Fractions Class 6 Notes CBSE Maths Chapter 7 (Free PDF Download)

What have we discussed?

• A fraction is a number that represents a portion of something larger. 

• A single object or a group of objects might make up the whole.

• It is necessary to guarantee that all parts are equal when stating a circumstance of counting components to write a fraction.

• As in example  $\dfrac{8}{3}$ , the numerator in this equation is $8$ and the denominator is $3$.

Fraction on the Number Line:

• A number line can be used to display fractions. 

• Every fraction has a number line point connected with it.

• For example, how can we  $\dfrac{9}{7}$ on a number line?

Solution:

The value of $\dfrac{9}{7}$ is $1.23$ which means that $\dfrac{9}{7}$ lies between $1$ and $2$ . We can show this on a number is as follows; 

Like fractions are those that have the same denominator.

For example, $\dfrac{9}{30},\dfrac{5}{30},\dfrac{1}{30}$ are the like fractions.

Unlike fractions are fractions having different denominators.

For example, $\dfrac{3}{56},\dfrac{2}{30},\dfrac{8}{5}$ are the unlike fractions.

Proper Fraction:

• A proper fraction is a fraction that represents a portion of a whole.

• The numerator of a proper fraction is less than the denominator. 

• For example, $\dfrac{3}{5},\dfrac{7}{30},\dfrac{6}{5}$ are the proper fractions.

Improper Fraction:

• Improper fractions are those in which the numerator is higher than the denominator. 

• When an improper fraction is stated as a mixture of a whole and a part, it is referred to as a mixed fraction.

• There are several equivalent fractions for each right or improper fraction.

• A mixed fraction can be expressed as an incorrect fraction in the following way:

$\dfrac{\left( \text{Whole  }\!\!\times\!\!\text{  Denominator} \right)\text{ + Numerator}}{\text{Denominator}}$

• For example,  $4\dfrac{2}{5},3\dfrac{1}{9},12\dfrac{7}{8}$ are the improper fractions.

We can multiply or divide both the numerator and the denominator of a given fraction by the same number to discover an equal fraction.

For example, Equivalent fraction of $\dfrac{14}{56}$ is;

$  \dfrac{14}{56}=\dfrac{14\div 14}{56\div 14} $

$ \dfrac{14}{56}=\dfrac{1}{4} $

If the numerator and denominator of a fraction have no common factor except $1$ , it is said to be in the simplest (or lowest) form.

When adding or subtracting fractions, the $LCM$of the denominators is selected as the common denominator.

Comparing Unlike Fractions:

• The fraction with the greater numerator is greater when two fractions have the same denominator.

For example, if we compare $\dfrac{9}{3}$ and $\dfrac{4}{3}$ fractions, we can conclude that

$\dfrac{9}{3}>\dfrac{4}{3}$

That is 

 $\Rightarrow \dfrac{9}{3}>\dfrac{4}{3} $

 $ \Rightarrow 3>1.33 $ 

• When the numerator is the same in two fractions then, the fraction with the smaller denominator is greater of the two.

For example, if we compare \[\dfrac{7}{6}\] and $\dfrac{7}{3}$ fractions, we can conclude that \[\dfrac{7}{6}<\dfrac{7}{3}\]

That is 

$\Rightarrow \dfrac{7}{6}<\dfrac{7}{3}  $

$\Rightarrow 1.16<2.33  $

How do we add or subtract mixed fractions?

• Mixed fractions can be expressed as a whole portion plus a proper fraction or as an improper fraction in its entirety.

• One method of adding (or subtracting) mixed fractions is to do the operation separately for each of the whole parts.

• The alternative method is to write the mixed fractions as improper fractions and then add (or subtract) them directly.

• Example:

$4\dfrac{2}{3}+3\dfrac{1}{4}$ , We will solve the given example by applying both the methods.

Solution:

Method $I$:

$   \text{4}\dfrac{\text{2}}{\text{3}}\text{+3}\dfrac{\text{1}}{\text{4}}\text{ = 4+}\dfrac{\text{2}}{\text{3}}\text{+3+}\dfrac{\text{1}}{\text{4}}$

$\text{4}\dfrac{\text{2}}{\text{3}}\text{+3}\dfrac{\text{1}}{\text{4}}\text{ = 7+}\dfrac{\text{2}}{\text{3}}\text{+}\dfrac{\text{1}}{\text{4}}\text{       -----}\left( \text{1} \right)  $

Here,

    $~~\frac{2}{3}+\frac{1}{4}\text{ }=\text{ }\frac{2\text{ }\times \text{ }4}{3\text{ }\times \text{ }4}+\frac{1\text{ }\times \text{ }3}{4\text{ }\times \text{ }3}~~$

$~~~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }=\text{ }\frac{8}{12}+\frac{3}{12}~$

$~~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }=\frac{11}{12}$

\[\dfrac{\text{11}}{\text{12}}\] can be written as;

        $~\text{ }\frac{11}{12}~\text{ }=\text{ }\frac{12-1}{12}~$

$~\text{ }~\text{ }~\text{ }~\text{ }=\text{ }1\text{ }-\text{ }\frac{1}{12}~$

Therefore, equation \[\left( 1 \right)\] becomes,

$~7+\frac{2}{3}+\frac{1}{4}\text{ }=\text{ }7\text{ }+\text{ }1-\frac{1}{12}~$

$~~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }=\text{ }8\text{ }-\text{ }\frac{1}{12}~~$

\[~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }=\text{ }\frac{\left( 12\text{ }\times \text{ }8 \right)\text{ }-\text{ }1}{12}\]

Therefore,

$~~7+\frac{2}{3}+\frac{1}{4}\text{ }=\text{ }\frac{96\text{ }-\text{ }1}{12}~$

$~7+\frac{2}{3}+\frac{1}{4}\text{ }=\text{ }\frac{95}{12}~$

Method $II$:

\[\text{4}\dfrac{\text{2}}{\text{3}}\text{+3}\dfrac{\text{1}}{\text{4}}\text{ }\]

\[\text{4}\dfrac{\text{2}}{\text{3}}\] can be written as

$4\frac{2}{3}=\frac{\left( 4\times 3 \right)+2}{3}~~$

$~4\frac{2}{3}=\frac{12+2}{3}$

$4\frac{2}{3}=\frac{14}{3}$

\[3\dfrac{1}{4}\] can be written as

$~~3\frac{1}{4}=\frac{\left( 3\times 4 \right)+1}{4}$

$~3\frac{1}{4}=\frac{12+1}{4}$

$~~3\frac{1}{4}=\frac{13}{4}~~$

Therefore,

  $~7+\frac{2}{3}+\frac{1}{4}\text{ }=\text{ }\frac{14}{3}+\frac{13}{4}~~$

$~~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }=\text{ }\frac{\left( 14\text{ }\times \text{ }4 \right)+\left( 13\text{ }\times \text{ }3 \right)}{\left( 3\text{ }\times \text{ }4 \right)}~$

$~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }~\text{ }=\text{ }\frac{56+39}{12}$

Since,

\[\left( \text{LCM of 3 and 4 = 12} \right)\]

Therefore,

\[\text{7+}\dfrac{\text{2}}{\text{3}}\text{+}\dfrac{\text{1}}{\text{4}}\text{ = }\dfrac{\text{95}}{\text{12}}\]

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