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Chapter 5: Problem 13

Eliza wants to sell a used car stereo online. From her research on the website she will post to, she found 8 similar stereos listed. She decides to list her stereo for 20\(\%\) less than the mean price of the stereos already for sale on the site. Let \(x\) represent the sum of the prices of the stereos she found in her research. Write an expression to calculate the price she will list as the cost of her stereo.

Short Answer

Expert verified
Eliza's selling price will be given by the expression \( p_s = x / 8 - 0.025x \).

Step by step solution

01

Calculation of Mean Price

The mean price of the stereos is calculated by dividing the sum \(x\) of all stereo prices by the number of stereos, which is 8, so, the mean price \(p_m\) would be \(p_m = x / 8\).
02

Find the Reduction Amount

Next, you have to find out how much needs to be reduced from the mean price. As Eliza wants to sell her stereo for 20\% less than the mean price, the reduction will be \(20\%\) of \(p_m\). The reduction amount \(r\) is thus \(r = 0.20 \cdot p_m = 0.20 \cdot x / 8 = 0.025x\).
03

Finding Eliza's Selling Price

You can now find the selling price of Eliza's stereo by subtracting the reduction \(r\) from the mean price \(p_m\), that gives the equation for selling price \(p_s\) as \(p_s = p_m - r = x / 8 - 0.025x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Price Calculation
Calculating the mean price is a straightforward yet essential method in determining an average from a set of numbers. Eliza begins by summing all the prices of the 8 used car stereos she found on the website, represented as \(x\). To find the mean price, or average, she divides the total sum \(x\) by the number of stereos. This calculation forms the mean price formula: \(p_m = \frac{x}{8}\).

The mean price gives a sense of what the average cost of these stereos is. Dividing the total price sum by 8 helps find this average, smoothing out any variations in the individual prices. Understanding the mean allows Eliza to know what a typical stereo costs on the website, setting a reference point for her sales strategy.
Percentage Reduction
Eliza wants to attract buyers by listing her stereo at a slightly lower price than average. Thus, she aims for a 20\(\%\) reduction from the mean price. To calculate the reduction amount, we take 20\(\%\) of the mean price \(p_m\). This percentage can be represented as a decimal 0.20.

Therefore, the reduction amount \(r\) is calculated by multiplying 0.20 by the mean price: \(r = 0.20 \times \frac{x}{8}\). This simplifies to \(r = 0.025x\).

The reduction gives a tangible value that will be subtracted from the mean price. It reflects the discount Eliza offers to make her listing more competitive. By displaying a lower price, there's a higher chance that potential buyers will consider her stereo.
Selling Price Determination
The final step is determining Eliza's selling price. This involves adjusting the mean price by subtracting the calculated reduction amount \(r\) from it.

The equation for finding the selling price \(p_s\) is \(p_s = p_m - r\). Substituting the calculated values, the expression becomes \(p_s = \frac{x}{8} - 0.025x\).

This expression neatly combines all previous calculations and provides an exact price point that Eliza should list her stereo at. The selling price \(p_s\) ensures that her stereo is priced competitively, leveraging the average market price but adjusted for an appealing discount. This strategy can increase her chances of a successful sale.

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Most popular questions from this chapter

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