Question

If y varies directly as $x$ and $y = 10$ when $x = 2,$ what is $y$ when $x = 3{\text{ ?}}$

Answer 1

Hint: In this question, there is a direct relationship given between the variables $x{\text{ and }}y.$ The meaning of the term direct relation or direct variation is that the value of one variables increases with increase in the value of another variable. On the other hand if the value of one variable decreases with increase in the value of another variable then it is called an inverse variation or inverse relationship. Direct relationships or direct variations are represented by : $y \propto x$ . We can change this relation into an equation by multiplying it with a constant of proportionality like: $y = cx$ , then we can find the value of proportionality constant c by the information given in the question about the variables $x{\text{ and }}y$ . Once we find out the value of the proportionality constant, we can find out the value of the variable $y$ for any value of the variable $x$ and vice versa.

Complete step-by-step solution:
The given statement is ;
$ \Rightarrow y \propto x{\text{ }}......\left( 1 \right)$
Now, to convert the above relation into an equation, we have to remove the proportionality sign and multiply with a constant of proportionality ( Let the proportionality constant is c ) ;
$ \Rightarrow y = cx{\text{ }}......\left( 2 \right)$
We can find the value of the proportionality constant c , by using the given information about the variables x and y provided in the question.
In the question it is given that ;
When $x = 2,{\text{ }}y = 10$
Put the values of the variables $x = 2{\text{ and }}y = 10$ in equation $2,$ we get;
$ \Rightarrow 10 = c \times 2$
$ \Rightarrow c = \dfrac{{10}}{2}$
$ \Rightarrow c = 5$
Hence, the value of the proportionality constant $c = 5$ ; hence we can rewrite equation $2$ as ;
$ \Rightarrow y = 5x{\text{ }}......\left( 3 \right)$
In the question, we are asked the value of $y{\text{ when }}x = 3$ ;
So, put the respective values in the equation $3,$ we get ;
$ \Rightarrow y = 5 \times 3$
$ \Rightarrow y = 15$
Therefore, the value of $y = 15,{\text{ when }}x = 3.$
So, the correct answer for this question is $y = 15.$

Note: This question was about two directly proportional quantities. Let’s discuss the inversely proportional case here. Two variables or quantities are said to be inversely proportional when the value of one variable decreases with increase in the value of another variable and vice versa. For example: If the two variables are x and y then the inverse relation is written as: $x \propto \dfrac{1}{y}{\text{ or }}y \propto \dfrac{1}{x}$ . The general equation for inverse relation can be written as: $x = \dfrac{c}{y}{\text{ or }}y = \dfrac{c}{x}$ ; where c is the constant of proportionality.