Adding Integers: A Step-by-Step Guide To 145 + (-27)

Hey there, math enthusiasts! Today, we're going to break down a fundamental concept in mathematics: adding integers. Specifically, we'll be tackling the problem of $145 + (-27)$. Don't worry if you find this a bit tricky at first; we'll go through it step by step, ensuring you understand not just the how but also the why behind the solution. So, let's dive in and master the art of integer addition!

Understanding Integers and the Number Line

Before we jump into the calculation, let's quickly recap what integers are and how they relate to the number line. Integers are whole numbers (no fractions or decimals) that can be positive, negative, or zero. Think of them as the set of numbers: ..., -3, -2, -1, 0, 1, 2, 3, ...

The number line is a visual representation of integers, extending infinitely in both positive and negative directions. Zero sits in the middle, positive numbers stretch to the right, and negative numbers stretch to the left. This visual aid is incredibly helpful when we're adding and subtracting integers, as it allows us to see the movement and direction involved.

When we add a positive integer, we move to the right on the number line. For example, 145 is a positive integer, so starting at zero and adding 145 would take us 145 units to the right. Conversely, when we add a negative integer, we move to the left on the number line. This is crucial to understanding our problem, $145 + (-27)$. We're starting at 145 and then moving 27 units to the left because we're adding a negative number.

It’s also helpful to think about this in terms of gains and losses. A positive integer can represent a gain or an increase, while a negative integer represents a loss or a decrease. So, $145 + (-27)$ can be interpreted as starting with 145 and then experiencing a loss of 27. Visualizing the number line and thinking in terms of gains and losses are both powerful tools for grasping integer addition.

Breaking Down the Problem: $145 + (-27)$

Now that we have a solid understanding of integers and the number line, let's tackle our specific problem: $145 + (-27)$. There are a couple of ways to approach this, and we'll explore both to give you a well-rounded understanding.

Method 1: Visualizing on the Number Line

Imagine our number line. We start at 145, which is a positive number located far to the right of zero. Now, we're adding -27, which means we need to move 27 units to the left. Picture yourself hopping 27 steps backward from 145. Where do you end up? This visual representation helps to conceptualize the subtraction that's inherently happening when we add a negative number.

Method 2: Converting to Subtraction

Adding a negative number is the same as subtracting the positive version of that number. This is a fundamental rule of integer arithmetic. So, $145 + (-27)$ is equivalent to $145 - 27$. This conversion makes the problem more straightforward for many people, as subtraction often feels more intuitive than adding a negative.

Now we have a simple subtraction problem: $145 - 27$. Let's solve this using the standard subtraction method.

Method 3: Using a Hundreds Chart

For those who benefit from visual aids, a hundreds chart can be a useful tool. While it doesn't directly show negative numbers, it can help in breaking down the subtraction. Imagine you have a chart that goes beyond 100 to 145. You can then count back 27 spaces, which will lead you to the correct answer.

Step-by-Step Calculation of $145 - 27$

Let's perform the subtraction $145 - 27$ step by step:

1. Set up the problem: Write the numbers vertically, aligning the ones, tens, and hundreds places:

   145
   - 27
   -----
   

2. Subtract the ones place: We have 5 ones minus 7 ones. Since 5 is smaller than 7, we need to borrow from the tens place. The 4 in the tens place becomes a 3, and we add 10 to the ones place, making it 15. Now we have 15 - 7 = 8. Write 8 in the ones place of the answer:

   1 3 15
   1 4 5
   - 2 7
   -----
       8
   

3. Subtract the tens place: We now have 3 tens minus 2 tens, which is 3 - 2 = 1. Write 1 in the tens place of the answer:

   1 3 15
   1 4 5
   - 2 7
   -----
    1 8
   

4. Subtract the hundreds place: We have 1 hundred minus nothing (zero hundreds in 27), which is 1 - 0 = 1. Write 1 in the hundreds place of the answer:

   1 3 15
   1 4 5
   - 2 7
   -----
   1 1 8
   

Therefore, $145 - 27 = 118$.

The Solution: $145 + (-27) = 118$

We've arrived at our solution! By visualizing the number line, converting to subtraction, and performing the calculation step by step, we've found that $145 + (-27) = 118$. Remember, the key is to understand the concept of adding negative numbers – it's essentially the same as subtraction.

So, there you have it! We've successfully navigated the world of integer addition. The answer to the problem $145 + (-27)$ is 118. Whether you prefer the number line visualization or the subtraction conversion method, the important thing is to find the approach that clicks with you. Keep practicing, and you'll become a pro at integer addition in no time! If you guys ever get stuck, just remember these steps and you'll be golden. Keep up the awesome work, and remember math can be fun!

Real-World Applications of Integer Addition

Understanding integer addition isn't just about solving textbook problems; it's a crucial skill that applies to various real-world scenarios. Let's explore some examples to see how this concept comes into play in our daily lives.

1. Finance and Budgeting

In personal finance, integers are used to track income and expenses. Positive integers represent income (money coming in), while negative integers represent expenses (money going out). Adding integers helps you calculate your net balance. For example, if you earn $145 (positive integer) and spend $27 (negative integer), adding these integers ($145 + (-27)$) shows you have a net balance of $118. This concept is vital for creating budgets, managing debt, and understanding your financial health.

