Integer Operations Simplifying Expressions And Solving Problems

In the realm of mathematics, integers play a crucial role in various calculations and problem-solving scenarios. Understanding how to perform operations with integers, such as addition, subtraction, and applying them in real-world contexts, is fundamental for mathematical proficiency. This article delves into three distinct problems involving integers, providing step-by-step solutions and explanations to enhance your comprehension. Let's embark on this mathematical journey to conquer integer operations!

Simplifying Integer Expressions: 73 + (-80) + (-20)

When simplifying integer expressions, the key is to understand the rules of addition and subtraction with negative numbers. In this particular problem, we are tasked with simplifying the expression: 73 + (-80) + (-20). To effectively tackle this, we'll break it down step-by-step, emphasizing the importance of sign conventions and the order of operations.

First, let's address the addition of 73 and -80. Adding a negative number is the same as subtracting the positive counterpart. Therefore, 73 + (-80) can be rewritten as 73 - 80. The result of this subtraction is -7. This is because we are subtracting a larger number (80) from a smaller number (73), which results in a negative difference. It's crucial to remember this rule when dealing with integers of different signs.

Now, we have -7 + (-20). Again, adding a negative number is equivalent to subtraction. So, we can rewrite this as -7 - 20. When subtracting a number from a negative number, we move further into the negative direction on the number line. In this case, we are moving 20 units to the left of -7. This gives us a result of -27. It's essential to visualize this on a number line to grasp the concept fully. Imagine starting at -7 and then moving 20 steps to the left; you will land on -27.

Therefore, the simplified form of the expression 73 + (-80) + (-20) is -27. This problem highlights the importance of understanding how to handle negative numbers in addition and subtraction. By breaking down the problem into smaller steps and applying the rules of integer operations, we arrive at the correct solution. This approach not only helps in solving similar problems but also builds a strong foundation for more complex mathematical concepts. Remember, practice is key to mastering these operations. The more you work with integers, the more intuitive these rules will become.

Subtracting Integers From a Sum A Detailed Explanation

The second problem we'll address involves both addition and subtraction of integers. The question asks us to subtract (-35) from the sum of (-28) and 53. This problem not only tests our ability to add and subtract integers but also our understanding of the order of operations. To solve this effectively, we'll break it down into two main parts: first, finding the sum of (-28) and 53, and second, subtracting (-35) from that sum.

Let's start by finding the sum of (-28) and 53. When adding integers with different signs, we essentially find the difference between their absolute values and use the sign of the number with the larger absolute value. In this case, we are adding a negative number (-28) and a positive number (53). The absolute value of -28 is 28, and the absolute value of 53 is 53. The difference between 53 and 28 is 25. Since 53 has a larger absolute value and is positive, the sum of -28 and 53 is positive 25. This step is crucial as it sets the stage for the next operation.

Now that we have the sum, 25, we need to subtract (-35) from it. Subtracting a negative number is the same as adding its positive counterpart. This is a fundamental rule in integer operations. So, 25 - (-35) becomes 25 + 35. This transformation is key to solving the problem correctly. Many students make mistakes by not properly handling the double negative. By understanding that subtracting a negative is equivalent to adding a positive, we simplify the problem significantly.

Adding 25 and 35 is a straightforward arithmetic operation. The sum of 25 and 35 is 60. Therefore, the final answer to the problem is 60. This problem underscores the importance of paying close attention to signs and understanding the rules of integer operations. It also highlights how a seemingly complex problem can be simplified by breaking it down into smaller, more manageable steps. By mastering these basic operations, you'll be well-prepared to tackle more advanced mathematical challenges.

Real-World Application of Integer Operations Temperature Changes

Our third problem presents a real-world scenario involving temperature changes, which provides a practical application of integer operations. The problem describes a freezing process where the room temperature needs to be lowered from 40°C at a rate of 5°C every hour. We are asked to determine the room temperature 10 hours after the process begins. This problem not only tests our understanding of integer operations but also our ability to apply mathematical concepts to real-life situations.

To solve this, we need to calculate the total temperature decrease over the 10-hour period. The temperature decreases at a rate of 5°C per hour. Therefore, over 10 hours, the total decrease in temperature will be 5°C/hour multiplied by 10 hours. This calculation gives us a total decrease of 50°C. It's important to recognize that this decrease is represented by a negative value since the temperature is going down. This is a crucial step in setting up the problem correctly.

Now, we need to subtract this total temperature decrease from the initial temperature. The initial room temperature is 40°C. We are subtracting a decrease of 50°C, which can be represented as subtracting the integer 50 from 40. So, the operation we need to perform is 40 - 50. This is a simple subtraction of integers with different signs. When subtracting a larger number from a smaller number, the result will be negative.

Performing the subtraction, 40 - 50, gives us -10. This means that the room temperature after 10 hours will be -10°C. The negative sign indicates that the temperature is below zero, which is consistent with the freezing process described in the problem. This final answer provides a clear and concise solution to the problem. It demonstrates how integer operations can be used to model and solve real-world problems involving changes in temperature.

This problem highlights the importance of understanding how integers are used to represent quantities that can increase or decrease. By applying integer operations to this real-world scenario, we can effectively calculate and predict temperature changes. It’s a powerful illustration of how mathematical concepts are not just abstract ideas but are tools we can use to understand and interact with the world around us.

In conclusion, mastering integer operations is essential for success in mathematics and beyond. By understanding the rules of addition, subtraction, and applying them in various contexts, we can solve a wide range of problems. The three problems we've explored in this article provide a solid foundation for working with integers. Remember to break down complex problems into smaller steps, pay close attention to signs, and practice regularly to build your skills. With dedication and a clear understanding of the concepts, you'll be well-equipped to tackle any integer-related challenge!