Problem 15 - Fundamentals of Mathematics

Question

Perform the following subtractions. $$ \begin{array}{r} 526 \\ -358 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result of the subtraction is 168.

Step 1: Align the Numbers

Write the numbers vertically, aligning them by their place values. The numbers are already aligned: 526 on top and 358 below it, with the subtraction sign in between.

Step 2: Subtract the Units Digit

Subtract the digits in the units place: 6 - 8. Since 6 is smaller than 8, we need to borrow 1 from the tens place. Reduce the 2 (tens place) to 1, and make the 6 a 16. Now, subtract: 16 - 8 = 8.

Step 3: Subtract the Tens Digit

Now subtract the tens digits: after borrowing, we have 1 - 5. Again, since 1 is smaller than 5, borrow 1 from the hundreds place. Reduce the 5 (hundreds place) to 4 and make the 1 a 11. Now subtract: 11 - 5 = 6.

Step 4: Subtract the Hundreds Digit

Subtract the digits in the hundreds place: 4 - 3 = 1. There's no need to borrow in this step.

Step 5: Compile the Result

Write down the results from each column to get the final answer. The numbers from each column are: hundreds – 1, tens – 6, and units – 8. Therefore, the result is 168.

Key Concepts

Borrowing in Subtraction

When performing subtraction, sometimes the top digit in a column is smaller than the bottom digit. In these cases, we use a technique called "borrowing". This is essential for obtaining correct results.

Understanding Borrowing

Borrowing involves taking one unit from the next left column. This is usually the closest larger place value. Here's how it works: - If you can't subtract because the top number is too small, you "borrow" from the next-digit-left column. - Subtract 1 from the digit of the borrowing column and add 10 to the current column.

Example Application

Consider trying to subtract 8 from 6: - 6 isn't enough, so borrow 1 from the tens column. - Change the 6 to 16, making subtraction possible. - Subtraction becomes 16 - 8 = 8. Pay attention to borrowing efficiently. Practice makes it easier to handle and understand!

Place Value Alignment

Aligning numbers correctly by their place values is crucial in subtraction. This proper alignment ensures that each digit only interacts with its corresponding place value digit from the second number.

The Basics of Alignment

- Write each place value under the other: units under units, tens under tens, hundreds under hundreds, etc. - This prevents calculation errors and confusion.

Significance of Alignment

Misalignment can lead to incorrect calculations, as you may subtract numbers from the wrong columns: - Always start lining numbers from the rightmost digit, the units. - Ensure each digit lines up with its corresponding place in the other number. Correct alignment is fundamental to ensuring subtraction accuracy. Practice aligning numbers directly and maintaining their positions!

Step-by-Step Subtraction

Subtraction can be broken down into manageable steps, making complex equations simpler.

Breaking Down the Steps

1. **Align the numbers:** Ensure they are placed vertically, which aligns their place values. 2. **Subtract the units column:** If borrowing is necessary, apply it to make subtraction feasible. 3. **Move to the tens column:** Repeat the borrowing if needed as subtraction proceeds. 4. **Finish with the hundreds column:** Draw from previous experience where less borrowing might be needed.

Compile the Result

Once all place values are calculated, combine the values for your final answer: - In our example, the hundreds, tens, and units values are 1, 6, and 8, respectively. - This makes the final answer of 168. Following these structured steps can significantly simplify the subtraction process, providing a solid foundation for solving even more complex problems.