Solving Mathematical Equations Step-by-Step A Comprehensive Guide

Hey guys! Let's tackle these math problems step by step. We'll break them down, making sure we understand each operation. Math can be fun when we approach it systematically, so let's dive in!

B. 392 - (42 - .) = .... - 12 = ....

Okay, in this first equation, the key is to work from the inside out. We have 392 minus something in parentheses, and inside the parentheses, we have 42 minus a mystery number. Then, the result of that subtraction is further subtracted by 12. This might look tricky, but let’s think about what we’re trying to achieve.

The first step is understanding the purpose of parentheses in mathematical operations. Parentheses, often called brackets, indicate a priority in the order of operations. In simple terms, whatever is inside the parentheses needs to be calculated first before anything else in the equation. This is a fundamental rule in mathematics, ensuring that we solve equations in a consistent and logical manner. Think of it like this: the parentheses are saying, “Hey, deal with me first!” Ignoring this rule can lead to incorrect answers, so it’s super important to pay attention to them.

Now, focusing on the expression inside the parentheses, we have 42 - .. The dot here signifies a missing number, something we need to figure out. To understand its role, let's consider the whole equation: 392 - (42 - .) = .... - 12 = ..... We're looking for a number that, when subtracted from 42, gives us a result that, when subtracted from 392, leaves us with a value that, when further reduced by 12, results in our final answer. This may sound complex, but it's all about peeling back the layers.

Let's strategically choose a number for the dot. Suppose we want the final subtraction of 12 to give us a clean, easy-to-work-with number. For instance, let's aim for the intermediate result before subtracting 12 to be something like 132. This is just a strategic choice to simplify our calculations. To get 132 after subtracting 12, we need the previous result to be 132 + 12 = 144. So now our equation looks like this: 392 - (42 - .) = 144 - 12 = 132. We're making progress!

Now we know that 392 - (42 - .) should equal 144. This means that the expression inside the parentheses, (42 - .), must equal the difference between 392 and 144. Let's calculate that: 392 - 144 = 248. So we have 42 - . = 248. Wait a minute! This reveals something interesting. We've stumbled upon a situation where subtracting a number from 42 results in 248. Mathematically, this implies that the missing number would have to be a negative number since the result is much larger than 42. This is a valuable insight because it highlights how choosing different values for the missing number can significantly affect the outcome and even lead to negative numbers.

Let's take a step back and rethink our strategy. Maybe aiming for 132 as the final result was too ambitious. Instead, let’s try to make the number inside the parentheses smaller, so the result of 392 - (42 - .) is closer to 392. This way, we might avoid negative numbers and keep things simpler. What if we want the result of the entire expression before subtracting 12 to be 380? This is a more conservative approach, aiming for a smaller reduction from the initial number.

If we want 380 before subtracting 12, then after subtracting 12, we would get 380 - 12 = 368. So our equation now looks like 392 - (42 - .) = 380 - 12 = 368. Now we need to find a value for the dot that makes 392 - (42 - .) equal to 380. This means that the expression inside the parentheses, (42 - .), must equal the difference between 392 and 380. Let’s calculate that: 392 - 380 = 12.

So now we have 42 - . = 12. This is much more manageable! To find the missing number, we need to figure out what we subtract from 42 to get 12. This is a simple subtraction problem: 42 - 12 = 30. So the missing number is 30! Now we can fill in the blanks in our equation: 392 - (42 - 30) = 380 - 12 = 368.

Therefore, the final solution for B is:

392 - (42 - 30) = 380 - 12 = 368

This exercise highlights the importance of strategic thinking and how choosing intermediate results can simplify the process of solving complex equations. Remember, math is like a puzzle, and sometimes you need to try a few different approaches to find the right fit!

C. (589 - 61) - 18 = .... - .... = ....

Alright, let's move on to equation C. In this equation, we're presented with a series of subtractions: (589 - 61) - 18 = .... - .... = ..... The beauty of this problem lies in its straightforward nature. We just need to follow the order of operations and subtract carefully.

Just like in the previous problem, parentheses play a crucial role here. They tell us which operation to perform first. In this case, we need to subtract 61 from 589 before we do anything else. This is a fundamental aspect of mathematical problem-solving: prioritizing operations within parentheses. Ignoring this could lead to a completely different answer, so it's important to respect the parentheses!

So, our first step is to tackle the subtraction inside the parentheses: 589 - 61. This is a standard subtraction problem. You can do it mentally, on paper, or even with a calculator – whatever works best for you. When we subtract 61 from 589, we get 528. So, we can replace the expression inside the parentheses with its result: 528. Now our equation looks like this: 528 - 18 = .... - .... = ..... We've already made significant progress by simplifying the first part of the equation.

Now, we need to subtract 18 from 528. This is another straightforward subtraction. Again, you can use whichever method you're most comfortable with. Subtracting 18 from 528 gives us 510. So, after performing the subtraction, our equation now looks like this: 510 = .... - .... = ..... We're getting closer to the final answer!

Now, let's focus on filling in the remaining blanks. The equation is structured as .... - .... = ...., which essentially asks us to express 510 as the result of a subtraction. There are infinite possibilities here! We could choose any two numbers that, when subtracted, give us 510. This is where the problem becomes a little more open-ended and allows for some creativity.

For example, we could choose 520 and 10. Subtracting 10 from 520 gives us 510. So, we could fill in the blanks like this: 520 - 10 = 510. This is a perfectly valid solution. Or, we could choose 600 and 90. Subtracting 90 from 600 also gives us 510. So, another valid solution would be 600 - 90 = 510. You see, there's no single right answer here; it's about finding a pair of numbers that fit the equation.

To make things simple and easy to follow, let's choose numbers that are relatively close to 510. This will keep the subtraction straightforward. How about 530 and 20? Subtracting 20 from 530 does indeed give us 510. So, this is a good choice. Now we can rewrite our equation with these numbers: 510 = 530 - 20 = 510.

Therefore, one possible solution for C is:

(589 - 61) - 18 = 530 - 20 = 510

Remember, this is just one possible solution. You could choose different numbers and still arrive at the correct answer. The key is to ensure that the subtraction is accurate and that the final result is indeed 510. This problem highlights that in mathematics, sometimes there isn't just one right answer, but rather a range of possibilities. It encourages us to think creatively and explore different options.

D. 769 - (.... - ....) = .... - 37 = ....

Okay, guys, let's tackle the final equation, D: 769 - (.... - ....) = .... - 37 = ..... This one looks a bit more challenging because it has missing numbers both inside and outside the parentheses. But don't worry, we'll break it down step by step. Just like before, the key is to think strategically and consider the order of operations.

First, let's remind ourselves of the significance of parentheses. They group operations together, telling us which calculations to perform first. In this equation, the parentheses enclose a subtraction: (.... - ....). This means we need to figure out what numbers to put in those blanks and perform that subtraction before we can subtract the result from 769. So, the expression inside the parentheses is our first priority.

Now, let's look at the rest of the equation: 769 - (.... - ....) = .... - 37 = ..... We have 769 minus the result of the subtraction inside the parentheses. This will give us an intermediate result, which is then subtracted by 37 to give us the final answer. Our goal is to fill in the blanks in a way that makes the equation true and the calculations manageable.

To approach this, let's think about what the result of the subtraction inside the parentheses should be. We want to subtract this result from 769, so let's consider what kind of number would be easy to subtract from 769. A good strategy is to aim for a round number, something that ends in zero. This will simplify the subsequent subtraction of 37.

For instance, let's aim for the intermediate result before subtracting 37 to be 737. This is a strategic choice because it's close to 769 and ends in 7, which will make the subtraction of 37 straightforward. To get 737 after subtracting something from 769, we need the result of the subtraction inside the parentheses to be the difference between 769 and 737. Let's calculate that: 769 - 737 = 32. So, we want the expression (.... - ....) to equal 32.

Now we know that (.... - ....) should equal 32. This is similar to what we did in equation C, where we need to find two numbers that, when subtracted, give us a specific result. There are multiple possibilities here. We could choose 42 and 10, since 42 - 10 = 32. Or, we could choose 50 and 18, since 50 - 18 = 32. The key is to pick numbers that are easy to work with.

Let's go with 42 and 10. This seems like a manageable choice. So now we can fill in the blanks inside the parentheses: (42 - 10). Our equation now looks like this: 769 - (42 - 10) = .... - 37 = ..... We've made significant progress by figuring out the values inside the parentheses.

Now, let's simplify the equation further. We know that 42 - 10 = 32, so we can replace the expression inside the parentheses with 32: 769 - 32 = .... - 37 = ..... This is much simpler! We've reduced the complexity of the equation by dealing with the parentheses first.

Next, we need to subtract 32 from 769. This is a standard subtraction problem. Performing the subtraction, we get 769 - 32 = 737. So now our equation looks like this: 737 = .... - 37 = ..... We're getting closer to the final answer!

Finally, we need to subtract 37 from 737. This will give us the final result. Performing the subtraction, we get 737 - 37 = 700. So the final answer is 700. Now we can fill in all the blanks in our equation:

769 - (42 - 10) = 737 - 37 = 700

Therefore, the solution for D is:

769 - (42 - 10) = 737 - 37 = 700

This problem highlights the importance of breaking down complex equations into smaller, more manageable steps. By strategically choosing intermediate results and working through the operations in the correct order, we were able to solve this equation successfully. Remember, math is like building with blocks; you need to lay the foundation before you can build the structure!

Wrapping Up

So there you have it, guys! We've worked through these math problems together, step by step. Remember, the key to solving math problems is to break them down, understand the order of operations, and think strategically. Keep practicing, and you'll become a math whiz in no time!