Question

There are several human beings and several dogs in a room. One-tenth of the humans have lost a leg. The total numbers of feet are 77. Then the number of dogs is
A) Not determined due to insufficient data
B) 4
C) 5
D) 6

Answer 1

Hint: First assume the number of human beings and dogs to be x and y respectively, as human beings have two legs and dogs have four legs multiply x with 2 and y with 4 respectively then make an equation which gives total no. Legs then use hit and trial method to find the number of dogs.

Complete step by step solution: Let suppose the number of human beings be \[x\]
And the number of dogs be \[y\]
As, dog have 4 legs and human beings have 2 legs,
Therefore, total legs \[ = 2x + 4y\]
We have given One tenth of the humans have lost a leg Thus, the total number of feet are
\[2x + 4y - \dfrac{x}{{10}} = 77\]
On simplifying we get,
\[\dfrac{{19x}}{{10}} + 4y = 77 \ldots \left[ 1 \right]\]
Both the terms, \[\dfrac{{19x}}{{10}}\] and \[4y\] must be natural numbers as they represent the number of legs of human beings and dogs respectively.
Therefore, \[x\] must be multiple of \[10\] for above equation \[\left[ 1 \right]\] to be satisfied.
If suppose \[x = 10\]
Putting \[x = 10\] equation \[\left[ 1 \right]\] we get,
\[\dfrac{{190}}{{10}} + 4y = 77\]
\[ \Rightarrow 4y = 58\]
\[ \Rightarrow y = \dfrac{{58}}{4}\]
\[ \Rightarrow y = \dfrac{{29}}{2}\], Which is not possible
\[ \Rightarrow x \ne 10\]
If suppose\[x = 20\]
Putting \[x = 20\]equation \[\left[ 1 \right]\] we get,
\[\dfrac{{380}}{{10}} + 4y = 77\]
\[ \Rightarrow 4y = 39\]
\[ \Rightarrow y = \dfrac{{39}}{4}\], Which is not possible
\[ \Rightarrow x \ne 20\]
If suppose \[x = 30\]
Putting\[x = 30\] equation \[\left[ 1 \right]\] we get,
\[\dfrac{{570}}{{10}} + 4y = 77\]
\[ \Rightarrow 4y = 20\]
\[ \Rightarrow y = \dfrac{{20}}{4}\]
\[ \Rightarrow y = 5\]
\[\therefore \]The number of dogs \[y = 5\]
And the number of human beings \[x = 30\]

Hence, option C. 5 is the correct answer.

Note:
Alternate Method:
Here it is given that there are several humans and dogs in a room.
Let x be the number of humans and y be the number of dogs in the room.
We know that humans have two feet, so the number of human feet is 2x and that of dogs have 4 feet, so the number of dog feet is 4y.
It is known that, one-tenth of the humans have one feet lost i.e. $\dfrac{x}{{10}}$ humans have one feet and also it is given that the total number of feet in the room is 77.
Thus, the equation can be given as,
\[2x + 4y - \dfrac{x}{{10}} = 77\]
\[\therefore \dfrac{{20x - x}}{{10}} + 4y = 77\]
\[\therefore \dfrac{{19x}}{{10}} + 4y = 77\]
Now, it is clear that the terms $\dfrac{{19x}}{{10}}$ and 4y must be natural numbers as they represent the number of feet of humans and dogs respectively.
Also, the value of x must be in the multiple of 10 as the term $\dfrac{{19x}}{{10}}$ needs to be a natural number.
So, by hit and trial method, let us first put x = 10 in the equation and check.
$\therefore \dfrac{{19\left( {10} \right)}}{{10}} + 4y = 77$
$\therefore 4y = 58$
$\therefore y = \dfrac{{58}}{4}$
This is not the required answer as y must be a natural number.
So, let us try the same by putting x = 20.
$\therefore \dfrac{{19\left( {20} \right)}}{{10}} + 4y = 77$
$\therefore 4y + 38 = 77$
$\therefore y = \dfrac{{39}}{4}$
This is not the required answer as y must be a natural number.
So, let us try the same by putting x = 30.
$\therefore \dfrac{{19\left( {30} \right)}}{{10}} + 4y = 77$
$\therefore 4y + 57 = 77$
$\therefore y = \dfrac{{20}}{4}$
$\therefore y = 5$
Here, we get 5 as the answer.
So, the correct option is option (C). Thus, the number of dogs in the room is 5.