Suppose that the price per pencil at an office supply store decreases as the number of the pencils bought increases. We are asked to determine whether the price per pencil varies inversely with the number of pencils bought. To do so, let's recall the definition of inverse variation.
Inverse Variation
The variables x and y vary inversely if their product is constant. This means that there exists a constant k such that xy=k.
If the price per pencil varies inversely with the number of pencils bought, the product of the price per pencil and the number of pencils bought will be constant.
number of pencils* price per pencil=constant
This equation means that no matter how many pencils we buy, we will pay the same amount of money. From the given information, we cannot conclude that such condition is met. Consider the following example.
Number of Pencils
Price per Pencil
Total Price
1
$ 0.15
1* $ 0.15=$ 0.15
2
$ 0.14
2* $ 0.14=$ 0.28
3
$ 0.10
3* $ 0.10=$ 0.30
4
$ 0.08
4* $ 0.08=$ 0.32
We can see that as the number of pencils increases, the price per pencil decreases. However, the total price of the pencils is not constant. Therefore, in our example, the price per pencil does not vary inversely with the number of pencils bought. Consequently, for the given the situation, the price per pencil does not necessarily vary inversely with the number of pencils bought.
Extra
A logical approach
Let's think about the equation that we created in the beginning of this explanation with a strictly logical approach.
number of pencils* price per pencil=constant
While it is possible for this situation to happen, it would be against the best interests of a store to make it where you could have as many as you wanted of something and always pay the same price.
