Question

y varies directly as \[x\] and \[y = 40\] when \[x = 15\], find the constant of variation.

Answer 1

Hint: Here, in the question, the statement is given “\[y\] varies directly as \[x\]”. This means when \[x\] increases, \[y\] increases by the same factor. And this mutual relationship can be written as \[y = kx\], where \[x\] and \[y\] are the variables and \[k\] is constant of variation. Now, we need to find out the value of \[k\] by considering the given values of \[x\] and \[y\].
Formula used:
\[y = kx\], where \[k\] is constant of variation and \[x\] and \[y\] are the variables.

Complete step-by-step solution:
Let us collect the given information:
\[x = 15\], and,
\[y = 40\]
As mentioned in the question, \[y\] varies directly as \[x\].
\[ \Rightarrow y \propto x\]
So, the equation will be written as
\[y = kx\]
Taking the values of \[x\] and \[y\], we get
\[40 = k \times 15\]
Simplifying it, we get,
\[v\Rightarrow k = \dfrac{{40}}{{15}} \\
   \Rightarrow k = \dfrac{8}{3} \]
\[ \Rightarrow k = 2.667\]
Hence, the constant of variation when \[y = 40\] when \[x = 15\] and \[y\] varies directly as \[x\] is \[2.667\].
Additional information: If a variable varies indirectly with another variable, then it means when \[x\] increases, \[y\] decreases by the same factor. In other words, the expression \[xy\] always remains constant. And this mutual relationship can be written as \[y = k \times \dfrac{1}{x}\], where \[x\] and \[y\] are the variables and \[k\] is constant of variation.

Note: Constant of variation is also known as constant of proportionality. One should not get confused if the term changes. While doing such types of questions, read the given conditions properly. There may be changes in the inverse variation equation, such as squares, cubes or square roots. To find the constant of variation or the value of\[k\], use the information given in the question carefully.