Question

One-fourth of a herd of cows is in the forest. Twice the square root of the herd has gone to mountains and the remaining 15 are on the banks of a river. The total number of cows is:
A. 6
B. 100
C. 63
D. 36

Answer 1

Hint: Let the total number of cows in the herd be \[x\]. So, one-fourth of the herd of cows will be \[\dfrac{x}{4}\] and twice of the square root of the herd of cows will be \[2\times \sqrt{x}\]. Also there is a herd of 15 cows separated by the rest. Add the last three expressions above as they are the statements of the question and equate their sum to \[x\].

Complete step-by-step answer:
Let us divide the question in simpler statements to make it easy to solve, there are total 4 statements,
Statement 1: One-fourth of a herd of cows is in the forest.
Mathematically, we can write it as
Number of cows in forest is \[\dfrac{x}{4}\]
Statement 2: Twice the square root of the herd has gone to mountains.
Mathematically, we will get it as
Number of cows on the mountains is \[2\sqrt{x}\]
Statement 3: Remaining 15 cows are on the banks of the river.
Mathematically, we can deduce that
Number of cows on the banks of the river are 15 which is also the remaining.
Now, we will add all the three statements presented above and then equate it to the total number of cows as all the cows are counted from every place and the remaining 15 were also mentioned. So the total must be all of them. Hence the equation becomes,
\[\dfrac{x}{4}+2\sqrt{x}+15=x\]
Now solving the equation, to make it easy to solve we must first get rid of the square root,
\[2\sqrt{x}=x-\dfrac{x}{4}-15\]
\[2\sqrt{x}=\dfrac{3x}{4}-15\]
\[2\sqrt{x}=\dfrac{3x-60}{4}\]
\[8\sqrt{x}=3x-60\]
Now squaring both sides, we get,
\[\begin{align}
  & 64x={{(3x-60)}^{2}} \\
 & 64x=9{{x}^{2}}+3600-360x \\
 & 9{{x}^{2}}-424x+3600=0 \\
\end{align}\]
Now solving the quadratic equation by using the quadratic formula. For a general quadratic equation like \[a{{x}^{2}}+bx+c=0\], we have
\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
But for the equation derived above it will be,
\[x=\dfrac{-(-424)\pm \sqrt{{{(-424)}^{2}}-4(9)(3600)}}{2(9)}\]
\[\begin{align}
  & x=\dfrac{424\pm \sqrt{179776-129600}}{18} \\
 & x=\dfrac{424\pm \sqrt{50176}}{18} \\
 & x=\dfrac{424\pm 224}{18} \\
\end{align}\]
So, we have two values of our variable,
\[x=\dfrac{424+224}{18}\] and \[x=\dfrac{424-224}{18}\]
\[x=\dfrac{648}{18}\] and \[x=\dfrac{200}{18}\]
\[x=36\] and \[x=11.111\]
There are two values of cows but we have to choose only one out of them.
So, now as the number of cows can only be a positive integer we are going to reject the value of 11.111 and choose the value of 36.
Therefore, the correct option is D. 36

Note: We can solve the quadratic equation by other methods like Completing the square method and factorization but factorization does not work every time and completing the square method is very time consuming so we go for the quadratic formula method. Sometimes one value of the variable might come out negative so do not get confused and choose the positive one if it is feasible with the question. Always move step by step and analyze the statements correctly and carefully, and also try to break the question into a number of statements as it will be easy for you.