Nick and Liz have decided to move from the city to the suburbs. This means that they will have to make the essential purchase of a car in order to get to work. They researched used 2-year-old cars of the same make, model, condition, and equipped with the same options. They found a website stating that the average price should be $19,200. These are the prices they were quoted: $15,500, $17,800, $19,900, $18,000, $21,400.
If they continued their search for one more price quote, what would that price have to be so that the mean of all seven of the car prices would be the same as the mean quoted on the website?

To find the price that would result in the mean of all seven car prices being equal to $19,200, we can follow these steps:

### Step 1: Calculate the sum of the six given prices.
The six prices given are: $15,500, $17,800, $19,900, $18,000, $21,400, and the unknown seventh price $ x $.

Let's first find the sum of the six known prices:

\[
15,500 + 17,800 + 19,900 + 18,000 + 21,400 = 92,600
\]

### Step 2: Set up the equation for the mean of all seven prices.
The mean of all seven prices should be $19,200. The mean is calculated by dividing the total sum of the prices by the number of prices (7 in this case):

\[
\frac{92,600 + x}{7} = 19,200
\]

### Step 3: Solve for $ x $.
Multiply both sides by 7 to eliminate the fraction:

\[
92,600 + x = 19,200 \times 7
\]

\[
92,600 + x = 134,400
\]

Subtract 92,600 from both sides to find $ x $:

\[
x = 134,400 - 92,600
\]

\[
x = 41,800
\]

### Final Answer:
The seventh price quote would need to be **$41,800** in order for the mean of all seven car prices to equal $19,200.

Would you like to explore more about how the mean is affected by each individual value? Here are five related questions to consider:

1. What happens to the mean if we increase or decrease one of the car prices significantly?
2. How would the median price compare to the mean in this scenario?
3. What if Nick and Liz found another price that was much lower or higher than the existing ones—how would that affect the mean?
4. How is the mean different from other measures of central tendency like the mode and median?
5. How does the range of car prices affect their decision in choosing the right car?

**Tip:** Understanding how to adjust values to meet a target mean can be a powerful tool, especially in budgeting and forecasting scenarios!

Learn how to calculate the mean of car prices and determine the seventh price needed to achieve a target mean. This problem involves statistical concepts like the arithmetic mean. Suitable for students in Grades 7-9.

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