Solving Math Problems Number Differences, Equations, And Algebraic Expressions

Let's delve into the world of number relationships and tackle this intriguing problem. At the heart of this question lies a simple yet fundamental concept the difference between two numbers. The difference between two numbers is 168 is our key piece of information. We're also given that the smaller number is 113. Our mission, should we choose to accept it, is to find the other number. To decipher this numerical enigma, we'll employ the power of basic arithmetic. Since we know the difference and the smaller number, we can determine the larger number by adding the difference to the smaller number. This approach leverages the inverse relationship between subtraction and addition. Think of it as undoing the subtraction that gave us the difference in the first place. We start with the smaller number, 113. We then add the difference, 168, to it. The equation representing this operation is 113 + 168 = ?. Performing this addition, we find that 113 + 168 equals 281. Therefore, the other number, the one we've been seeking, is 281. Now, let's examine the answer choices provided. We have A. 223, B. 271, C. 281, and D. 291. Our calculated answer, 281, perfectly matches option C. Thus, we can confidently conclude that the correct answer is C. 281. This problem underscores the importance of understanding the relationship between mathematical operations and how to apply them to solve real-world puzzles. It's a testament to the power of basic arithmetic in unraveling numerical mysteries. The beauty of mathematics lies in its ability to provide clear and concise solutions, and this problem serves as a shining example of that principle. By carefully analyzing the given information, we were able to identify the appropriate operation and arrive at the correct answer. This approach can be applied to a wide range of similar problems, making it a valuable tool in any mathematician's arsenal. So, the next time you encounter a problem involving the difference between two numbers, remember the power of addition and the elegance of mathematical reasoning.

Now, let's embark on a journey into the realm of algebra and conquer this equation. This problem presents us with an algebraic equation, a statement that two expressions are equal. Our goal is to solve the equation: 2x - 3(x - 1) = 6. This means we need to find the value of the variable 'x' that makes the equation true. To achieve this, we'll employ a series of algebraic manipulations, carefully following the order of operations and the properties of equality. First, we need to simplify the equation by distributing the -3 across the parentheses. This means multiplying -3 by both 'x' and -1. This gives us: 2x - 3x + 3 = 6. Notice how the negative sign in front of the 3 changes the sign of both terms inside the parentheses. This is a crucial step in correctly solving the equation. Next, we combine like terms on the left side of the equation. We have 2x and -3x, which combine to give us -x. So, our equation now looks like this: -x + 3 = 6. Now, we want to isolate the variable 'x' on one side of the equation. To do this, we subtract 3 from both sides of the equation. This is a valid algebraic manipulation because it maintains the equality. Subtracting 3 from both sides gives us: -x = 3. Finally, we need to solve for 'x', not '-x'. To do this, we multiply both sides of the equation by -1. This gives us: x = -3. Therefore, the solution to the equation is x = -3. Let's check this solution by substituting it back into the original equation: 2(-3) - 3(-3 - 1) = 6. Simplifying this, we get: -6 - 3(-4) = 6. Further simplification yields: -6 + 12 = 6, which is indeed true. This confirms that our solution is correct. Now, let's examine the answer choices provided. We have A. 3, B. -3, C. -7, and D. -9. Our calculated answer, -3, perfectly matches option B. Thus, we can confidently conclude that the correct answer is B. -3. This problem demonstrates the importance of following the order of operations and applying algebraic principles correctly. By carefully simplifying the equation and isolating the variable, we were able to arrive at the solution. This approach can be applied to a wide range of algebraic equations, making it a valuable skill for any aspiring mathematician.

In this section, we'll focus on the translation of verbal statements into mathematical expressions. This is a fundamental skill in algebra, as it allows us to represent real-world situations using mathematical symbols and equations. The question asks us to represent the sum of 5 and x divided by 4. This is a classic example of a problem that requires us to carefully consider the order of operations. Let's break down the phrase step by step. First, we have "the sum of 5 and x". This translates directly to the algebraic expression 5 + x. The word "sum" indicates addition, and we are adding 5 and the variable x. Next, we have "divided by 4". This means we are taking the sum we just calculated (5 + x) and dividing it by 4. In algebraic notation, this is written as (5 + x) / 4. It's crucial to use parentheses here to indicate that the entire sum (5 + x) is being divided by 4. If we wrote 5 + x / 4 without parentheses, the order of operations would dictate that we divide x by 4 first and then add 5, which is not what the problem statement intends. This problem highlights the importance of paying attention to the wording of a mathematical statement and using parentheses correctly to ensure the expression accurately represents the intended meaning. The ability to translate verbal statements into algebraic expressions is a crucial skill in mathematics, as it allows us to model and solve a wide range of problems. By carefully analyzing the words and identifying the mathematical operations involved, we can construct accurate and meaningful expressions. This skill is not only essential for success in algebra but also for applying mathematical concepts to real-world situations. So, the next time you encounter a verbal statement that needs to be translated into a mathematical expression, remember to break it down step by step, identify the operations involved, and use parentheses when necessary to ensure accuracy.