Question

haley and bob are moving to woonsocket from providence and so they need to buy a used car. the kind of car theyre looking for should have an average price of $25,000 based on the internet. these are the prices they were quoted:

$21,000 $23,000 $22,000 $19,000 $26,000 $22,000

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 $25,000?

Explanation:

Step1: Recall the formula for the mean

The mean (average) of a set of numbers is calculated by the sum of the numbers divided by the count of the numbers. Let the new price be \( x \). The total number of prices after adding the new one will be 7, and the mean should be \( \$25,000 \). So the formula for the mean is \( \text{Mean} = \frac{\text{Sum of all prices}}{n} \), where \( n = 7 \) and \( \text{Mean} = 25000 \).

Step2: Calculate the sum of the existing prices

First, we sum up the given prices: \( 21000 + 23000 + 22000 + 19000 + 26000 + 22000 \).
Calculating step by step:
\( 21000 + 23000 = 44000 \)
\( 44000 + 22000 = 66000 \)
\( 66000 + 19000 = 85000 \)
\( 85000 + 26000 = 111000 \)
\( 111000 + 22000 = 133000 \)

Step3: Set up the equation for the mean with the new price

The sum of all seven prices will be \( 133000 + x \), and the mean is \( \frac{133000 + x}{7} = 25000 \).

Step4: Solve for \( x \)

Multiply both sides of the equation by 7: \( 133000 + x = 25000 \times 7 \)
Calculate \( 25000 \times 7 = 175000 \)
Then, subtract 133000 from both sides: \( x = 175000 - 133000 \)
\( x = 42000 \)

Answer:

\( \$42,000 \)