Solving a Cost Puzzle: Hat and Pencil Equation

Let's explore a classic example of a cost-based algebra problem. This puzzle involves a hat and a pencil, and it serves as an excellent introduction to solving cost-based equations. The problem is presented as follows: a hat and a pencil cost a total of $2.10, with the hat costing $2 more than the pencil. The goal is to find the individual costs of the hat and the pencil.

Introduction to the Problem

This type of problem is a common exercise in algebra, where a set of relationships is described, and the task is to find the values that satisfy these relationships. In this case, we are given the total cost and the difference in cost between two items, and we must determine each item's price.

Solving the Puzzle

Let's denote the cost of the pencil as x dollars. Based on the problem statement, the hat costs $2 more than the pencil, which can be expressed as:

hat x 2

The total cost of the hat and the pencil is given as $2.10. Therefore, we can set up the following equation:

x (x 2) 2.10

Simplifying the equation:

2x 2 2.10

Subtracting 2 from both sides of the equation:

2x 0.10

Dividing both sides by 2:

x 0.05

Therefore, the pencil costs $0.05. Now, to find the cost of the hat:

hat x 2 0.05 2 2.05

This means the hat costs $2.05.

Another Approach to the Same Problem

Let's solve the problem again using a similar method to ensure the solution is consistent and to demonstrate multiple approaches to solving the same type of problem. In this version, we still denote the cost of the pencil as x dollars.

The hat's cost is expressed as x 2 dollars. The total cost for both items is $2.10. Using this information, we can set up the following equation:

x (x 2) 2.10

Which simplifies to:

2x 2 2.10

Subtracting 2 from both sides:

2x 0.10

Dividing both sides by 2:

x 0.05

Thus, the pencil costs $0.05. Following this, the cost of the hat is:

2 - 0.05 2.05

Therefore, the hat still costs $2.05.

Conclusion

Through this problem, we have learned how to solve a cost-based algebraic problem. We used basic algebra to set up and solve the equation, and both methods provided the same answer, confirming the accuracy of our solution. This type of problem-solving technique is valuable for developing analytical skills and understanding the relationships between different quantities.

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