Questions: A video store manager observes that the number of videos sold seems to vary inversely as the price per video. If the store sells 510 videos per week when the price per video is 17.40, how many does he expect to sell if he lowers the price to 16 ? Round your answer to the nearest integer if necessary.

Solution Steps

To solve this problem, we need to use the concept of inverse variation. Inverse variation means that as one variable increases, the other decreases proportionally. The relationship can be expressed as \( x \cdot y = k \), where \( k \) is a constant. Given the initial conditions, we can find \( k \) and then use it to find the number of videos sold at the new price.

1. Use the given data to find the constant \( k \) using the formula \( k = \text{price} \times \text{number of videos} \).
2. Use the constant \( k \) to find the number of videos sold at the new price by rearranging the formula to \( \text{number of videos} = \frac{k}{\text{new price}} \).

Given that the number of videos sold varies inversely with the price per video, we can express this relationship as:

\[ k = \text{price} \times \text{number of videos} \]

Substituting the initial values:

\[ k = 17.40 \times 510 = 8874.0 \]

Using the constant \( k \), we can find the expected number of videos sold at the new price of \( 16.00 \):

\[ \text{number of videos} = \frac{k}{\text{new price}} = \frac{8874.0}{16.00} \]

Calculating this gives:

\[ \text{number of videos} = 555.875 \]

Rounding \( 555.875 \) to the nearest integer results in:

\[ \text{number of videos} = 556 \]

Final Answer