Question

Both $x$ and $y$ are in direct proportion, then $\dfrac{1}{x}$ and $\dfrac{1}{y}$ are:
(A) in direct proportion
(B) in inverse proportion
(C) neither indirect not in inverse proportion
(D) sometimes indirect and sometimes in inverse proportion

Answer 1

Hint: Start with expressing the relationship of proportionality using an equation and the coefficient of proportionality, i.e. $x = ky$ . Now divide both sides with ‘xy’ and simplify the equation further. This will give you an equation that represents the relation between the numbers $\dfrac{1}{x}$ and $\dfrac{1}{y}$ . Now choose the appropriate option from the given choices.

Complete step-by-step answer:
Here in this problem, we are given a condition for two numbers $x$ and $y$ that they are directly proportional to each other. With this information, we need to find the correct answer among the given four options for the relationship between $\dfrac{1}{x}$ and $\dfrac{1}{y}$ .
Before starting with the solution we should understand the concept of proportionality. In mathematics, two varying quantities are said to be in a relation of proportionality, if they are multiplicatively connected to a constant; that is when either their ratio or their product yields a constant. The value of this constant is called the coefficient of proportionality or proportionality constant.
According to the question, $x$ and $y$ have a relationship of direct proportion
$ \Rightarrow x \propto y$ ; where the symbol $' \propto '$ represents the relation of proportionality
This proportionality can be replaced by the equality sign using the coefficient of proportionality as follows:
$ \Rightarrow x \propto y \Rightarrow x = k \times y$ ; here ‘k’ is the proportionality constant
Let’s now divide the above equation by $xy$ :
$ \Rightarrow \dfrac{x}{{x \times y}} = \dfrac{{k \times y}}{{x \times y}}$
We can further solve this equation as:
\[ \Rightarrow \dfrac{x}{{x \times y}} = \dfrac{{k \times y}}{{x \times y}} \Rightarrow \dfrac{1}{y} = \dfrac{k}{x}\]
Therefore, we get the equation \[\dfrac{1}{y} = \dfrac{k}{x}\] , which shows the relationship of numbers $\dfrac{1}{x}{\text{ and }}\dfrac{1}{y}$ of direct proportion.
Considering ‘k’ as the proportionality constant, we can remove the coefficient of proportionality and write the given equation as:
\[ \Rightarrow \dfrac{1}{y} = \dfrac{k}{x} \Rightarrow \dfrac{1}{y} \propto \dfrac{1}{x}{\text{ or }}\dfrac{1}{x} \propto \dfrac{1}{y}\]
Therefore, the expression \[\dfrac{1}{x} \propto \dfrac{1}{y}\] represents the relation of direct proportion between $\dfrac{1}{x}$ and $\dfrac{1}{y}$ .
Hence, the option (A) is the correct answer.

Note: In the above problem, there was an assumption that the variable $x{\text{ and }}y$ cannot take a value $0$ . Since the operation of division by zero in mathematics is not defined. An alternative approach to this problem can be to transpose both ‘x’ and ‘y’ to the opposite side and you will have both of them in the denominator.