Operation on the Fraction
Category : 5th Class
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.
Add \[\frac{12}{15}\] and \[\frac{27}{12}\]and choose the correct option.
(a) \[\frac{39}{20}\]
(b) \[\frac{61}{12}\]
(c) \[\frac{61}{15}\]
(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) \[\frac{1472}{315}\]
(c) \[\frac{1876}{315}\]
(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) \[\frac{2}{3}\]
(c) \[\frac{3}{4}\]
(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) \[\frac{57}{100}\]
(c) \[\frac{13}{350}\]
(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
(c) 105
(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) \[\frac{189}{680}\]
(c) \[\frac{784}{458}\]
(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) \[\frac{14}{9}\]
(c) \[\frac{14}{18}\]
(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 \[\frac{3}{40}\]?
(a) 1
(b) 2
(c) 3
(d) 4
(e) None of these
Answer: (c)
Jack multiplies two unit fractions and finds the following conclusions. Which one is not true?
(a) Value of the resultant fraction increases
(b) Value of the resultant fraction decreases
(c) The resultant fraction is also a unit fraction
(d) The resultant fraction is also a proper fraction
(e) None of these
Answer: (a)
In which one of the following figures shaded part represents equivalentfraction of the product of \[\frac{2}{7}\times \frac{35}{13}\]?
(a)
(b)
(c)
(d)
(e) None of these
Answer: (a)
Product of a whole number and a fraction is \[\frac{26}{7}.\]If the whole number is 9, find the fraction.
(a) \[\frac{234}{7}\]
(b) \[\frac{26}{63}\]
(c) \[\frac{63}{26}\]
(d) \[\frac{7}{234}\]
(e) None of these
Answer: (b)
Find the quotient when A is divided by B.
(a) \[\frac{3}{5}\]
(b)\[\frac{3}{16}\]
(c) \[\frac{5}{16}\]
(d) \[\frac{3}{80}\]
(e) None of these
Answer: (a)
Explanation
A represents \[=\frac{3}{16}\] and B represents \[\frac{5}{16}\]
\[\text{A }\!\!\div\!\!\text{ B=}\frac{\text{3}}{\text{16}}\text{ }\!\!\div\!\!\text{ }\frac{\text{5}}{\text{16}}\]
\[\Rightarrow \frac{3}{16}\times \frac{16}{5}\Rightarrow \frac{3}{5}.\]
\[\frac{x}{y}\]is a fraction and z is a whole number. Which one of the following is not correct?
(a) if \[x=z,\frac{x}{y}\div z=\frac{1}{y}\]
(b) if \[y=z,\frac{x}{y}\div z=\frac{1}{{{y}^{2}}}\]
(c) If \[x=y=z,\]quotient of \[\frac{x}{y}\div z\]is an improper fraction
(d) If \[\frac{x}{y}\] is a unit fraction and \[z=y,z\]is reciprocal of \[\frac{x}{y}\]
(e) None of these
Answer: (c)
Represent the shaded part in the following figures as a fraction and solve
(a) \[\frac{1}{8}\]
(b) \[\frac{3}{4}\]
(c) \[\frac{1}{4}\]
(d) \[\frac{1}{6}\]
(e) None of these
Answer: (d)
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