2. Temperature Measurement

Temperature is another area where integers are commonly used. Temperatures above zero are positive integers, while temperatures below zero are negative integers. Adding integers can help you calculate temperature changes. For instance, if the temperature starts at 145 degrees Fahrenheit and then drops by 27 degrees, the new temperature can be calculated as $145 + (-27) = 118$ degrees Fahrenheit. This is particularly useful in meteorology and understanding climate patterns.

3. Altitude and Elevation

Altitude, or elevation above sea level, is often represented using integers. Sea level is considered zero, heights above sea level are positive integers, and depths below sea level are negative integers. Adding integers can help determine the relative heights of different locations. If an airplane climbs from an altitude of 145 feet to a new altitude that is 27 feet lower than that, the calculation $145 + (-27) = 118$ feet would give the new altitude. This concept is essential in aviation, geography, and mapping.

4. Game Scoring

Many games use positive and negative integers to represent points. Gaining points is represented by positive integers, while losing points is represented by negative integers. Adding integers helps calculate a player's total score. Imagine a player starts with 145 points and then loses 27 points. Their new score would be $145 + (-27) = 118$ points. This applies to various games, from board games to video games.

5. Stock Market

The stock market uses integers to track price fluctuations. An increase in stock price is represented by a positive integer, while a decrease is represented by a negative integer. Adding integers helps investors calculate their gains and losses. If a stock's price starts at $145 and then drops by $27, the new price can be calculated as $145 + (-27) = $118. This is a fundamental concept for anyone involved in trading and investment.

6. Scientific Measurements

In scientific fields, integers are used to represent various measurements, such as changes in volume, mass, or electrical charge. Positive integers represent increases, while negative integers represent decreases. Adding integers is crucial for calculating net changes in experiments. For example, if a solution starts with a volume of 145 milliliters and then 27 milliliters are removed, the new volume would be $145 + (-27) = 118$ milliliters.

These real-world examples demonstrate that understanding integer addition is more than just an academic exercise. It's a practical skill that helps us make sense of the world around us. From managing our finances to understanding weather patterns and navigating the stock market, integers play a vital role. So, mastering this concept opens doors to a deeper understanding of various aspects of life. Keep exploring and you'll be amazed at how often you encounter integers in your daily experiences!

Common Mistakes to Avoid When Adding Integers

When working with integers, it's easy to make a few common mistakes, especially when dealing with negative numbers. Let's go over these pitfalls and how to steer clear of them. This way, you'll be able to tackle integer addition with confidence and accuracy.

1. Forgetting the Sign

The most frequent mistake is overlooking the negative sign. Remember, a negative sign changes the direction on the number line. Adding a negative number is the same as subtracting its positive counterpart. For example, $145 + (-27)$ is the same as $145 - 27$. If you forget the negative sign, you might incorrectly add 27 to 145 instead of subtracting it.

How to Avoid: Always pay close attention to the signs of the numbers you are adding. If you see a plus sign followed by a negative number, mentally convert it to a subtraction problem. Writing it down explicitly as a subtraction problem can also help.

2. Incorrectly Applying the Rules of Addition and Subtraction

Another common error is mixing up the rules for adding and subtracting integers. Adding two positive integers is straightforward, but when negative integers come into play, it can get confusing. Remember:

• Adding two positive integers: The result is positive. (e.g., $5 + 3 = 8$)
• Adding two negative integers: The result is negative, and you add the absolute values of the numbers. (e.g., $-5 + (-3) = -8$)
• Adding a positive and a negative integer: Subtract the smaller absolute value from the larger absolute value. The result takes the sign of the number with the larger absolute value. (e.g., $-5 + 3 = -2$, $5 + (-3) = 2$)

How to Avoid: Practice the rules of integer addition until they become second nature. Use the number line as a visual aid to reinforce the concept. Thinking in terms of gains and losses can also help clarify the process.

3. Not Borrowing Correctly in Subtraction

Since adding a negative number often turns into a subtraction problem, errors in subtraction can also lead to incorrect answers. One common mistake is not borrowing correctly when the digit in the ones place of the first number is smaller than the digit in the ones place of the second number. For example, in $145 - 27$, you need to borrow from the tens place to subtract 7 from 5.

How to Avoid: Review the borrowing process in subtraction. Write out the steps clearly, showing how you borrow from the next place value. Double-check your work to ensure you've borrowed correctly.

4. Rushing Through the Problem

Rushing can lead to careless errors, such as misreading numbers or skipping steps. It's important to take your time and work through the problem methodically, especially when dealing with multiple steps or negative numbers.

How to Avoid: Slow down and focus on each step. Break the problem down into smaller, manageable parts. Write out each step clearly and double-check your work as you go.

5. Not Using Visual Aids

Visual aids like the number line can be incredibly helpful when adding integers, especially when you're first learning the concept. Not using these tools can make it harder to visualize the process and avoid mistakes.

How to Avoid: Use the number line to visualize integer addition. Draw it out if necessary. Picture yourself moving along the number line as you add the integers. This will help solidify your understanding and reduce errors.

6. Failing to Check Your Answer

It's always a good idea to check your answer to ensure accuracy. This is especially important when working with integers, as small errors in sign or calculation can lead to incorrect results.

How to Avoid: After you've solved the problem, take a moment to check your answer. You can do this by working backward (e.g., adding 27 to 118 to see if you get 145) or by using a calculator to verify your result.

By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy when adding integers. Remember, practice makes perfect! Keep working at it, and you'll become a master of integer addition in no time.