Twice A Number Divided By 6 Is 42

Twice a Number Divided by 6 is 42: Unraveling the Mathematical Mystery

This seemingly simple math problem, "Twice a number divided by 6 is 42," hides a wealth of learning opportunities, extending far beyond the immediate solution. We'll not only solve this equation but also explore the underlying concepts, delve into different approaches to solving it, and examine its applications in real-world scenarios. This exploration will demonstrate how seemingly simple problems can offer profound insights into mathematical thinking and problem-solving strategies.

Understanding the Problem

Before diving into the solution, let's break down the problem statement: "Twice a number divided by 6 is 42." This phrase translates directly into a mathematical equation. Let's represent "a number" with the variable x. The problem then becomes:

(2x) / 6 = 42

This equation states that twice a number (2x), when divided by 6, equals 42. Our goal is to find the value of x.

Solving the Equation: A Step-by-Step Approach

Solving this equation involves a series of algebraic manipulations to isolate x. Here's a step-by-step approach:

1. Multiply both sides by 6:

To eliminate the division by 6, we multiply both sides of the equation by 6:

6 * (2x / 6) = 42 * 6

This simplifies to:

2x = 252

2. Divide both sides by 2:

To isolate x, we divide both sides of the equation by 2:

2x / 2 = 252 / 2

This simplifies to:

x = 126

Therefore, the number is 126.

Verifying the Solution

It's crucial to verify our solution by substituting x = 126 back into the original equation:

(2 * 126) / 6 = 42

252 / 6 = 42

42 = 42

The equation holds true, confirming that our solution, x = 126, is correct.

Alternative Methods of Solving the Equation

While the previous method is straightforward, let's explore alternative approaches to solving the same equation. This demonstrates the flexibility and adaptability of mathematical problem-solving.

1. Simplifying before Solving:

We can simplify the equation before solving it. Notice that (2x)/6 can be simplified to x/3. The equation then becomes:

x / 3 = 42

Multiplying both sides by 3 directly gives:

x = 126

This method streamlines the process, reducing the number of steps.

2. Using Inverse Operations:

We can also solve this using inverse operations. The equation involves multiplication and division. To isolate x, we perform the inverse operations in reverse order. First, we undo the division by multiplying by 6, and then we undo the multiplication by 2 by dividing by 2. This method reinforces the understanding of inverse operations and their role in solving equations.

Real-World Applications

While this problem seems purely mathematical, it's crucial to understand its real-world applications. The ability to translate word problems into mathematical equations and solve them is a fundamental skill applicable across numerous fields.

• Business and Finance: This type of problem could represent profit calculations, where twice the number of units sold (2x) divided by the production cost (6) equals the total profit (42). Solving for x would tell you the number of units sold.

• Engineering and Physics: Similar equations can represent relationships between variables in physical systems. For instance, the equation could model the relationship between force, mass, and acceleration.

• Everyday Life: Proportion problems are common in everyday situations. Dividing a recipe for two people to feed six would involve similar calculations.

Expanding the Problem: Introducing Variations

Let's now consider variations of the problem to further deepen our understanding and problem-solving skills.

1. Twice a number divided by y is 42:

Instead of dividing by 6, let's introduce a variable y. The equation becomes:

(2x) / y = 42

This introduces an additional layer of complexity. To solve for x, we need the value of y. For example, if y = 7, we solve as follows:

2x = 42 * 7 = 294 x = 294 / 2 = 147

2. a times a number divided by 6 is 42:

We can generalize further by replacing "twice" with a variable a. The equation becomes:

(ax) / 6 = 42

Solving for x requires the value of a. If a = 3, then:

3x = 42 * 6 = 252 x = 252 / 3 = 84

The Importance of Word Problems in Mathematics

This problem highlights the importance of word problems in mathematics education. They bridge the gap between abstract mathematical concepts and their real-world applications. They encourage critical thinking, logical reasoning, and the ability to translate real-world scenarios into mathematical models. Solving these types of problems develops essential skills that are crucial for success in various academic and professional fields.

Conclusion: Beyond the Numbers

The seemingly simple problem, "Twice a number divided by 6 is 42," offers a rich learning experience. Through solving it and exploring its variations, we've touched upon fundamental algebraic concepts, alternative problem-solving strategies, and real-world applications. More importantly, we've demonstrated how a simple equation can lead to a deeper understanding of mathematical thinking and its relevance in various aspects of life. The ability to dissect and solve such problems is not just about getting the right answer; it's about developing crucial analytical and problem-solving skills that are valuable far beyond the classroom. This problem serves as a microcosm of the broader mathematical world, showcasing the beauty and practicality of mathematical reasoning. So, the next time you encounter a seemingly simple math problem, remember the potential for learning and discovery it holds.