Five Times The Sum Of A Number And 4 Is

Five Times the Sum of a Number and 4 Is... A Mathematical Exploration

This seemingly simple phrase, "five times the sum of a number and 4 is," opens the door to a world of mathematical exploration. It's a foundation upon which we can build complex equations, explore different solution methods, and even delve into the practical applications of algebra in everyday life. This article will dissect this phrase, examining its different interpretations, solving various equations based on it, and exploring the broader mathematical concepts involved.

Understanding the Phrase: Deconstructing the Problem

The phrase "five times the sum of a number and 4 is" is a concise way of expressing a mathematical relationship. Let's break it down step-by-step:

• A number: This represents an unknown value, which we typically denote with a variable, such as x, y, or n.
• The sum of a number and 4: This means we are adding 4 to our unknown number. Mathematically, this is written as x + 4.
• Five times the sum: This indicates we are multiplying the previous expression (x + 4) by 5. This gives us 5(x + 4) or 5x + 20.
• Is: This signifies an equality. The expression 5(x + 4) will be equal to some other value.

Therefore, the complete mathematical representation of the phrase depends on what the phrase "is" is followed by. For instance, "Five times the sum of a number and 4 is 35" translates to the equation: 5(x + 4) = 35.

Solving Equations: Different Approaches

Let's explore how to solve equations based on the core phrase, using different methods and varying "is" values.

Example 1: Five times the sum of a number and 4 is 35

Our equation is: 5(x + 4) = 35

Method 1: Distributive Property

1. Distribute the 5: Expand the equation using the distributive property: 5x + 20 = 35
2. Subtract 20: Subtract 20 from both sides: 5x = 15
3. Divide by 5: Divide both sides by 5: x = 3

Therefore, the number is 3.

Method 2: Divide First

1. Divide by 5: Divide both sides of the original equation by 5: x + 4 = 7
2. Subtract 4: Subtract 4 from both sides: x = 3

This method demonstrates that sometimes a different approach can simplify the solution.

Example 2: Five times the sum of a number and 4 is equal to twice the number plus 30

This presents a slightly more complex equation: 5(x + 4) = 2x + 30

1. Distribute: 5x + 20 = 2x + 30
2. Subtract 2x: 3x + 20 = 30
3. Subtract 20: 3x = 10
4. Divide by 3: x = 10/3 or x ≈ 3.33

This example shows how the phrase can lead to equations involving multiple variables and requiring multiple steps to solve.

Example 3: Five times the sum of a number and 4 is less than 40

This introduces an inequality: 5(x + 4) < 40

1. Distribute: 5x + 20 < 40
2. Subtract 20: 5x < 20
3. Divide by 5: x < 4

This solution demonstrates that x can be any number less than 4. This highlights the versatility of the phrase in representing a range of values, not just a single solution.

Exploring Applications: Real-World Scenarios

The seemingly abstract nature of this mathematical phrase finds practical application in various real-world scenarios. Consider these examples:

• Calculating earnings: Suppose you earn $4 per hour plus a $20 bonus. The total earnings after 5 hours could be represented as: 5(4 + 20) = 120
• Geometric problems: The area of a rectangle with a length of 5 units and a width of (number + 4) units could be written as: 5(x + 4)
• Financial planning: If you save a certain amount (x) each week plus $4, and want to save at least $40 in 5 weeks: 5(x + 4) ≥ 40

These scenarios demonstrate that the simple algebraic expression derived from the phrase holds practical value in a variety of contexts.

Beyond the Basics: Expanding Mathematical Concepts

The fundamental expression, 5(x + 4), provides a springboard for exploring further mathematical concepts:

• Order of Operations (PEMDAS/BODMAS): The equation reinforces the importance of following the order of operations (parentheses/brackets, exponents/orders, multiplication and division, addition and subtraction).
• Linear Equations: The resulting equations are often linear equations, which have numerous applications in various fields including physics, engineering, and economics.
• Inequalities: As seen in Example 3, the phrase can readily be adapted to create inequalities, representing scenarios where a value must be greater than, less than, or greater than or equal to a certain number.
• Graphical Representation: Linear equations can be graphically represented as straight lines, allowing for visual understanding of the relationship between variables.

Conclusion: The Power of Simplicity

The phrase "five times the sum of a number and 4 is" might seem deceptively simple, but it holds significant mathematical weight. Through its various interpretations and applications, it provides a valuable tool for understanding and practicing core algebraic principles. Its ability to represent both equations and inequalities demonstrates its adaptability in solving a wide array of problems, from basic arithmetic to complex real-world scenarios. This exploration underlines the power of even the simplest mathematical expressions in conveying complex relationships and providing practical solutions. The journey from this simple phrase to the advanced mathematical concepts it unveils highlights the beauty and practicality of mathematics in everyday life. Mastering such fundamental concepts paves the way to tackling more complex mathematical challenges with confidence and precision.