Mastering Integer Arithmetic Subtracting Sums And Finding Integers

In this section, we will delve into the intricacies of subtracting the sum of two positive integers from a negative integer. Understanding the order of operations is paramount in solving this problem accurately. We will break down the steps involved, ensuring clarity and comprehension. This foundational concept is crucial for mastering more complex arithmetic problems. First, let's define the key components of this problem. We have two positive integers, 13 and 8, and a negative integer, -32. The task is to subtract the sum of 13 and 8 from -32. To begin, we need to find the sum of 13 and 8. This is a straightforward addition operation. 13 plus 8 equals 21. So, the sum of 13 and 8 is 21. Now, we need to subtract this sum, 21, from -32. This is where the concept of subtracting a positive number from a negative number comes into play. When we subtract a positive number from a negative number, we are essentially moving further into the negative number line. In other words, we are adding the negative of the number we are subtracting. So, subtracting 21 from -32 is the same as adding -21 to -32. -32 plus -21 equals -53. Therefore, subtracting the sum of 13 and 8 from -32 results in -53. This problem highlights the importance of understanding the number line and how operations affect the position on the number line. When subtracting a positive number from a negative number, the result will always be a more negative number. This is a fundamental concept in integer arithmetic and serves as a building block for more advanced mathematical concepts.

Next, we'll tackle the subtraction of the sum of two negative integers from a positive integer. This problem introduces the concept of adding negative numbers and then subtracting the result from a positive number. This requires a solid grasp of negative number operations. We will explore the mechanics of this operation in detail. The problem involves two negative integers, -18 and -13, and a positive integer, 38. The objective is to subtract the sum of -18 and -13 from 38. The first step is to find the sum of -18 and -13. When adding two negative numbers, we simply add their absolute values and then apply the negative sign. The absolute value of -18 is 18, and the absolute value of -13 is 13. 18 plus 13 equals 31. So, the sum of -18 and -13 is -31. Now, we need to subtract this sum, -31, from 38. Subtracting a negative number is the same as adding its positive counterpart. So, subtracting -31 from 38 is the same as adding 31 to 38. 38 plus 31 equals 69. Therefore, subtracting the sum of -18 and -13 from 38 results in 69. This problem illustrates the concept of subtracting a negative number, which is equivalent to addition. It's crucial to remember this rule when dealing with integer arithmetic. Subtracting a negative number effectively moves us in the positive direction on the number line. This concept is fundamental to understanding how negative numbers interact with subtraction.

This section focuses on subtracting the sum of a positive and a negative integer from another negative integer. This scenario further expands our understanding of integer arithmetic by introducing a combination of addition and subtraction involving both positive and negative numbers. The key here is to carefully follow the order of operations and pay close attention to the signs of the numbers. Let's break down the problem step by step. We have a positive integer, 20, a negative integer, -18, and another negative integer, -17. The task is to subtract the sum of 20 and -18 from -17. The first step is to find the sum of 20 and -18. Adding a negative number to a positive number is the same as subtracting the absolute value of the negative number from the positive number. So, 20 plus -18 is the same as 20 minus 18, which equals 2. The sum of 20 and -18 is 2. Now, we need to subtract this sum, 2, from -17. Subtracting a positive number from a negative number results in a more negative number. So, subtracting 2 from -17 is the same as adding -2 to -17. -17 plus -2 equals -19. Therefore, subtracting the sum of 20 and -18 from -17 results in -19. This problem reinforces the concept of adding integers with different signs and the effect of subtracting a positive number from a negative number. It highlights the importance of paying attention to the signs of the numbers and applying the correct operations to arrive at the correct answer. Understanding these nuances is crucial for building a strong foundation in integer arithmetic.

Now, let's explore the scenario where we subtract the sum of two negative integers from another negative integer. This problem provides another opportunity to reinforce the rules of integer arithmetic, particularly when dealing with negative numbers and subtraction. We'll break down the problem into manageable steps to ensure clarity and accuracy. The problem involves two negative integers, -8 and -11, and another negative integer, -18. The objective is to subtract the sum of -8 and -11 from -18. The first step is to find the sum of -8 and -11. As we discussed earlier, adding two negative numbers involves adding their absolute values and then applying the negative sign. The absolute value of -8 is 8, and the absolute value of -11 is 11. 8 plus 11 equals 19. So, the sum of -8 and -11 is -19. Now, we need to subtract this sum, -19, from -18. Subtracting a negative number is the same as adding its positive counterpart. So, subtracting -19 from -18 is the same as adding 19 to -18. -18 plus 19 equals 1. Therefore, subtracting the sum of -8 and -11 from -18 results in 1. This problem demonstrates the principle of subtracting a negative number, which is equivalent to addition. It further solidifies the understanding of how negative numbers interact with subtraction and addition. This is a critical concept for mastering integer arithmetic and solving more complex mathematical problems.

Finally, we will address a problem that involves finding an unknown integer given its sum with another integer. This problem shifts the focus from subtraction to a slightly different perspective of integer arithmetic. It requires us to think algebraically and apply our understanding of inverse operations. Let's dissect the problem. The problem states that the sum of two integers is 18. One of the integers is -13, and we need to find the other integer. Let's represent the unknown integer as 'x'. According to the problem, x plus -13 equals 18. To find 'x', we need to isolate it on one side of the equation. We can do this by performing the inverse operation of adding -13, which is adding 13, to both sides of the equation. So, x plus -13 plus 13 equals 18 plus 13. This simplifies to x equals 31. Therefore, the other integer is 31. This problem demonstrates the use of algebraic thinking in solving integer arithmetic problems. It highlights the importance of understanding inverse operations and how they can be used to solve for unknown values. This skill is fundamental to algebra and higher-level mathematics. Understanding how to manipulate equations and isolate variables is a crucial skill for any aspiring mathematician.

In conclusion, mastering integer arithmetic requires a solid understanding of the rules of addition and subtraction, particularly when dealing with negative numbers. By carefully following the order of operations and paying attention to the signs of the numbers, we can confidently solve a wide range of integer arithmetic problems. These foundational skills are essential for success in more advanced mathematical concepts.