Question

Half of a herd of deer are grazing in the field and three fourth of the remaining are playing nearby. The rest 9 are drinking water from the pond. Find the number of deer in the herd.

Answer 1

Hint: Suppose the total number of deer’s as a variable and try to get the values of deer’s for grazing, playing in the term of that variable. Equate the total number of deer’s supposed and calculated from the given information in the question and form an equation in terms of the supposed variable. Solve it to get the value of the variable.

Complete step by step answer:
Let us assume the total number of deer in the herd is ‘x’. Now, it is given that half of the deer are grazing the field, it means half of x i.e. total deer are present there. So, we get total number of deer’s for grazing
$\dfrac{1}{2}\times x=\dfrac{x}{2}...............\left( i \right)$
And another three fourth of the remaining deer are grazing in the field, so we can calculate the remaining deer’s by the difference of total number of deer’s and $\dfrac{x}{2}$ deer’s in the field. Hence the deer’s playing nearby can be calculated by taking $\dfrac{3}{4}th$ of the remaining deer’s. So, we get
Remaining deer’s after $\dfrac{x}{2}$ goes for grazing
$=\dfrac{x}{1}-\dfrac{x}{2}=\dfrac{2x-x}{2}=\dfrac{x}{2}$
Remaining deer’s after $\dfrac{x}{2}$ goes for grazing
$=\dfrac{x}{2}$
So, deer’s playing in nearby be
$=\dfrac{3}{4}\times \dfrac{x}{2}=\dfrac{3x}{8}$
Now, it is given that the remaining deer in the herd are drinking water and the number of them is 9. So, the number of resting deer’s for drinking water is 9. So, we can get equation as
Total deer’s for grazing + Total deer’s playing nearby + Total deer’s for drinking water = Total deer’s in the herd.
Hence, we can write the above equation in mathematical form as
$\begin{align}
  & \dfrac{x}{2}+\dfrac{3x}{8}+9=x \\
 & \Rightarrow \dfrac{4x+3x}{8}+9=x \\
 & \Rightarrow \dfrac{7x}{8}+9=x \\
 & \Rightarrow 9=x-\dfrac{7x}{8}=\dfrac{8x-7x}{8}=\dfrac{x}{8} \\
 & \Rightarrow \dfrac{9}{1}=\dfrac{x}{8} \\
\end{align}$
On cross multiplying the above equation we get
72 = x or
x = 72.
As ‘x’ represents the total deer in the herd. It means there are 72 deer’s in the herd

Note: One may from the equation by following way as well:
Total number of deer’s playing + Total deer’s for drinking water = half of the total deer’s
$\dfrac{3x}{8}+9=\dfrac{x}{2}$
As, we can ignore $\dfrac{x}{2}$ deer’s for grazing and can equate the remaining deer's in the herd as well.
Don’t take three fourth of total deer’s for calculating the number of deer’s for playing nearby. As, it will equal to the $\dfrac{3}{4}th$ of the remaining deer's, not of the total deer’s. So don’t get confused with that part. Writing the given problem in words to the mathematical equation is the key point of the question.