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Chapter 3: Problem 11

Michael and Kevin want to buy chocolates. They can't agree on whether they want chocolatecovered almonds, chocolate-covered peanuts, or chocolatecovered raisins. They agree to create a mix. They bought 4 pounds of chocolate-covered almonds at \(\$ 3.50\) per pound, 3 pounds of chocolate-covered peanuts for \(\$ 2.75\) per pound, and 2 pounds of chocolate-covered raisins for \(\$ 2.25\) per pound. Determine the cost per pound of the mix.

Short Answer

Expert verified
The cost per pound is \$ 2.97.

Step by step solution

01

Calculate the total cost of chocolate-covered almonds

Michael and Kevin bought 4 pounds of chocolate-covered almonds at \( \$ 3.50 \) per pound. Multiply the number of pounds by the cost per pound: \( 4 \, \text{pounds} \times \$ 3.50 / \text{pound} = \$ 14.00 \).
02

Calculate the total cost of chocolate-covered peanuts

They bought 3 pounds of chocolate-covered peanuts at \( \$ 2.75 \) per pound. Multiply the number of pounds by the cost per pound: \( 3 \, \text{pounds} \times \$ 2.75 / \text{pound} = \$ 8.25 \).
03

Calculate the total cost of chocolate-covered raisins

They bought 2 pounds of chocolate-covered raisins at \( \$ 2.25 \) per pound. Multiply the number of pounds by the cost per pound: \( 2 \, \text{pounds} \times \$ 2.25 / \text{pound} = \$ 4.50 \).
04

Calculate the total weight of the mix

Add the weight of all the types of chocolates together: \( 4 \, \text{pounds of chocolate-covered almonds} + 3 \, \text{pounds of chocolate-covered peanuts} + 2 \, \text{pounds of chocolate-covered raisins} = 9 \, \text{pounds} \).
05

Calculate the total cost of the mix

Add the costs of all the types of chocolates together: \( \$ 14.00 \text{(chocolate-covered almonds)} + \$ 8.25 \text{(chocolate-covered peanuts)} + \$ 4.50 \text{(chocolate-covered raisins)} = \$ 26.75 \).
06

Calculate the cost per pound of the mix

Divide the total cost by the total weight: \( \$ 26.75 \text{(total cost)} \/ 9 \text{(total weight)} = \$ 2.97 \text{(cost per pound)} \).

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Key Concepts

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

Weighted Average
When Michael and Kevin decided to create a mix of different chocolate-covered treats, they essentially created a weighted average. A weighted average helps to find a mean value when different items contribute unequally to a total. Here, different weights (4 pounds of almonds, 3 pounds of peanuts, 2 pounds of raisins) and their respective prices create varied influences on the final cost. In this exercise, the weights are the amounts of chocolates, and each type of chocolate鈥檚 cost contributes to the overall average cost based on its weight. This is different from a simple average where each item would contribute equally.
Cost Analysis
Cost analysis involves understanding the components of the total cost. In our example, we need to determine the overall cost of several types of chocolate-covered treats. By breaking it down:
  • The total cost of chocolate-covered almonds is calculated as 4 pounds \times \(3.50 per pound, which gives us \)14.
  • For chocolate-covered peanuts, it is 3 pounds \times \(2.75 per pound, resulting in \)8.25.
  • Finally, the chocolate-covered raisins cost 2 pounds \times \(2.25 per pound, totaling \)4.50.
    • The next step is adding all these individual costs to get the total cost of the chocolates (almonds + peanuts + raisins), which is \(14 + \)8.25 + \(4.50 = \)26.75.
    Understanding cost analysis helps in managing budgets and expenses efficiently.
Arithmetic Operations
Arithmetic operations are basic mathematical calculations used to find the total cost and weight of the chocolate mix. They include multiplication, addition, and division. First, we find the total weight:
  • Add the number of pounds of each type: 4 pounds (almonds) + 3 pounds (peanuts) + 2 pounds (raisins) to get 9 pounds.

Then, we find the total cost using addition:
  • Add the individual costs: \(14 (almonds) + \)8.25 (peanuts) + \(4.50 (raisins) equaling \)26.75.

Finally, calculate the cost per pound using division:
  • Divide the total cost by the total weight: \(26.75 (total cost) / 9 (total weight) = \)2.97 per pound.

These simple steps using arithmetic operations can help solve many real-life problems.

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