- Identify the formula for the mean: \(\text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}}\).
- Set up the equation using the given mean of \(\$0.73\) and the five cookie costs: \(\frac{0.73 + 0.82 + 0.70 + x + 0.86}{5} = 0.73\).
- Simplify the sum and solve the resulting linear equation: \(3.11 + x = 3.65\).
- Determine the value of \(x\) is \(\boxed{\text{\$0.54}}\).
Explanation
- Understanding the Problem and Setting Up the Equation
To find the cost of a sugar cookie, we analyze the given data about the cookie types and their respective costs. We have five types of cookies with costs \$0.73, \$0.82, \$0.70, \$x\$, and \$0.86. The mean cost per cookie for all five types is \$0.73. The formula for the mean of a set of values is the sum of the values divided by the number of values. Given that there are 5 types of cookies, the formula is:
\[ \text{Mean} = \frac{0.73 + 0.82 + 0.70 + x + 0.86}{5} \]
We are given that the mean is \$0.73, so we set up the equation:
\[ 0.73 = \frac{0.73 + 0.82 + 0.70 + x + 0.86}{5} \]
- Solving for the Unknown Cost \(x\)
To solve for \(x\), we first simplify the numerator by summing the known values:
\[ 0.73 + 0.82 + 0.70 + 0.86 = 3.11 \]
Substituting this back into the equation, we get:
\[ 0.73 = \frac{3.11 + x}{5} \]
Next, we multiply both sides of the equation by 5 to isolate the numerator:
\[ 0.73 \times 5 = 3.11 + x \]
Calculating the product:
\[ 3.65 = 3.11 + x \]
- Conclusion and Final Answer
Now, we subtract 3.11 from both sides to find \(x\):
\[ x = 3.65 - 3.11 \]
\[ x = 0.54 \]
Thus, the cost per sugar cookie is \$0.54. We can verify this by calculating the mean with \(x = 0.54\):
\[ \frac{0.73 + 0.82 + 0.70 + 0.54 + 0.86}{5} = \frac{3.65}{5} = 0.73 \]
The calculation is correct. The cost for a sugar cookie is \$0.54.
Examples
This problem demonstrates how to calculate an unknown value when the average of a dataset is known. In real-life scenarios, such as managing a business, you might use this to determine the pricing of an individual product component to reach a target average cost. For example, if a bakery wants to maintain a specific average cost for a variety of products to achieve a desired profit margin, they can calculate the required cost of a new item using this method.