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At a certain time in a deer park, the number of heads and number of legs of deer and human visitors was counted and it was found that there were 39 heads and 132 legs. Find the number of deer and human visitors in the park.

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Last updated date: 01st Mar 2024
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IVSAT 2024
Answer
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Hint: Here we need to find the number of deer and human visitors in the park. For that, we will first assume the number of legs of human visitors and number of legs of deer to be any variable. Then we will add both of them and then we will equate the sum with the given number of legs. From there, we will get the first equation including all the variables. Then we will form the second equation using the information given in the question. We will solve these two equations to get the values of the number of humans and deer in the park.

Complete step by step solution:
Let the total number of legs of deer be \[x\] and let the total number of legs of human visitors be \[y\].
It is given that the sum of the number of legs of deer and human visitors is 132. Therefore,
\[x + y = 132\] ……… \[\left( 1 \right)\]
We know that 1 deer has 4 legs and 1 head and 1 human visitor has 2 legs and 1 head. So,
Number of heads of deer \[ = \dfrac{x}{4}\]
Number of heads of human visitors \[ = \dfrac{y}{2}\]
It is given that the sum of the number of heads of deer and human visitors is 39.
\[ \Rightarrow \dfrac{x}{4} + \dfrac{y}{2} = 39\] …….. \[\left( 2 \right)\]
Now, we will multiply 4 on both sides of equation \[\left( 2 \right)\].
\[ \Rightarrow x + 2y = 39 \times 4\]
On multiplying the terms, we get
\[ \Rightarrow x + 2y = 156\] …………. \[\left( 3 \right)\]
On subtracting equation \[\left( 1 \right)\] from equation \[\left( 3 \right)\], we get
\[\begin{array}{l}x + 2y - \left( {x + y} \right) = 156 - 132\\ \Rightarrow x + 2y - x - y = 24\end{array}\]
Adding and subtracting like terms, we get
\[ \Rightarrow y = 24\]
Now, we will substitute the value of \[y\] in equation \[\left( 1 \right)\].
 \[x + 24 = 132\]
Now, we will subtract 24 from both sides. Therefore, we get
\[\begin{array}{l} \Rightarrow x + 24 - 24 = 132 - 24\\ \Rightarrow x = 108\end{array}\]
Now, substituting the value of \[x\] and \[y\] the equation of number of heads of deer and humans, we get
Number of heads of deer \[ = \dfrac{x}{4} = \dfrac{{108}}{4} = 27\]
Number of heads of human visitors \[ = \dfrac{y}{2} = \dfrac{{24}}{2} = 12\]

Hence, the number of deer and human visitors in the park are 27 and 12 respectively.

Note:
Here, we have framed two linear equations in two variables based on the given information. Linear equations are the equations that have the highest degree of the variable as 2. Linear equation in two variables means there are two distinct variables and both have the highest degree 1. We have solved the two linear equations to find the value of the two variables.
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