Unveiling Mathematical Truth Question 12 Analysis

In the realm of mathematics, discerning the truth from falsehood is a fundamental skill. Question 12 presents us with a trio of equations, each a potential pathway to understanding mathematical principles. Our mission is to meticulously analyze each statement, unraveling the operations and relationships within, to pinpoint the one that holds the key to accuracy. This endeavor will not only test our computational prowess but also our ability to interpret and apply the rules that govern the world of numbers.

Statement A: $2 imes (4+2) - 6 - 14 = (36 imes 2) + 4$ - A Critical Analysis

Our mathematical journey begins with statement A: $2 imes (4+2) - 6 - 14 = (36 imes 2) + 4$. To determine its veracity, we must embark on a step-by-step evaluation of both sides of the equation. The left-hand side (LHS) presents us with a series of operations that demand our attention. First, we encounter the expression within the parentheses: $(4+2)$. According to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), we must resolve this inner sanctum before proceeding further. The sum of 4 and 2 is a straightforward 6, transforming our expression to $2 imes 6 - 6 - 14$.

Next, we encounter the multiplication operation: $2 imes 6$. This yields 12, and our equation now reads $12 - 6 - 14$. The remaining operations are subtractions, which we perform sequentially from left to right. Subtracting 6 from 12 gives us 6, leading to $6 - 14$. Finally, subtracting 14 from 6 results in -8. Thus, the LHS of our equation simplifies to -8.

Now, let's turn our attention to the right-hand side (RHS) of the equation: $(36 imes 2) + 4$. Here, we first encounter the multiplication operation: $36 imes 2$. This product is 72, transforming our expression to $72 + 4$. Adding 4 to 72 gives us 76. Therefore, the RHS of our equation simplifies to 76.

Comparing the simplified LHS (-8) and RHS (76), we find a stark disparity. -8 is emphatically not equal to 76. This discrepancy leads us to the inescapable conclusion that statement A is false. The equation presented does not hold true under the scrutiny of mathematical evaluation. This highlights the importance of meticulous calculation and adherence to the order of operations in determining the validity of mathematical statements.

Statement B: 1 imes ig(rac{1}{2} imes 0ig) = 6 - 8 - ig(rac{1}{4} imes 16ig) + 2 - A Quest for Accuracy

Our exploration continues with statement B: 1 imes ig(rac{1}{2} imes 0ig) = 6 - 8 - ig(rac{1}{4} imes 16ig) + 2. As with statement A, we must dissect both sides of the equation, meticulously unraveling the operations to reveal the underlying truth. Let's begin with the left-hand side (LHS): 1 imes ig(rac{1}{2} imes 0ig). Within the parentheses, we encounter the expression rac{1}{2} imes 0. A fundamental principle of mathematics dictates that any number multiplied by zero yields zero. Therefore, rac{1}{2} imes 0 equals 0. Our expression now simplifies to $1 imes 0$.

Again, we encounter multiplication by zero. Applying the same principle, $1 imes 0$ equals 0. Thus, the LHS of our equation gracefully simplifies to 0. This seemingly simple side of the equation underscores the profound impact of zero in mathematical operations.

Now, let's turn our attention to the right-hand side (RHS) of the equation: 6 - 8 - ig(rac{1}{4} imes 16ig) + 2. The expression within the parentheses demands our immediate attention: rac{1}{4} imes 16. Multiplying rac{1}{4} by 16 is equivalent to dividing 16 by 4, which results in 4. Our equation now reads $6 - 8 - 4 + 2$.

We are left with a series of additions and subtractions, which we perform sequentially from left to right. Subtracting 8 from 6 gives us -2, leading to $-2 - 4 + 2$. Subtracting 4 from -2 results in -6, transforming our expression to $-6 + 2$. Finally, adding 2 to -6 yields -4. Therefore, the RHS of our equation simplifies to -4.

Comparing the simplified LHS (0) and RHS (-4), we find a clear divergence. 0 is definitively not equal to -4. This discrepancy compels us to conclude that statement B is false. The equation presented fails to maintain equality under mathematical scrutiny. This reinforces the importance of careful calculation and attention to detail in navigating the world of mathematical expressions.

Statement C: 6 + rac{20}{5} = 10 - The Verdict

Our final contender is statement C: 6 + rac{20}{5} = 10. This statement appears deceptively simple, yet it holds the potential for profound insight. To ascertain its truth, we must meticulously evaluate the expression, adhering to the established order of operations.

The left-hand side (LHS) of the equation presents us with 6 + rac{20}{5}. Here, we encounter both addition and division. According to PEMDAS, division takes precedence over addition. Thus, we must first resolve the fraction rac{20}{5}. Dividing 20 by 5 yields 4. Our expression now simplifies to $6 + 4$.

Adding 4 to 6 gives us 10. Therefore, the LHS of our equation elegantly simplifies to 10. Now, let's turn our attention to the right-hand side (RHS), which is simply 10.

Comparing the simplified LHS (10) and RHS (10), we discover a harmonious agreement. 10 is indeed equal to 10. This congruence leads us to the triumphant conclusion that statement C is true. The equation presented stands as a testament to mathematical accuracy, validating the principles that govern numerical relationships. This successful verification underscores the importance of understanding the order of operations and applying it diligently in mathematical evaluations.

Conclusion: The Triumph of Truth

Our mathematical expedition has led us through a landscape of equations, each presenting a unique challenge. Through meticulous analysis and unwavering adherence to the principles of mathematics, we have successfully navigated the complexities and arrived at the truth. Statements A and B, initially promising, ultimately succumbed to the scrutiny of calculation, revealing their falsity. Statement C, however, emerged as the victor, its inherent accuracy shining brightly.

This exercise serves as a potent reminder of the power of mathematical reasoning and the importance of precision in our calculations. The world of numbers is governed by immutable laws, and our ability to understand and apply these laws is the key to unlocking the secrets they hold. As we continue our mathematical journey, let us carry with us the lessons learned, embracing the challenge and celebrating the triumph of truth.