Question

How do you find the value of $y$ for a given value of $x$, if $y$ varies directly with $x$. If $y = 166$ when $x = 83$, what is $y$ when $x = 237$ ?

Answer 1

Hint: We have been given in the question that $y$ varies directly with $x$ so, $y$ is directly proportional to $x$. Then, we will find the constant of variation in the equation and then will find the value of constant. Put the value of $x = 237$ in the formed equation and find the value of $y$.

Complete Step by Step Solution:
From the question, we know that, $y$ varies directly with $x$, so $y$ is directly proportional to $x$. Therefore, the above statement in the form of mathematical expression as –
$ \Rightarrow y\alpha x$
According to the question, we have been given with us –
$y = 166$
$x = 83$
We have to find the value when $x = 237$ so, we need an equation in which we can put the value of $x = 237$ and then find the value of $y$.
Hence, it is given that –
$ \Rightarrow y\alpha x$
To convert the above expression into an equation, we have to multiply $k$ with the above expression where, $k$ is the constant of variation. Therefore, the above expression can be written as –
$ \Rightarrow y = kx \cdots \left( 1 \right)$
Now, putting the data which is given in the question in the above equation to find the value of $k$ where, $y = 166$ and $x = 83$ , we get –
$ \Rightarrow 166 = k \times 83$
Shifting 83 to another side of the equation, we get –
$ \Rightarrow k = \dfrac{{166}}{{83}}$
Dividing 166 by 83, we get 2 as quotient, therefore, the value of $k$ is –
$ \Rightarrow k = 2$
Hence, the value of $k$ is equal to 2 in the equation (1), then the equation becomes –
$ \Rightarrow y = 2x$
Now, we have to find the value of $y$ when $x = 237$, so, put the value of $x = 237$ in the above equation, we get –
$
   \Rightarrow y = 2 \times 237 \\
   \Rightarrow y = 474 \\
 $

Hence, the value of $y$ is 474 when the value of $x$ is 237.

Note:
The students can usually make mistakes while making the equation for the question. Try to understand the question and carefully read it and use the data carefully according to the question. When $y$ varies directly with $x$ so, we can write that $y$ is directly proportional to $x$.