$$=\dfrac{4}{9}$$

Hence, option (C) is correct.

### 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

$$\dfrac{5}{9}$$

Option C is Correct