5th Class Mathematics Operation on Fractions Operation on the Fraction

Operation on the Fraction

Category : 5th Class

Operation on the Fraction

Addition and Subtraction of Like Fractions

Like fractions have same denominator. In the operation of addition, numerators of the like fractions are added and their sum become the numerator for the required fraction and their common denominator becomes denominator. For the example:

\[\frac{P}{Q}+\frac{R}{Q}+\frac{P+R}{Q}=\frac{S}{Q}\] (Where\[S=P+R\]). In the operation of subtraction, difference of numerators is found

Ex: \[\frac{P}{Q}-\frac{R}{Q}=\frac{P-R}{Q}=\frac{S}{Q}\] (Where \[S=P-R\])

Add \[\frac{15}{7}\] and \[\frac{9}{7}\]

Explanation

Addition of \[\frac{15}{7}\] and \[\frac{9}{7}=\frac{15}{7}+\frac{9}{7}\]

\[=\frac{9+15}{7}=\frac{24}{7}.\]

Subtract \[\frac{9}{7}\] from \[\frac{15}{7}.\]

Solution:

\[\frac{15}{7}-\frac{9}{7}=\frac{15-9}{7}=\frac{6}{7}.\]

Addition and Subtraction of Unlike Fractions

In the operation of addition of unlike fractions, LCM of denominators is found. The LCM becomes denominator for the required fraction. Now the LCM is divided each of the denominators and quotient is multiplied with the respective numerates Sum of the products becomes numerator for the required fraction. For the example,

\[\frac{\text{P}}{\text{Q}}\text{+}\frac{\text{R}}{\text{S}}=\frac{(T\div Q)P+(T\div S)R}{\text{T}}=\frac{\text{Z}}{\text{T}}\]

[Where T is LCM of Q and S and \[\text{Z=(T }\!\!\div\!\!\text{ Q)P+(T }\!\!\div\!\!\text{ S)R}\,\text{ }\!\!]\!\!\text{ }\]

In case of subtraction, difference of the product becomes numerator for the required fraction

\[\frac{\text{P}}{\text{Q}}-\frac{\text{R}}{\text{S}}=\frac{(T\div Q)P-(T\div S)R}{\text{T}}=\frac{\text{Z}}{\text{T}}\]

[Where T is LCM of Q and S and \[\text{z=(T }\!\!\div\!\!\text{ Q)P-(T }\!\!\div\!\!\text{ S)R}\,\text{ }\!\!]\!\!\text{ }\]

Find \[\frac{7}{15}+\frac{8}{20}\]

Explanation

LCM of 15 and 20 =60

Thus \[\frac{7}{15}+\frac{8}{20}=\frac{(60\div 15)7+(60\div 20)8}{60}\]

\[=\frac{4\times 7+3\times 8}{60}=\frac{52}{60}=\frac{13}{15}\]

Find \[\frac{7}{15}-\frac{8}{20}.\]

Solution:

\[\frac{7}{15}-\frac{8}{20}=\frac{(60\div 15)7-(60\div 20)8}{60}\]

\[\frac{4\times 7-3\times 8}{60}=\frac{4}{60}=\frac{1}{15}.\]

Addition and Subtraction of Mixed Fractions

Mixed fractions are changed into improper fractions and then improper fractions are added or subtracted as per the given problems.

Add \[4\frac{1}{15}\] and \[5\frac{5}{12}.\]

Explanation

\[4\frac{1}{15}=\frac{61}{15}\]and\[5\frac{5}{12}=\frac{65}{12}\]

\[\frac{61}{15}+\frac{15}{12}=\frac{(60\div 15)61+(60\div 12)65}{60}=\frac{569}{60}.\]

Multiplication of a Fraction and a Whole Number

Let \[\frac{\text{P}}{\text{Q}}\] is a fraction and R is a whole number. Their product \[\frac{\text{P}}{\text{Q}}\text{ }\!\!\times\!\!\text{ R}\] can also be written as \[\frac{\text{P}}{\text{Q}}\text{ }\!\!\times\!\!\text{ }\frac{\text{R}}{1}.\]Now multiply numerator to numerator and denominator to denominator.

