Finding the Cost of Each Pencil

Oct 13, 2025 by vinooalluri.com 94 views

Introduction

In this article, we will delve into the world of mathematics and solve an equation to find the cost of each pencil. The scenario is as follows: Alan bought 3 pencils and a notebook, with the notebook costing $6 and the total expenditure being $12. We will use the variable $p$ to represent the cost of each pencil.

Understanding the Problem

To solve this problem, we need to understand the relationship between the cost of the pencils and the total expenditure. Since Alan bought 3 pencils and a notebook, we can represent the total expenditure as the sum of the cost of the pencils and the cost of the notebook.

Setting Up the Equation

Let's use the variable $p$ to represent the cost of each pencil. Since Alan bought 3 pencils, the total cost of the pencils can be represented as $3p$ . The cost of the notebook is given as $6. Therefore, the total expenditure can be represented as:

3p + 6 = 12

Solving the Equation

To solve for $p$ , we need to isolate the variable on one side of the equation. We can do this by subtracting 6 from both sides of the equation:

3p = 12 - 6

3p = 6

Finding the Cost of Each Pencil

Now that we have isolated the variable, we can solve for $p$ by dividing both sides of the equation by 3:

p = \frac{6}{3}

p = 2

Conclusion

Therefore, the cost of each pencil is $2. This means that Alan spent $2 on each of the 3 pencils he bought, and the remaining $6 was spent on the notebook.

Real-World Applications

This problem may seem simple, but it has real-world applications in various fields such as finance, economics, and business. For example, a company may need to calculate the cost of producing a certain product, and the cost of raw materials may be represented by the variable $p$ . By solving the equation, the company can determine the cost of each raw material and make informed decisions about production.

Tips and Tricks

When solving equations, it's essential to follow the order of operations (PEMDAS):

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

By following these steps, you can ensure that you're solving equations correctly and accurately.

Common Mistakes

When solving equations, it's easy to make mistakes. Here are some common mistakes to avoid:

Not following the order of operations: Failing to follow the order of operations can lead to incorrect solutions.
Not isolating the variable: Failing to isolate the variable on one side of the equation can make it difficult to solve for the variable.
Not checking the solution: Failing to check the solution can lead to incorrect answers.

Practice Problems

To practice solving equations, try the following problems:

A book costs $15, and a pen costs $5. If the total expenditure is $25, how much did each book cost?
A shirt costs $20, and a pair of pants costs $15. If the total expenditure is $40, how much did each shirt cost?

By practicing these problems, you can improve your skills in solving equations and become more confident in your math abilities.

Conclusion

In conclusion, solving equations is an essential skill in mathematics and has real-world applications in various fields. By following the order of operations, isolating the variable, and checking the solution, you can ensure that you're solving equations correctly and accurately. Practice problems can help you improve your skills and become more confident in your math abilities.

Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to solve them correctly. In this article, we'll address some frequently asked questions about solving equations, providing you with a better understanding of this concept.

Q: What is an equation?

A: An equation is a mathematical statement that expresses the equality of two expressions. It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), which are separated by an equal sign (=).

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I isolate the variable?

A: To isolate the variable, you need to get the variable on one side of the equation and the constant on the other side. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 5 is a linear equation. A quadratic equation is an equation in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation.

Q: What is the difference between a system of equations and a single equation?

A: A system of equations is a set of two or more equations that are solved simultaneously. A single equation is a single equation that is solved independently.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use the following methods:

Substitution method: Substitute the expression for one variable from one equation into the other equation.
Elimination method: Add or subtract the equations to eliminate one variable.
Graphing method: Graph the equations on a coordinate plane and find the point of intersection.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear inequality is an inequality in which the highest power of the variable is 1. For example, 2x + 3 > 5 is a linear inequality. A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 5 is a linear equation.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can use the following methods:

Graphing method: Graph the inequality on a coordinate plane and shade the region that satisfies the inequality.
Test point method: Choose a test point that satisfies the inequality and check if it is true.

Conclusion

In conclusion, solving equations is a fundamental concept in mathematics, and it's essential to understand how to solve them correctly. By following the order of operations, isolating the variable, and using the correct methods for solving equations, you can become proficient in solving equations and tackle more complex problems.

Practice Problems

To practice solving equations, try the following problems:

Solve the equation 2x + 5 = 11.
Solve the system of equations x + y = 4 and 2x - 3y = -2.
Solve the linear inequality 2x - 3 > 5.

By practicing these problems, you can improve your skills in solving equations and become more confident in your math abilities.