Solving Pencil and Eraser Cost Problems: A Comprehensive Guide

Introduction to the Problem

In this article, we will explore a common algebraic problem involving pencils and erasers. By using basic algebra, we can determine the individual costs of pencils and erasers. Let's dive into the solution step-by-step to ensure a clear understanding.

Problem Statement

Given the following two scenarios: 12 pencils and 12 erasers cost $16.08. 12 pencils and 4 erasers cost $6.80. We are tasked with determining the cost of 1 pencil and 1 eraser. Follow along to solve this problem methodically.

Initial Equations

Let P represent the cost of one pencil and E represent the cost of one eraser. We can write the given conditions as: 12P 12E 16.08 12P 4E 6.80

Solving for the Cost of Eraser

Subtract the second equation from the first equation to eliminate the pencils and focus on the erasers.

12P 12E - (12P 4E) 16.08 - 6.80

12E - 4E 9.28

8E 9.28

E frac{9.28}{8} 1.16

Therefore, the cost of one eraser (E) is $1.16.

Solving for the Cost of Pencil

Now that we know the cost of one eraser, we can substitute it back into one of the original equations to solve for the cost of a pencil (P). Let's use the second condition:

12P 4E 6.80

12P 4(1.16) 6.80

12P 4.64 6.80

12P 6.80 - 4.64

12P 2.16

P frac{2.16}{12} 0.18

So, the cost of one pencil (P) is $0.18.

Final Cost of One Pencil and One Eraser

Adding the cost of one pencil and one eraser:

1P 1E 0.18 1.16 1.34

Thus, the combined cost of one pencil and one eraser is $1.34.

Conclusion

Understanding and solving algebraic problems involving pencils and erasers can be straightforward once you set up the correct equations. By using basic algebraic techniques, we have determined that the cost of one pencil and one eraser is $1.34. This method can be applied to similar problems to find the cost of other combinations of items.

Frequently Asked Questions (FAQs)

Q: Why is it necessary to use algebra to solve this problem?
A: Algebra helps us organize and solve complex problems systematically by breaking them down into equations. This approach ensures accuracy and provides a clear, step-by-step solution. Q: Can the method be applied to similar problems with different items?
A: Yes, the method can be applied to similar problems. Simply set up the equations based on the given information and solve for the unknowns using algebraic techniques. Q: Are there any other common algebraic problems involving pencils and erasers?
A: Yes, there are many. For example, you might be given a scenario where a different number of pencils and erasers are involved, and their total cost needs to be determined. The approach remains the same: set up the equations and solve using algebra.