Find the product of 8 and \[\frac{\text{P}}{\text{Q}}\text{ }\!\!\times\!\!\text{ }\frac{\text{R}}{1}.\]

Explanation

\[8\times \frac{22}{75}\]or\[\frac{8}{1}\times \frac{22}{75}=\frac{176}{75}.\]

Multiplication of Fractions

Numerator is multiplied with numerator and denominator is multiplied with denominator.

For the example ,\[\frac{\text{P}}{\text{Q}}\text{ }\!\!\times\!\!\text{ }\frac{\text{R}}{\text{S}}\text{=}\frac{\text{P }\!\!\times\!\!\text{ R}}{\text{Q }\!\!\times\!\!\text{ S}}.\]

Find the product of \[\frac{5}{17}\] and \[\frac{32}{85}.\]

Explanation

\[\frac{5}{17}\times \frac{32}{85}=\frac{5\times 32}{17\times 85}\]

\[=\frac{160}{1445}=\frac{32}{289}.\]

Multiplication of a Fraction and a Mixed Fraction

Step 1: Mixed fraction is changed into an improper fraction

Step 2: Numerator is multiplied with numerator and denominator is multiplied withdenominator

Multiply\[\frac{72}{100}\] and \[7\frac{12}{13}.\]

Explanation

\[7\frac{12}{13}=\frac{103}{13}\]

And \[\frac{103}{13}\times \frac{72}{100}=\frac{7416}{1300}.\]

Division of a Fraction by a Whole Number and Vice Versa

Let \[\frac{\text{P}}{\text{Q}}\] is a fraction and R is a whole number. The fraction \[\frac{\text{P}}{\text{Q}}\]is divided by the whole number \[\text{R}\Rightarrow \frac{\text{P}}{\text{Q}}\text{ }\!\!\div\!\!\text{ R}\]

Step 1: Whole number is written as a fraction by taking 1 as denominator \[\frac{\text{P}}{\text{Q}}\text{ }\!\!\div\!\!\text{ }\frac{\text{R}}{\text{1}}.\]

Step2: Reverse the order of divisor so that denominator becomes numerator and numerator becomes denominator, and put the sign of multiplication in place of division.

\[\frac{\text{P}}{\text{Q}}\times \frac{1}{\text{R}}.\]

Step3: Multiply numerator with numerator and denominator with denominator\[\frac{\text{P }\!\!\times\!\!\text{ 1}}{\text{Q }\!\!\times\!\!\text{ R}}\]

Divide \[\frac{7}{24}\]by 4.

Explanation

\[\frac{7}{24}\div 4\Rightarrow \frac{7}{24}\div \frac{4}{1}\Rightarrow \frac{7}{24}\times \frac{1}{24}\Rightarrow \frac{7\times 1}{24\times 4}\Rightarrow \frac{7}{96}.\]

Divide 8 by \[\frac{256}{27}\]

Solution:

\[8\div \frac{256}{27}\]

\[8\times \frac{256}{27}\]

\[=\frac{27}{32}\]

Division of Fractions

Let \[\frac{\text{p}}{\text{q}}\] and \[\frac{\text{r}}{\text{s}}\] are two fractions and \[\frac{\text{p}}{\text{q}}\] is divided by \[\frac{\text{r}}{\text{s}}\Rightarrow \frac{\text{p}}{\text{q}}\text{ }\!\!\div\!\!\text{ }\frac{\text{r}}{\text{s}}\text{.}\]

Step 1: Reverse the order of divisor fraction and put the sign of multiplication in place of division \[\frac{\text{p}}{\text{q}}\times \frac{\text{r}}{\text{s}}\text{.}\]

Step 2: Multiply numerator with numerator and denominator with denominator

\[\frac{\text{p}}{\text{q}}\text{ }\!\!\times\!\!\text{ }\frac{\text{s}}{\text{r}}\text{.}\]

Divide \[\frac{17}{15}\] by \[\frac{12}{23}.\]

Explanation

\[\frac{17}{15}\div \frac{12}{23}\Rightarrow \frac{17}{15}\times \frac{23}{12}\Rightarrow \frac{17\times 23}{15\times 12}\Rightarrow \frac{391}{180}.\]

Note: If there is mixed fraction as a divisor or as a dividend, the mixed fraction is first changed into an improper fraction and the above process is followed.

