Problem 31 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ 117+(-79) $$

Step-by-Step Solution

Answer

117 + (-79) equals 38.

Step 1: Identify the Problem

The operation we need to perform is adding two numbers: 117 and -79.

Step 2: Conceptual Understanding

Understand that adding a negative number is the same as subtracting its absolute value from the other number. Thus, the expression 117 + (-79) can be rewritten as 117 - 79.

Step 3: Perform the Subtraction

Now, subtract 79 from 117. Start by subtracting the ones: 7 - 9, which we can't do easily, so we borrow 1 from the tens column. The 7 becomes 17. Then, 17 - 9 = 8. Now, handle the tens column: 1 (which became 0 after borrowing) minus 7, which is not possible again, so borrow 1 from the hundreds column. The hundreds column goes from 1 to 0. Then in the tens' column we have 10 - 7 = 3. So, after accounting for the borrowed amount, we have 38.

Step 4: Finalize the Answer

The result after performing the subtraction is 38. Therefore, 117 + (-79) equals 38.

Key Concepts

Conceptual Understanding of Addition

When we talk about adding two numbers, it's important to grasp the underlying concept. Addition means combining two quantities to achieve a total. However, things get interesting with negative numbers, which brings us to the idea of adding a negative number. Adding a negative number is essentially the same as subtracting its absolute value. This is because a negative number represents a decrease or removal. For example, instead of thinking about 117 + (-79), we can reframe it as 117 - 79. This reframing helps because subtraction is often more intuitive in the context of removing or decreasing a value. To sum up, understanding this fundamental principle makes dealing with integer addition involving negative numbers more intuitive and less intimidating.

Subtracting with Borrowing

When subtracting two numbers and the situation arises where the first number's digit is smaller than the second's, borrowing is an essential technique. In our example, when subtracting 9 from 7, direct subtraction isn't possible. Here's the breakdown:

• Since 7 is less than 9, we borrow 1 from the tens column.
• This turns the 7 into 17, allowing us to subtract 9 effectively, giving us 8.
• Then, since we borrowed, the tens column now requires another look; adjust the 1, turning it to 0.
• When subtracting in the tens column, we need to borrow again from the hundreds column.
• Finally, after resolving all columns, we'd have a remainder of 38.

Using borrowing ensures accuracy in subtraction, confirming that even when direct subtraction isn't possible, the process is still achievable with simple adjustments.

Negative Numbers

Negative numbers play a unique role in math. They represent values below zero, often signifying a loss or debt in real-life contexts. Understanding them is key to solving many mathematical problems, especially in addition and subtraction. Think of negative numbers as steps backward. In our problem, -79 means moving 79 steps backward from 117. Instead, we handle it as 117 - 79, making it simpler through the lens of subtraction. Key points about negative numbers include:

• They invert the direction of operations when added, acting as subtraction.
• When subtracted, they do the opposite and add the absolute value back in.
• Negative numbers are depicted with a minus symbol (-), crucial for distinguishing them from positive numbers.

Grasping the nature of negative numbers removes the fear of complex operations and allows for smoother calculation handling.