Question

Out of a number of saras birds, one-fourth of the number are moving about in lots, \[ \dfrac{1}{9}th \] coupled with \[ \dfrac{1}{4}th \] 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 numbers of birds?

Answer 1

Hint: In this question first we assume total number of birds to be constant number and then we will add all the numbers of Saras birds which are involved in different activities together and this will be equal to the total number of birds, hence we will get the total number of bird by solving the obtained equation.

Complete step-by-step answer:
Let the total number of Saras birds be x
Number of birds moving about in lots is \[ = \dfrac{x}{4} - - (i) \]
Now it is also been given that \[ \dfrac{1}{9}th \] of the Saras bird is coupled with \[ \dfrac{x}{4} \] of birds then,
total birds coupled \[ = \dfrac{x}{9} + \dfrac{x}{4} - - (ii) \]
Also it is given that 7 times the square root of the number of birds move on a hill \[ = 7 \sqrt x - - (iii) \]
Total numbers of birds remain in the vakula trees \[ = 56 - - (iv) \]
Now since we have a total number of x Saras birds, hence we can write the equation by adding (i), (ii), (iii) & (iv) as
 \[ \dfrac{x}{4} + \dfrac{x}{4} + \dfrac{x}{9} + 7 \sqrt x + 56 = x \]
Now let us assume \[x = {y^2} \] , hence we can rewrite the equation as
 \[ \dfrac{{{y^2}}}{4} + \dfrac{{{y^2}}}{4} + \dfrac{{{y^2}}}{9} + 7y + 56 = {y^2} \]
By further solving this equation we can write
 \[
 \Rightarrow \dfrac{{9{y^2} + 9{y^2} + 4{y^2} + 252y + 2016}}{{36}} = {y^2} \\
 \Rightarrow 22{y^2} + 252y + 2016 = 36{y^2} \\
 \Rightarrow 14{y^2} - 252y - 2016 = 0 \\
 \Rightarrow {y^2} - 18y - 144 = 0 \
  \]
Now we have got the above equation in quadratic form so we will find its two roots by factorizing its middle term, hence we can write
  \[
 \Rightarrow {y^2} - 18y - 144 = 0 \\
 \Rightarrow {y^2} - 24y + 6y - 144 = 0 \\
 \Rightarrow y \left( {y - 24} \right) + 6 \left( {y - 24} \right) = 0 \\
 \Rightarrow \left( {y - 24} \right) \left( {y + 6} \right) = 0 \
  \]
So we can write
 \[y = 24,y = - 6 \]
Now since we are finding the number of birds and it’s can never be negative hence we will exclude \[y = - 6 \] , so we can write \[y = 24 \]
Since we have taken \[x = {y^2} \] , where \[y = 24 \] , hence we can write
 \[ \Rightarrow x = {y^2} = { \left( {24} \right)^2} = 576 \]
Therefore the total numbers of birds \[ = 576 \]
So, the correct answer is “576”.

Note: In this type of problems, the key factor is to develop the mathematical equation such that it should satisfy all the given conditions of the problem. Moreover, we should always try to minimize the number of the parameter so that in less number of the equations, all the parameter’s value could be determined.