Question

Out of a number of Saras birds, one fourth the number are moving about it lotus plants, $\dfrac{1}{9}{\text{th}}$ to coupled(along) with $\dfrac{1}{4}{\text{th}}$ as well as $7$ times the square root of the number move on a hill, $56$ birds remain in vakula trees. What is the total number of birds?
A) 36
B) 576
C) both (a) & (b)
D) None of these

Answer 1

Hint: First we have to find the total number of birds, for that we can represent the information in a system of linear equations. It will help us to find the required total number of birds. Using the multiple variable, we can construct the linear equation. By factoring the linear equation we can get the value of the multiple variable. Then related it to our, we will get the final answer.

Complete step-by-step answer:
It is given that , Out of a number of Saras birds, one fourth the number are moving about it lotus plants, $\dfrac{1}{9}{\text{th}}$ to coupled (along) with $\dfrac{1}{4}{\text{th}}$ as well as $7$ times the square root of the number move on a hill, $56$ birds remain in vakula trees.
Now consider we have that,
Let the number of birds be ${{\text{x}}^2}$
Number of birds moving about lotus plant $ = \dfrac{{{{\text{x}}^2}}}{4}$
Number of birds coupled along $ = \dfrac{{{{\text{x}}^2}}}{9}$
Number of bird moves on hill $ = 7\sqrt {{{\text{x}}^2}} = 7{\text{x}}$
Number of birds remaining on trees $ = 56$
Now,
$\dfrac{{{{\text{x}}^2}}}{4} + \dfrac{{{{\text{x}}^2}}}{4} + \dfrac{{{{\text{x}}^2}}}{9} + 7{\text{x + 56 = }}{{\text{x}}^2}$
Simplifying we get,
$\dfrac{{11{{\text{x}}^2}}}{{18}} + 7{\text{x + 56 = }}{{\text{x}}^2}$
Rearranging the terms,
$7{\text{x + 56 = }}\dfrac{7}{{18}}{{\text{x}}^2}$
Solving the terms, we get
$ \Rightarrow 7{{\text{x}}^2}{\text{ - }}136{\text{x - 1008 = 0}}$
Dividing by $7$, we get
$ \Rightarrow {{\text{x}}^2}{\text{ - 18 x - 144 = 0}}$
Now, we factorized the terms
$ \Rightarrow {{\text{x}}^2}{\text{ - }}24{\text{ x + 6 x - 144}}$
Splitting by middle term method,
$ \Rightarrow {\text{x}}\left( {{\text{x - 24}}} \right) + 6\left( {{\text{x - 24}}} \right) = 0$
Simplifying we get,
$ \Rightarrow \left( {{\text{x + 6}}} \right)\left( {{\text{x - 24}}} \right) = 0$
Hence the value of x,
$ \Rightarrow {\text{x = - 6 , 24}}$
$ \Rightarrow {\text{x = 24}}$${\text{(x = - 6}}$ is rejected. Because number of birds cannot be a negative value$)$
$\therefore $ Number of birds $ = {{\text{x}}^2} = {24^2} = 576$

Hence option B is the correct answer.

Note: For these kinds of problems consider the unknown variable as a quantity and solve the problem. We use linear equations in one variable if we have one unknown quantity and we use linear equations in two variables if we have two unknown quantities. Also don’t use linear equations in two variables if you have to find only one variable.