Four Times The Sum Of A Number And 3

Four Times the Sum of a Number and 3: Exploring Mathematical Expressions and Their Applications

This article delves into the mathematical expression "four times the sum of a number and 3," exploring its various interpretations, applications, and extensions. We will examine how to translate this phrase into algebraic notation, solve equations involving it, and discuss its relevance in various mathematical contexts. We'll also look at real-world applications and explore more complex variations of this fundamental concept.

Understanding the Expression: From Words to Algebra

The phrase "four times the sum of a number and 3" can seem daunting at first, but breaking it down step by step reveals its straightforward nature. Let's analyze each component:

• A number: This represents an unknown value, typically denoted by a variable, most commonly x.

• The sum of a number and 3: This translates to x + 3. We are adding 3 to our unknown number.

• Four times the sum of a number and 3: This means we are multiplying the expression (x + 3) by 4. This gives us the final algebraic expression: 4(x + 3).

It's crucial to understand the order of operations (PEMDAS/BODMAS). The parentheses indicate that the addition within the parentheses must be performed before the multiplication.

Solving Equations with the Expression

The expression 4(x + 3) forms the basis of many algebraic equations. Let's explore a few examples:

Example 1: A Simple Equation

Let's say the expression is equal to 28. This gives us the equation:

4(x + 3) = 28

To solve for x, we follow these steps:

1. Distribute the 4: 4(x + 3) becomes 4x + 12. The equation now reads: 4x + 12 = 28.

2. Subtract 12 from both sides: This isolates the term with x: 4x = 16.

3. Divide both sides by 4: This solves for x: x = 4.

Therefore, the number is 4. We can verify this by substituting 4 back into the original equation: 4(4 + 3) = 4(7) = 28.

Example 2: A More Complex Equation

Consider a slightly more challenging equation:

4(x + 3) + 5 = 37

Here's how we solve it:

1. Subtract 5 from both sides: This simplifies the equation to 4(x + 3) = 32.

2. Distribute the 4: This gives us 4x + 12 = 32.

3. Subtract 12 from both sides: This results in 4x = 20.

4. Divide both sides by 4: This gives us the solution: x = 5.

Again, we can verify this solution by substituting it back into the original equation.

Example 3: Equations with Variables on Both Sides

Let's examine an equation with the expression on both sides:

4(x + 3) = 2x + 26

The solution process involves:

1. Distribute the 4: The left side becomes 4x + 12. The equation is now 4x + 12 = 2x + 26.

2. Subtract 2x from both sides: This simplifies the equation to 2x + 12 = 26.

3. Subtract 12 from both sides: This gives 2x = 14.

4. Divide both sides by 2: This yields the solution: x = 7.

Verification confirms the accuracy of the solution.

Real-World Applications

The expression "four times the sum of a number and 3" isn't just a theoretical concept; it has practical applications in various real-world scenarios:

Geometry

Imagine a rectangle where the length is 3 units more than the width (x). The perimeter of the rectangle is given by 2(length + width) = 2(x + x + 3) = 2(2x + 3) = 4x + 6. If we increase the perimeter by 12 units (perhaps by adding a border), the new perimeter is 4x + 18. While not exactly "four times the sum of a number and 3", it demonstrates the application of similar algebraic thinking in a geometric context.

Finance

Suppose you invest a certain amount of money (x) and earn 3% interest. Then the total amount is x + 0.03x = 1.03x. If you quadrupled that amount, you would have 4(1.03x), reflecting a similar mathematical structure to the expression under consideration. While not a direct application, it illustrates the relevance of scaling and multiplying expressions involving sums.

Physics

In physics, many formulas involve scaling and adding quantities. For example, consider a scenario where the force is directly proportional to the sum of mass and an additional constant, and this is then further increased by a factor. Although not identical, the principle of combining addition and multiplication is analogous to our expression.

Extending the Concept: More Complex Variations

The fundamental concept can be extended to more complex expressions. For instance:

• Five times the sum of a number and 5: This translates to 5(x + 5).

• Three times the difference between a number and 2: This translates to 3(x - 2). Notice the subtraction instead of addition.

• The square of the sum of a number and 2, then multiplied by 4: This becomes 4(x + 2)². This introduces the concept of exponents.

These variations require a thorough understanding of order of operations and algebraic manipulation to solve.

Conclusion: Mastering Algebraic Expressions

The seemingly simple expression "four times the sum of a number and 3" serves as a foundation for understanding more complex algebraic concepts. By breaking down the phrase into its components and mastering the steps involved in solving equations, you build a strong base for tackling more advanced mathematical problems. The examples and applications presented in this article illustrate the practical relevance of this fundamental concept across various disciplines, showcasing its importance beyond abstract mathematical exercises. Remember that consistent practice and a clear understanding of order of operations are crucial for success in algebra and related fields. As you progress, you'll find that mastering these fundamental concepts opens doors to a much broader understanding of mathematical principles and their applications in the real world.