Question

If \[y\] varies inversely as \[x\] , and \[y = 9\] when \[x = 2\] ,find \[y\] when \[x = 3\].

Answer 1

Hint: In this question, it is given that \[y\] varies inversely as \[x\] .We will first use the proportionality relation between \[y\] and \[x\] .Then we will remove the proportionality sign by introducing a constant \[k\] .After that we will substitute the given values of \[x\] and \[y\] to find out the value of constant. And finally, we will find the value of \[y\] by substituting the value of constant and the value of \[x\] in the relation.
Formula used: If two quantities \[a\] and \[b\] are given and they are inversely proportional to each other then,
\[a{\text{ }} \propto {\text{ }}\dfrac{1}{b}\]

Complete step-by-step answer:
We are given that \[y\] varies inversely as \[x\] which means that \[y\] is proportional to the inverse of \[x\]
Mathematically, we can write it as
\[{\text{y }} \propto {\text{ }}\dfrac{1}{x}{\text{ }} - - - \left( 1 \right)\]
Now when we have proportionality of the type \[{\text{a }} \propto {\text{ b}}\] ,then to obtain equality between \[a\] and \[b\] we have to remove the sign of proportionality by introducing a constant.
So, let us take the constant of proportionality as \[k\]
\[\therefore \] equation \[\left( 1 \right)\] becomes,
\[{\text{y = }}\dfrac{k}{x}{\text{ }} - - - \left( 2 \right)\]
Now we have to find the value of \[k\]
So, we will do this by substituting the given values
i.e., \[y = 9\] when \[x = 2\] in equation \[\left( 2 \right)\]
Thus, we get
\[{\text{9 = }}\dfrac{k}{2}{\text{ }}\]
On multiplying both sides by \[2\] ,we get
\[9 \times 2 = k\]
\[ \Rightarrow k = 18\]
Now substitute the value of \[k\] in equation \[\left( 2 \right)\] we get
\[{\text{y = }}\dfrac{{18}}{x}{\text{ }}\]
Now we are supposed to find the value of \[y\] when \[x = 3\]
\[\therefore {\text{y = }}\dfrac{{18}}{3}{\text{ }}\]
\[ \Rightarrow {\text{y = }}6{\text{ }}\]
Hence, the value of \[{\text{y = }}6{\text{ }}\] when \[x = 3\]

Note: Proportionality is of two types i.e., direct proportional and inversely proportional. Two variables are said to be directly proportional if both the variables change in the same proportion, i.e., if one increases, then the other variable also increases and if one decreases, then the other variable also decreases. And if two variables are inversely proportional, then both variables change in the opposite proportion, i.e., if one increases, then the other variable decreases and vice-versa.