Six Times The Sum Of A Number And 15 Is

Six Times the Sum of a Number and 15 Is... A Mathematical Exploration

This seemingly simple phrase, "Six times the sum of a number and 15 is...", opens up a world of mathematical possibilities. It's a gateway to exploring algebraic expressions, solving equations, and understanding the power of symbolic representation. This article will delve deep into this seemingly simple phrase, exploring its various interpretations, applications, and the underlying mathematical principles.

Understanding the Phrase: Deconstructing the Problem

The phrase itself is a concise description of an algebraic relationship. Let's break it down step-by-step:

• "a number": This represents an unknown quantity, which we typically denote with a variable, most commonly x.

• "the sum of a number and 15": This translates directly to the algebraic expression x + 15. We're adding 15 to our unknown number.

• "Six times the sum of a number and 15": This means we're multiplying the previous expression by 6, resulting in the algebraic expression 6(x + 15). The parentheses are crucial; they indicate that the entire sum (x + 15) is multiplied by 6.

• "is...": This signifies an equation; it indicates that the expression 6(x + 15) is equal to something else. The phrase is incomplete without specifying what this expression equals.

Solving for 'x': Different Scenarios and Equations

The incompleteness of the phrase provides a fantastic opportunity to explore various scenarios. Let's consider a few examples:

Scenario 1: Six times the sum of a number and 15 is 90

This gives us the complete equation: 6(x + 15) = 90

Solving the Equation:

1. Distributive Property: Distribute the 6 to both terms inside the parentheses: 6x + 90 = 90

2. Subtract 90: Subtract 90 from both sides of the equation: 6x = 0

3. Solve for x: Divide both sides by 6: x = 0

Therefore, in this scenario, the number is 0.

Scenario 2: Six times the sum of a number and 15 is 126

This leads to the equation: 6(x + 15) = 126

Solving the Equation:

1. Distributive Property: 6x + 90 = 126

2. Subtract 90: 6x = 36

3. Solve for x: x = 6

Here, the number is 6.

Scenario 3: Six times the sum of a number and 15 is equal to twice the number plus 138

This scenario introduces a more complex equation: 6(x + 15) = 2x + 138

Solving the Equation:

1. Distributive Property: 6x + 90 = 2x + 138

2. Subtract 2x from both sides: 4x + 90 = 138

3. Subtract 90 from both sides: 4x = 48

4. Solve for x: x = 12

In this case, the number is 12.

Expanding the Scope: Real-World Applications

While seemingly abstract, these types of algebraic expressions find practical applications in various fields:

• Finance: Calculating compound interest, determining loan repayments, and analyzing investment returns often involve similar algebraic manipulations.

• Physics: Formulas used in kinematics (the study of motion) or in analyzing forces and energy often utilize similar structures.

• Engineering: Designing structures, calculating stresses and strains, and modeling systems frequently require solving complex equations resembling the ones we've explored.

• Computer Science: Programming algorithms, especially those involving loops and conditional statements, often necessitates an understanding of algebraic relationships.

• Everyday Life: Simple tasks like calculating discounts, splitting bills, or figuring out the best deal at the grocery store can be approached with similar mathematical reasoning.

Beyond the Basics: Exploring More Complex Scenarios

Let's consider scenarios that involve inequalities:

Scenario 4: Six times the sum of a number and 15 is greater than 100

This translates to the inequality: 6(x + 15) > 100

Solving the Inequality:

1. Distributive Property: 6x + 90 > 100

2. Subtract 90: 6x > 10

3. Solve for x: x > 10/6 or x > 5/3

This means any number greater than 5/3 (approximately 1.67) satisfies the inequality.

Scenario 5: Six times the sum of a number and 15 is less than or equal to 75

This gives us the inequality: 6(x + 15) ≤ 75

Solving the Inequality:

1. Distributive Property: 6x + 90 ≤ 75

2. Subtract 90: 6x ≤ -15

3. Solve for x: x ≤ -15/6 or x ≤ -5/2 or x ≤ -2.5

Here, any number less than or equal to -2.5 satisfies the inequality.

The Importance of Parentheses and Order of Operations

The use of parentheses in the expression 6(x + 15) is absolutely critical. Without them, the expression would be interpreted differently, leading to an incorrect result. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Without parentheses, multiplication would be performed before addition, resulting in a completely different algebraic expression.

Conclusion: A Simple Phrase, Limitless Possibilities

The seemingly simple phrase, "Six times the sum of a number and 15 is...", serves as a powerful illustration of the fundamental concepts of algebra. By exploring different scenarios and solving various equations and inequalities, we've gained a deeper understanding of algebraic manipulation and its relevance in numerous contexts. This simple phrase unlocks a universe of mathematical problem-solving and real-world applications, highlighting the importance of understanding fundamental mathematical principles. The key takeaway is that even the simplest mathematical statement can lead to a rich exploration of concepts and their practical uses. This highlights the beauty and power of mathematics in its ability to model and solve problems across numerous disciplines.