Question

How do you determine whether the situation is an example of an inverse or direct variation: Nicole earns $\$14$ for babysitting \[2\] hours, and $\$21$ for babysitting 3 hours?

Answer 1

Hint: From the question we have been asked to find whether the situation is an example of an inverse or direct variation. To solve this question we will use the definition of direct and indirect variation. For this question we will check that whether any of these \[y=kx\] or \[y=\dfrac{k}{x}\] are satisfied by the given values in the question and by using this we will conclude whether the situation is an example of an inverse or direct variation.

Complete step-by-step solution:
Firstly, we know that direct variation can always be written as \[y=kx\]. Here, k is the constant of variation.
In our question let us assume that,
In the first case we have,
\[\Rightarrow y=14\]
\[\Rightarrow x=2\]
So, for the first case we use the direct variation form mentioned above and after using that we get,
\[\Rightarrow y=kx\]
\[\Rightarrow 14=2k\]
\[\Rightarrow k=7\]
Now, similarly for the second case we have,
\[\Rightarrow y=21\]
\[\Rightarrow x=3\]
So, for the second case we use the direct variation form and we get,
\[\Rightarrow y=kx\]
\[\Rightarrow 21=3k\]
\[\Rightarrow k=7\]
Here we got the constant of variation as \[\Rightarrow k=7\].
Here we got that both cases obey the form of the direct variation.

Note: Students must be very careful in doing the calculations. For the question we can tell that the situation will be a direct variation because both the quantities move in the same direction. That is, whether it is whether it increases or decreases. For the inverse variation one case must be increasing and the other case must be decreasing. So, we can clearly say it as a direct variation situation.