Answer

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**Hint:**Here, we will assume the total number of Saras birds to be square of some variable. Then we will use the given information and find the fractions of the birds present at different places. Then we will add them together and equate it to the number of birds to form an equation. We will then simplify the equation to get the total number of birds.

**Complete step-by-step answer:**Let the total number of Saras birds be \[{x^2}\] ………………………… \[\left( 1 \right)\]

According to the question,

One fourth the number of Saras birds are moving about in lotus plants.

Therefore, number of birds moving about Lotus plants \[ = \dfrac{{{x^2}}}{4}\]

Also, it is given that, \[{\dfrac{1}{9}^{{\rm{th}}}}\] coupled (along) with \[{\dfrac{1}{4}^{{\rm{th}}}}\]

Hence, the number of birds who coupled (along) \[ = \dfrac{{{x^2}}}{9}\] with \[\dfrac{{{x^2}}}{4}\]

Also, 7 times the square root of the number move on a hill

Hence, the number of birds moving on a hill \[ = 7\sqrt {{x^2}} = 7x\]

The number of birds remaining in Vakula trees \[ = 56\]

Now, we are required to find the total number of birds.

Hence, we will add all the above information together. Therefore,

Total number of birds \[ = \dfrac{{{x^2}}}{4} + \dfrac{{{x^2}}}{4} + \dfrac{{{x^2}}}{9} + 7x + 56\]…………………………. \[\left( 2 \right)\]

Now from \[\left( 1 \right)\], we know that the total number of birds are \[{x^2}\]. So,

\[ \Rightarrow {x^2} = \dfrac{{{x^2}}}{2} + \dfrac{{{x^2}}}{9} + 7x + 56\]

Adding the like terms, we get

\[ \Rightarrow {x^2} = \dfrac{{9{x^2} + 2{x^2}}}{{18}} + 7x + 56\]

\[ \Rightarrow {x^2} = \dfrac{{11{x^2}}}{{18}} + 7x + 56\]

Rewriting the equation, we get

\[ \Rightarrow {x^2} - \dfrac{{11{x^2}}}{{18}} - 7x - 56 = 0\]

Taking LCM and solving further, we get

\[ \Rightarrow 18{x^2} - 11{x^2} - 126x - 1008 = 0\]

\[ \Rightarrow 7{x^2} - 126x - 1008 = 0\]

Dividing both sides by 7, we get

\[ \Rightarrow {x^2} - 18x - 144 = 0\]

The above equation is a quadratic equation, so will factorize the equation to find the value of \[x\].

Now, splitting the middle term, we get

\[ \Rightarrow {x^2} - 24x + 6x - 144 = 0\]

\[ \Rightarrow x\left( {x - 24} \right) + 6\left( {x - 24} \right) = 0\]

Factoring out the common terms, we get

\[ \Rightarrow \left( {x + 6} \right)\left( {x - 24} \right) = 0\]

Now, by zero product property, we get

\[\begin{array}{l} \Rightarrow x + 6 = 0\\ \Rightarrow x = - 6\end{array}\]

Or

\[\begin{array}{l} \Rightarrow x - 24 = 0\\ \Rightarrow x = 24\end{array}\]

But the value of \[x\] can’t be negative because the number of birds will always be positive.

Hence, \[x = 24\]

Therefore, the total number of birds \[ = {x^2} = {\left( {24} \right)^2} = 576\]

**Hence, the total number of required birds was 576.**

**Note:**In this question, we can also check whether our answer is correct or not.

Hence, in order to know whether the total number of birds were actually 576 or not,

We will substitute the value \[{x^2} = 576\] in equation \[\left( 2 \right)\]. Therefore, we get

Total number of birds \[ = \dfrac{{576}}{4} + \dfrac{{576}}{4} + \dfrac{{576}}{9} + 7\left( {24} \right) + 56\]

\[ \Rightarrow \] Total number of birds \[ = \dfrac{{576}}{2} + 64 + 7\left( {24} \right) + 56\]

Hence, solving further, we get

\[ \Rightarrow \] Total number of birds \[ = 288 + 120 + 168 = 576\]

\[ \Rightarrow \] Total number of birds \[ = 576 = {\left( {24} \right)^2} = {x^2}\]

Hence, verified

Thus, this shows that we have solved this question correctly and the total number of Saras birds are 576.

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