Question

There are only $232$ pets. It’s only cats and dogs. How many cats are there if the ratio of cats to dogs is $3:1$?

Answer 1

Hint: We can let the number of cats be $x$ and the number of dogs be $y$. According to the information given in the question the total number of cats and the dogs is equal to $232$. So we can have the equation $x+y=232$. Also, the ratio of the cats to the dogs is given in the question to be equal to $3:1$. From this information, we can have the equation $\dfrac{x}{y}=3$. The two equations obtained are the linear equations in the two variables $x$ and $y$, which can be solved for the variable $x$ so as to get the required number of cats.

Complete step by step solution:
Let the number of cats be $x$ and the number of dogs be $y$.
In the above question, we have been given that there are a total of $232$ pets. Also, it is stated that there are only cats and the dogs among all of the $232$ pets. Therefore, the sum of $x$ and $y$ will be equal to $232$, from which we can write the below equation.
$\Rightarrow x+y=232........\left( i \right)$
Now, it is also given in the above question that the ratio of the cats to the dogs is equal to $3:1$. From this we can write
\[\Rightarrow \dfrac{x}{y}=3\]
Multiplying by \[y\] both the sides, we get
\[\begin{align}
  & \Rightarrow \dfrac{x}{y}\times y=3y \\
 & \Rightarrow x=3y......\left( ii \right) \\
\end{align}\]
Putting (ii) in (i) we get
$\begin{align}
  & \Rightarrow 3y+y=232 \\
 & \Rightarrow 4y=232 \\
\end{align}$
Dividing by $4$ both the sides, we get
$\begin{align}
  & \Rightarrow \dfrac{4y}{4}=\dfrac{232}{4} \\
 & \Rightarrow y=58 \\
\end{align}$
Substituting this in (ii), we get
$\begin{align}
  & \Rightarrow x=3\left( 58 \right) \\
 & \Rightarrow x=174 \\
\end{align}$
According to our assumption, the number of cats is $x$, which is found to be equal to $174$.
Hence, the number of cats is equal to $174$.

Note: Do not forget that we have to calculate the number of cats, and not the number of dogs. In the above solution, we first have obtained the value of $y$, which is the number of dogs. So do not forget to solve for $x$. To avoid this mistake, we can substitute $y$ in terms of $x$ from either of the two equations into the other equation.