Question

x varies inversely as y and y = 60 when x = 1.5. Find x when y = 4.5.

Answer 1

Hint: To solve this question, we will write x and y inversely proportional to each other and then convert them in the equality using a constant. Finally, we will try to obtain the value of the constant ‘k’ using x = 1.5 and y = 60. After obtaining ‘k’ we will get x using y = 4.5.

Complete step-by-step solution:
We are given that x varies inversely as y.
\[\Rightarrow x\propto \dfrac{1}{y}\]
Or we can also write this statement as
\[\Rightarrow y\propto \dfrac{1}{x}\]
Let us consider, \[x\propto \dfrac{1}{y}.\]
Now, when we have proportionality of the type \[a\propto b\] then to make or obtain equality between a and b, we have to adjust a constant ‘k’ between and k as
\[\Rightarrow a\propto b\]
\[\Rightarrow a=kb\]
Or, \[\Rightarrow ka=b\]
Here, we have, \[x\propto \dfrac{1}{y}.\]
Inserting a constant ‘k’ in the above equation, we have,
\[x\propto \dfrac{1}{y}\]
\[\Rightarrow x=\dfrac{k}{y}\]
We are given in the question that when x = 1.5, then y = 60. Substituting x = 1.5 and y = 60 in the above equation, we have,
\[\Rightarrow x=\dfrac{k}{y}\]
\[\Rightarrow 1.5=\dfrac{k}{60}\]
Now, cross multiplying both the equations, we have,
\[\Rightarrow 1.5\times 60=k\]
\[\Rightarrow k=90\]
Hence, the value of the constant k is 90. Therefore, our equation \[x=\dfrac{k}{y}\] becomes
\[\Rightarrow x=\dfrac{90}{y}\]
Now, we have to calculate the value of x when y = 4.5.
\[\Rightarrow x=\dfrac{90}{y}\]
Substituting y = 4.5 in the above equation, we get,
\[\Rightarrow x=\dfrac{90}{4.5}\]
\[\Rightarrow x=20\]
Therefore, the value of x when y is 4.5 is given as 20.

Note: Another method to solve this question can be considering \[y\propto \dfrac{1}{x}\] to get the result. We will assume a constant ‘t’ in this case.
\[\Rightarrow y\propto \dfrac{1}{x}\]
\[\Rightarrow y=\dfrac{t}{x}\]
When x = 1.5 and y = 60, we get,
\[\Rightarrow 60=\dfrac{t}{1.5}\]
\[\Rightarrow t=60\times 1.5\]
\[\Rightarrow t=90\]
Again,
\[y=\dfrac{90}{x}\]
\[\Rightarrow 4.5=\dfrac{90}{x}\]
\[\Rightarrow x=\dfrac{90}{4.5}\]
\[\Rightarrow x=20\]
Therefore, in any case, the answer remains the same.