Problem 48 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ 816-1140 $$

Step-by-Step Solution

Answer

Answer: -324

Step 1: Identify the numbers

Determine the two numbers that need to be subtracted: 816 and 1140.

Step 2: Set up the subtraction

Write down the subtraction, with the larger number (1140) on top of the smaller number (816) and subtract, making sure the digits are aligned: $$ \begin{array}{c@{\;}c@{}c@{}c} & 1 & 1 & 4 & 0 \\ - & & 8 & 1 & 6 \end{array} $$

Step 3: Subtract from right to left

Begin subtracting the digits, starting from the rightmost column (ones place). Since we cannot subtract 6 from 0 in the ones place, we need to borrow from the tens place. $$ \begin{array}{c@{\;}c@{}c@{}c} & \cancelto{0}{1} & \cancelto{13}{1} & 4 & 0 \\ - & & 8 & 1 & 6 \end{array} $$ Now, we subtract the numbers in the tens place. Since 8 is greater than our current value of 13, we need to borrow again, this time from the hundreds place $$ \begin{array}{c@{\;}c@{}c@{}c} & \cancelto{0}{0} & \cancelto{3}{13} & 4 & 0 \\ - & & 8 & 1 & 6 \end{array} $$

Step 4: Complete subtraction

Now that we have borrowed from the hundreds place, we can complete the subtraction: $$ \begin{array}{c@{\;}c@{}c@{}c} & 0 & 3 & 4 & 0 \\ - & & 8 & 1 & 6 \\ \cline{2-4} & 3 & 2 & 4 \end{array} $$

Step 5: Interpret the result

The answer we got was 324, which means that 816 subtracted from 1140 equals -324, as the subtraction was in reverse order. So, the final answer is: $$ 816 - 1140 = -324 $$

Key Concepts

Subtracting Negative Results

Subtraction in mathematics can sometimes lead to negative results. This often occurs when a smaller number is subtracted from a larger number. In our example, the subtraction of 816 from 1140, we naturally reach a negative outcome since 816 is less than 1140. To interpret this result, we recognize that subtracting a smaller number from a larger one reflects the difference in value between them, with the sign indicating which number is greater.

Writing out the equation gives us \( 816 - 1140 = -324 \). The negative sign in front of the 324 indicates that 816 is 324 units less than 1140. It's essential to understand that negative numbers represent a difference in direction or deficit in arithmetic, and when we are subtracting a larger number from a smaller one, we are effectively finding how much less the smaller number is. This is a fundamental concept when dealing with subtraction across the real numbers.

Borrowing in Subtraction

When subtracting two numbers with multiple digits, we sometimes face a scenario where the digit in the smaller number is larger than the corresponding digit in the larger number. When this happens, we must 'borrow' from the next higher place value in order to proceed with the subtraction. In our exercise, the subtraction \( 816 - 1140 \) requires borrowing because we can't subtract 6 from 0 in the unit's place.

Here's what happens during borrowing:

• Move to the left and reduce the next higher place value by 1.
• Increase the current place value by 10 (since we're working in base 10).
• Proceed with the subtraction as normal.

If you need to borrow from a zero, continue moving left until you find a non-zero digit to borrow from. Understanding the concept of borrowing is essential for performing subtraction operations accurately and is particularly important when dealing with large numbers.

Subtracting Large Numbers

Subtracting large numbers follows the same basic principles as subtracting smaller numbers, but it usually involves more steps and greater attention to detail. This is because there are more digits to work with, and the possibility of borrowing increases. In the exercise given, we subtract large numbers 816 and 1140.

Key tips for subtracting large numbers include:

• Write the numbers vertically and align the digits by place value.
• Begin subtracting from the rightmost digit (ones place) and advance to the left.
• Use borrowing wherever necessary, as discussed earlier.

After all borrowing and subtracting steps are carefully executed, you arrive at the final answer, which could potentially be a large negative number if the top number is smaller than the bottom number.

Arithmetic Operations

Subtraction is one of the four fundamental arithmetic operations, the others being addition, multiplication, and division. Each operation has specific rules and applications in mathematics. Subtraction, in particular, is used to determine the difference between numbers.

The process of subtraction can include:

• Simple subtraction where no borrowing is needed.
• Complex subtraction of large numbers or instances requiring borrowing.
• Subtraction that results in negative numbers, as seen when subtracting a larger number from a smaller one.

Understanding subtraction within the context of all arithmetic operations is vital as it not only develops math skills but also contributes to logical reasoning and problem-solving abilities in everyday life. Always ensure to work through the operations methodically and check the work to avoid any potential errors.