Five Times The Difference Of A Number And 3.

Five Times the Difference of a Number and 3: A Deep Dive into Mathematical Expressions

This article explores the mathematical expression "five times the difference of a number and 3," dissecting its meaning, exploring its applications, and demonstrating how to manipulate it algebraically. We'll delve into various scenarios, solving equations, and even touching upon real-world applications to solidify your understanding. This comprehensive guide aims to provide a clear and engaging explanation suitable for various levels of mathematical proficiency.

Understanding the Expression

The phrase "five times the difference of a number and 3" translates directly into a mathematical expression. Let's break it down step-by-step:

• A number: We typically represent this unknown number with a variable, most commonly 'x'.
• The difference of a number and 3: This implies subtraction. The expression becomes 'x - 3'.
• Five times the difference: This means we multiply the previous expression by 5. Therefore, the complete expression is 5(x - 3).

It's crucial to understand the order of operations (PEMDAS/BODMAS). The parentheses indicate that the subtraction within the parentheses must be performed before the multiplication by 5.

Expanding the Expression

The expression 5(x - 3) can be simplified using the distributive property of multiplication over addition (or subtraction). The distributive property states that a(b + c) = ab + ac. Applying this to our expression:

5(x - 3) = 5 * x - 5 * 3 = 5x - 15

This simplified form, 5x - 15, is equivalent to the original expression 5(x - 3) and is often easier to work with in algebraic manipulations.

Solving Equations Involving the Expression

Let's explore how this expression can be used within equations. Consider the following examples:

Example 1: Finding the value of x

Let's say the expression "five times the difference of a number and 3" equals 20. We can write this as an equation:

5(x - 3) = 20

To solve for x, we can follow these steps:

1. Expand the expression: 5x - 15 = 20
2. Add 15 to both sides: 5x = 35
3. Divide both sides by 5: x = 7

Therefore, the number is 7. We can check our answer by substituting x = 7 back into the original equation: 5(7 - 3) = 5(4) = 20. Our solution is correct.

Example 2: A more complex equation

Consider a slightly more complex equation:

2(5(x - 3) + 10) = 40

To solve this:

1. Expand the inner parentheses: 2(5x - 15 + 10) = 40
2. Simplify within the parentheses: 2(5x - 5) = 40
3. Expand the remaining parentheses: 10x - 10 = 40
4. Add 10 to both sides: 10x = 50
5. Divide both sides by 10: x = 5

Again, we can check our answer: 2(5(5 - 3) + 10) = 2(5(2) + 10) = 2(20) = 40. The solution is correct.

Real-World Applications

While this might seem like a purely abstract mathematical concept, the principles behind "five times the difference of a number and 3" have real-world applications. Let's consider a few scenarios:

Scenario 1: Profit Calculation

Imagine a small business sells handmade crafts. Let's say 'x' represents the number of crafts sold, and each craft costs $3 to produce. The profit per craft is the selling price minus the production cost. If each craft sells for $8, the profit per craft is 8 - 3 = $5. If the business sold 'x' crafts, the total profit would be 5x. If they want to know the profit after selling all but 3 crafts, the equation becomes 5(x - 3).

Scenario 2: Temperature Conversion

Suppose we're working with a temperature scale where the baseline is 3 degrees. If we need to calculate a value five times the difference between a temperature reading ('x') and this baseline, we would use the expression 5(x - 3). This might be relevant in scientific experiments or engineering applications where a specific reference temperature is used.

Scenario 3: Discount Calculations

Consider a store offering a discount. Let's say 'x' represents the original price of an item, and the store offers a discount of $3. If the customer buys five such items, the total discount would be 5(x - 3). This allows us to easily calculate the total discount based on the number of items purchased.

Advanced Concepts and Extensions

The expression 5(x - 3) serves as a foundation for understanding more complex algebraic manipulations. We can extend this concept in several ways:

• Inequalities: We could use the expression within inequalities, such as 5(x - 3) > 10, to find the range of values of x that satisfy the condition.
• Functions: We can define a function, f(x) = 5(x - 3), which maps input values of x to output values based on the expression. Analyzing the properties of such a function (domain, range, etc.) provides further insights into its behavior.
• Graphing: Graphing the function f(x) = 5(x - 3) allows for a visual representation of the relationship between x and the value of the expression. This visual aid can enhance understanding, particularly when dealing with multiple expressions or comparing different functions.
• Systems of Equations: The expression could be incorporated into a system of equations, where it interacts with other expressions to find the solutions for multiple variables.

Conclusion

The simple expression "five times the difference of a number and 3," while seemingly basic, offers a rich foundation for understanding algebraic concepts and their applications. By breaking down the expression, expanding it, solving equations involving it, and exploring its real-world applications, we've gained a comprehensive understanding of its significance. Understanding this fundamental concept lays the groundwork for tackling more complex mathematical problems and real-world scenarios, solidifying your skills in algebra and problem-solving. Remember to practice solving various equations and explore different applications to further enhance your understanding and build your confidence in tackling mathematical challenges. The key is consistent practice and application of the principles discussed here. Keep exploring, keep learning, and remember that even the simplest mathematical concepts can unlock a world of possibilities.