When a fraction is multiplied or divided by 1, the fraction remains same.
A unit fraction is always a proper fraction.
Sum of two unit fraction is not a unit fraction.

In addition of like fractions, sum of their numerators is numerators is numerator and the common denominator is the denominator for the resultant fraction.
In subtraction of like fractions, difference of their numerators is the numerator and the comoon denominator is the denominator for the resultant fraction.
In addition of mixed fractions, mixed fractions are converted into improper fractions then they are added.
In multiplication of fractions, numerator is multiplied with numerator and denominator is multiplied with denominator.
In division of fractions, the order of numerator and denominator of the divisor fraction is reversed.

Add \[\frac{12}{15}\] and \[\frac{27}{12}\]and choose the correct option.

(a) \[\frac{39}{20}\]

(b) \[\frac{61}{12}\]

(d) \[\frac{61}{20}\]

(e) None of these

Answer: (d)

Explanation

\[\frac{12}{15}=\frac{4}{5}\]and\[\frac{27}{12}=\frac{9}{4}\]

\[\frac{4}{5}+\frac{9}{4}\frac{(20\div 5)4+(20\div 4)9}{20}\]

\[=\frac{61}{20}\]

What fraction should be added to \[\frac{7}{45}\]toget \[\frac{45}{7}\]?

(a) \[\frac{1976}{315}\]

(b) \[\frac{1472}{315}\]

(d) \[\frac{1972}{315}\]

(e) None of these

Answer: (a)

Explanation

\[\frac{45}{7}-\frac{7}{45}=\frac{(315\div 7)45-(315\div 45)7}{315}\]\[=\frac{1976}{315}.\]

Represent the shaded part in the following figures as a fraction and find their sum:

(a) 1

(b) \[\frac{2}{3}\]

(d) \[\frac{4}{5}\]

(e) None of these

Answer: (a)

By how much \[\frac{32}{70}\] is greater than \[\frac{42}{100}\]?

(a) \[\frac{45}{94}\]

(b) \[\frac{57}{100}\]

(d) \[\frac{13}{700}\]

(e) None of these

Answer: (c)

\[\frac{4}{5}+\frac{6}{7}+\frac{19}{3}=7\frac{\text{A}}{\text{105}}\]which one of the following numbers should come in place of A?

(a) 104

(b) 94

(d) 44

(e) None of these

Answer: (a)

Which one of the following is the product of \[\frac{42}{68}\]and \[\frac{72}{160}\]?

(a) \[\frac{456}{680}\]

(b) \[\frac{189}{680}\]

(d) \[\frac{698}{680}\]

(e) None of these

Answer: (b)

Explanation

\[\frac{42}{68}\times \frac{72}{160}\]

\[=\frac{3024}{10880}=\frac{189}{680}\]

A and B are two points on the following number line. Each of them represents a fraction. Find their product.

(a) \[\frac{14}{81}\]

(b) \[\frac{14}{9}\]

(d) \[\frac{13}{9}\]

(e) None of these

Answer: (a)

Explanation

A represents the fraction \[\frac{2}{9}\] and B represents the fraction \[\frac{7}{9}\] and their product

\[=\frac{2}{9}\times \frac{7}{9}\]

\[=\frac{14}{81}.\]

How many boxes in the figure (2) should be shaded so that product of the fractional representation for the shaded part in the following figures is \[\frac{3}{40}\]?