Five Times The Sum Of A Number And 4 Is

Five Times the Sum of a Number and 4 Is... A Deep Dive into Algebraic Expressions

This seemingly simple phrase, "five times the sum of a number and 4," opens a door to a world of mathematical exploration. It's a foundational concept in algebra, laying the groundwork for understanding more complex equations and problem-solving. This article will delve into the intricacies of this expression, exploring its different interpretations, applications, and how it relates to broader mathematical principles. We'll also examine how to solve problems involving this expression, and look at some real-world scenarios where it might be used.

Understanding the Expression: Deconstructing the Phrase

The phrase "five times the sum of a number and 4" is an algebraic expression. Let's break it down step-by-step:

• A number: This represents an unknown quantity, which we typically denote with a variable, often x.

• The sum of a number and 4: This translates to x + 4. The word "sum" indicates addition.

• Five times the sum of a number and 4: This means we multiply the previous expression (x + 4) by 5. This gives us the final algebraic expression: 5(x + 4). Notice the use of parentheses; they are crucial to indicate that the entire sum (x + 4) is being multiplied by 5. Without parentheses, the expression would be interpreted differently (5x + 4), which is a completely different expression with a different meaning and outcome.

Expanding the Expression: The Distributive Property

The expression 5(x + 4) can be simplified using the distributive property, a fundamental concept in algebra. The distributive property states that for any numbers a, b, and c: a(b + c) = ab + ac. Applying this to our expression:

5(x + 4) = 5 * x + 5 * 4 = 5x + 20

This simplified expression, 5x + 20, is equivalent to 5(x + 4). Both expressions represent the same mathematical concept, but 5x + 20 is often more useful for solving equations or performing other algebraic manipulations.

Solving Equations: Putting the Expression to Work

The expression 5(x + 4) or its simplified form 5x + 20 becomes part of an equation when it's equated to another value. For example, consider the equation:

5(x + 4) = 35

To solve for x, we can follow these steps:

1. Expand the expression (if not already expanded): We've already done this, resulting in 5x + 20 = 35.

2. Isolate the term with x: Subtract 20 from both sides of the equation: 5x = 35 - 20 = 15.

3. Solve for x: Divide both sides by 5: x = 15 / 5 = 3.

Therefore, the solution to the equation 5(x + 4) = 35 is x = 3.

Let's try another example:

5x + 20 = 45

1. Isolate the term with x: Subtract 20 from both sides: 5x = 45 - 20 = 25.

2. Solve for x: Divide both sides by 5: x = 25 / 5 = 5.

Therefore, the solution to the equation 5x + 20 = 45 is x = 5.

Real-World Applications: Where Does This Expression Appear?

While this might seem like an abstract mathematical concept, the expression "five times the sum of a number and 4" finds practical application in various real-world scenarios. Consider these examples:

• Calculating total cost: Imagine you're buying 5 identical items, each costing x dollars more than $4. The total cost would be 5(x + 4).

• Determining earnings: Suppose you earn a base salary of $4 per hour plus a bonus of x dollars per hour. If you work for 5 hours, your total earnings would be 5(x + 4).

• Analyzing geometric shapes: The expression could represent the perimeter or area of certain geometric shapes, depending on the context and values of x.

• Modeling Growth: In some growth models, particularly those involving compound interest, this expression or a variation of it could represent the total accumulation after a certain period.

Expanding the Concept: Variations and Extensions

The core idea of "five times the sum of a number and 4" can be extended to more complex scenarios. We can change the numbers, the operation (from addition to subtraction, multiplication, or division), or even introduce more variables. For instance:

• Three times the sum of a number and 7: This would be represented as 3(x + 7) or 3x + 21.

• Seven times the difference between a number and 2: This would be 7(x - 2) or 7x - 14.

• The product of a number and 6, increased by 8: This translates to 6x + 8.

Mastering the basic principle of this initial expression helps build a strong foundation for tackling more complex algebraic equations and word problems.

Advanced Applications and Problem Solving Strategies

Let’s delve into some more sophisticated problems involving variations of the expression "five times the sum of a number and 4":

Problem 1: Word Problem involving the expression

"John earns $4 per hour plus a bonus of x dollars per hour. After working 5 hours, he receives a total of $35. Find the value of x (his bonus per hour)."

Solution: This problem can be modeled using the equation 5(x + 4) = 35. Following the steps outlined previously, we find that x = 3. John's bonus per hour is $3.

Problem 2: Solving a more complex equation:

Solve for x: 5(x + 4) + 10 = 50

Solution:

1. Simplify: 5x + 20 + 10 = 50
2. Combine like terms: 5x + 30 = 50
3. Isolate x: 5x = 50 - 30 = 20
4. Solve for x: x = 20 / 5 = 4

Therefore, the solution to the equation is x = 4.

Problem 3: Inequalities

Instead of an equation, we can have an inequality. For example:

5(x + 4) > 25

Solution:

1. Expand: 5x + 20 > 25
2. Subtract 20 from both sides: 5x > 5
3. Divide by 5: x > 1

This means that any value of x greater than 1 will satisfy the inequality.

Conclusion: From Simple Expression to Powerful Tool

The simple algebraic expression "five times the sum of a number and 4" serves as a gateway to understanding more complex mathematical concepts. By mastering the principles of expanding expressions, applying the distributive property, and solving equations and inequalities, you've laid a strong foundation for further advancements in algebra and its numerous applications in various fields. The ability to translate word problems into mathematical expressions is a crucial skill that opens doors to problem-solving in diverse real-world scenarios. Remember to practice regularly and explore different variations of this expression to solidify your understanding and build confidence in your algebraic abilities. The seemingly simple phrase is, in fact, a powerful tool in your mathematical arsenal.