Answer
Verified
396.9k+ views
Hint:Fraction is a number which is of the form \[\dfrac{p}{q}\] where \[q\] is not equal to zero. That is, \[\dfrac{p}{q},(q \ne 0)\]
An Equivalent fraction is a fraction which has different numerators and denominators that represents the same value or proportion of whole. Here we make many ways to find the equivalent fraction of the given fraction.
Complete step-by-step answer:
It is given that the fraction \[\dfrac{7}{9}\]
We can write it as,
\[\dfrac{7}{9} = 0.777....\]
That is approximate to \[\dfrac{7}{9} = 0.\overline 7 \]
We can solve many way in this problem,
Here we can solve one way,
That is, \[7\]multiply by \[a\]and \[9\] multiply by \[b\]where \[a \ne 0,b \ne 0\] we can
We take \[a\] and \[b\]has variables
\[ \Rightarrow \dfrac{{7 \times a}}{{9 \times b}} = \dfrac{{7a}}{{9b}}.... \to \left( 1 \right)\]
Now we have some cases,
If \[(i)\] $\text{a = b}$
\[(ii)\] $\text{a} \ne\text{b}$
Case \[(i)\] $\text{a = b}$
We can take a number from 1 to 9, and put into a and b
Suppose \[a = b = 3\]
\[ \Rightarrow \dfrac{7}{9} \times \dfrac{3}{3} = \dfrac{{21}}{{27}}\]
\[ \Rightarrow \dfrac{{21}}{{27}} = 0.777..\]
\[ \Rightarrow \dfrac{{21}}{{27}} = 0.\overline 7 \]
Suppose we take \[a = b = 4\] substitute in equation \[\left( 1 \right)\] and we get \[\dfrac{{28}}{{36}}\]
Now we find,
\[\dfrac{{28}}{{36}} = 0.77..\]
\[\dfrac{{28}}{{36}} = 0.\overline 7 \]
Similarly, for any number \[a( = b) = \{ 1,2,3,4,5,6,7,8,9\} \]
It gives the same value, \[0.\overline 7 \]
Now we take second case
Case \[(ii)\] $\text{a} \ne\text{b}$
Put \[a = 6,b = 5\]
\[\dfrac{{7 \times 6}}{{9 \times 5}} = \dfrac{{42}}{{45}}\]
\[\dfrac{{42}}{{45}} \ne 0.\overline 7 \]
Here we can take any number for \[a\] and any number for \[b\](which is not the same number) is not equal to the given fraction value.
It is satisfied only when \[a\] and \[b\] are equal.
Then we take any arbitrary values for a and b (is equal)
In the sense, take \[a( = b) = \{ 2,5,9\} \]
We get,
\[ \Rightarrow \dfrac{{7a}}{{9b}} = \dfrac{{14}}{{18}} = 0.\overline 7 \]
\[ \Rightarrow \dfrac{{7a}}{{9b}} = \dfrac{{35}}{{45}} = 0.\overline 7 \]
\[ \Rightarrow \dfrac{{7a}}{{9b}} = \dfrac{{63}}{{81}} = 0.\overline 7 \]
$\therefore $The three fractions are \[\dfrac{{14}}{{18}},\dfrac{{35}}{{45}},\dfrac{{63}}{{81}}\] equivalent fraction to the fraction\[\dfrac{7}{9}\]
Note:If we notice, we use only a single digit in both cases and all through the sum, if you want a double digit or more than it will also be in the way of our solving.
If we take \[a = 7,b = 9\] in the given fraction that same as a numerator and denominator value for \[a\] and \[b\] we get the same result,
That is
\[ \Rightarrow \dfrac{{7a}}{{9b}} = \dfrac{{7 \times 7}}{{9 \times 9}} = \dfrac{{49}}{{81}}\]
\[ \Rightarrow \dfrac{{49}}{{81}} = 0.\overline 7 \]
This is the way to approach another method.
An Equivalent fraction is a fraction which has different numerators and denominators that represents the same value or proportion of whole. Here we make many ways to find the equivalent fraction of the given fraction.
Complete step-by-step answer:
It is given that the fraction \[\dfrac{7}{9}\]
We can write it as,
\[\dfrac{7}{9} = 0.777....\]
That is approximate to \[\dfrac{7}{9} = 0.\overline 7 \]
We can solve many way in this problem,
Here we can solve one way,
That is, \[7\]multiply by \[a\]and \[9\] multiply by \[b\]where \[a \ne 0,b \ne 0\] we can
We take \[a\] and \[b\]has variables
\[ \Rightarrow \dfrac{{7 \times a}}{{9 \times b}} = \dfrac{{7a}}{{9b}}.... \to \left( 1 \right)\]
Now we have some cases,
If \[(i)\] $\text{a = b}$
\[(ii)\] $\text{a} \ne\text{b}$
Case \[(i)\] $\text{a = b}$
We can take a number from 1 to 9, and put into a and b
Suppose \[a = b = 3\]
\[ \Rightarrow \dfrac{7}{9} \times \dfrac{3}{3} = \dfrac{{21}}{{27}}\]
\[ \Rightarrow \dfrac{{21}}{{27}} = 0.777..\]
\[ \Rightarrow \dfrac{{21}}{{27}} = 0.\overline 7 \]
Suppose we take \[a = b = 4\] substitute in equation \[\left( 1 \right)\] and we get \[\dfrac{{28}}{{36}}\]
Now we find,
\[\dfrac{{28}}{{36}} = 0.77..\]
\[\dfrac{{28}}{{36}} = 0.\overline 7 \]
Similarly, for any number \[a( = b) = \{ 1,2,3,4,5,6,7,8,9\} \]
It gives the same value, \[0.\overline 7 \]
Now we take second case
Case \[(ii)\] $\text{a} \ne\text{b}$
Put \[a = 6,b = 5\]
\[\dfrac{{7 \times 6}}{{9 \times 5}} = \dfrac{{42}}{{45}}\]
\[\dfrac{{42}}{{45}} \ne 0.\overline 7 \]
Here we can take any number for \[a\] and any number for \[b\](which is not the same number) is not equal to the given fraction value.
It is satisfied only when \[a\] and \[b\] are equal.
Then we take any arbitrary values for a and b (is equal)
In the sense, take \[a( = b) = \{ 2,5,9\} \]
We get,
\[ \Rightarrow \dfrac{{7a}}{{9b}} = \dfrac{{14}}{{18}} = 0.\overline 7 \]
\[ \Rightarrow \dfrac{{7a}}{{9b}} = \dfrac{{35}}{{45}} = 0.\overline 7 \]
\[ \Rightarrow \dfrac{{7a}}{{9b}} = \dfrac{{63}}{{81}} = 0.\overline 7 \]
$\therefore $The three fractions are \[\dfrac{{14}}{{18}},\dfrac{{35}}{{45}},\dfrac{{63}}{{81}}\] equivalent fraction to the fraction\[\dfrac{7}{9}\]
Note:If we notice, we use only a single digit in both cases and all through the sum, if you want a double digit or more than it will also be in the way of our solving.
If we take \[a = 7,b = 9\] in the given fraction that same as a numerator and denominator value for \[a\] and \[b\] we get the same result,
That is
\[ \Rightarrow \dfrac{{7a}}{{9b}} = \dfrac{{7 \times 7}}{{9 \times 9}} = \dfrac{{49}}{{81}}\]
\[ \Rightarrow \dfrac{{49}}{{81}} = 0.\overline 7 \]
This is the way to approach another method.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Summary of the poem Where the Mind is Without Fear class 8 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write an application to the principal requesting five class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE