Integer Arithmetic: Solving -42, -19, -5 And More!

Hey guys! Let's dive into a fun math problem today. We're going to tackle an integer arithmetic puzzle. Specifically, we are given the expression: 16 - (-7) - (-14) - (-21). Our task? Figure out which of the provided sums cannot be obtained by adding any three of the numbers we get from this expression. Let's break it down step by step, shall we?

Understanding the Problem: Unveiling the Integers

Alright, the first thing we need to do is simplify the expression. Remember that subtracting a negative number is the same as adding its positive counterpart. So, let's rewrite our expression to make it easier to work with. The expression 16 - (-7) - (-14) - (-21) is equivalent to 16 + 7 + 14 + 21. When we calculate this, we get four integers: 16, 7, 14, and 21. Now, the core of our question asks us to determine which of the given options cannot be obtained by adding any three of these numbers together. Therefore, we must consider all the possible sums we can make. This part of the problem is all about understanding and applying the order of operations. We must identify the individual integers created and then test each set of possible summations. This kind of question tests our fundamental arithmetic skills and logical thinking as well.

Let's consider each of the options and figure out which one doesn't fit. We're aiming to find the impossible sum, the one we can't create. First, we need to find all possible combinations of adding three numbers. The combinations we have are: 16 + 7 + 14, 16 + 7 + 21, 16 + 14 + 21, and 7 + 14 + 21. So let's calculate the sum for each combination. Calculating each sum carefully will help ensure that we come to the right conclusion. Making mistakes here will lead to the wrong answer. Careful attention to detail is necessary to be accurate. The correct method will result in the right answer. Using a calculator or doing it manually is up to us.

Step-by-Step Solution: Calculating the Sums

Let's calculate the potential sums to see which of the options can be obtained. We'll use the numbers 16, 7, 14, and 21 to find different combinations and determine what sums are possible. Let's begin! Firstly, let's calculate the sum of 16 + 7 + 14. Adding these numbers together, we get 37. This gives us our first possible sum. Next up, let's find the sum of 16 + 7 + 21. If we add these, we get 44. This is our second possibility. Now, let's combine 16 + 14 + 21. This combination totals 51. And finally, we can take 7 + 14 + 21. This yields 42. By now, we have four sums that can be obtained by combining three numbers from the original set of numbers. So, the possible sums are 37, 44, 51, and 42. Let's analyze the options provided: We're looking for a sum that doesn't match any of these values.

Now, let's check the answer choices. We're given -42, -19, and -5. We will be comparing these with our calculated values. Since we calculated the possible sums to be 37, 44, 51, and 42, none of the provided options directly matches the sums that we just calculated using the original expression and integer combinations. However, we are looking for the sum that cannot be obtained. We need to make sure that each number is not possible. The question requires us to identify the number that cannot be obtained by adding three numbers from our original set. To check, let's ensure we did our calculations correctly. Making sure we did not make a mistake in our addition or subtraction, or in transposing the numbers, is crucial.

Evaluating the Options: Finding the Impossible Sum

We need to match our calculated sums with the answer choices provided. Here's where the fun begins. We have already calculated all possible sums by combining three integers at a time. We also simplified our original expression, so we have a solid basis to solve this problem. Our task now is to examine each option (-42, -19, and -5) and see if any of these values can be obtained using the sums we have. We have identified 37, 44, 51, and 42 as possible sums by adding three numbers at a time. Let's examine these options and compare them with our derived sums. Our task is to determine which of the given options cannot be obtained by summing up any three numbers from our original set.

Now, carefully compare our calculated sums with the given answer options. None of the sums we have calculated, 37, 44, 51, and 42, are the same as -42, -19, or -5. However, the values are positive and none of the sums is negative. It's crucial to recognize that the problem doesn't ask us to find the sum, but rather to determine which cannot be achieved. Therefore, we are searching for an option that doesn't match any of the sums we've calculated. Looking at the choices, it's clear that all our sums are positive, so negative numbers like -42, -19, and -5 cannot be achieved. But since the question is which cannot be, we need to choose an option from the given options. Now we can determine which of the options cannot be achieved. The options -42, -19, and -5 are not attainable because we derived positive values from our initial calculation. Since none of the options match our results from adding up three numbers, we can definitively state that all are impossible.

Conclusion: The Unreachable Sum

So, which of the given options cannot be obtained? Well, from our step-by-step solution, we've determined that none of the provided sums can be obtained. It is important to remember that we have to choose one answer. After we simplify the original expression and combine all three numbers at a time, the possible sums are positive integers. The provided options, -42, -19, and -5, are all negative, which means that none of these sums can be achieved. Therefore, the correct answer is any of the three options presented, as they are all impossible to obtain through the calculations we performed. The critical idea is that we simplified the initial expression to find the correct integer combination, then we evaluated the sums using the available options.

I hope this explanation helps you understand the process! Keep practicing, and you'll get better at these types of problems. If you have any questions, feel free to ask! Keep up the great work, and keep learning!