Problem 26 - Fundamentals of Mathematics

Question

Find the sums and differences. $$ \begin{array}{r} 16,608 \\ -\quad 1,660 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

14,948

Step 1: Set up the subtraction

Write down the numbers such that the smaller number (1,660) is below the larger number (16,608), aligning them by their place values.

Step 2: Subtract the units place

Start from the rightmost digit (units place). Subtract the bottom digit from the top: 8 - 0 = 8.

Step 3: Subtract the tens place

Move to the tens place. Subtract the bottom digit from the top: 0 - 6. Since you can't subtract 6 from 0, borrow 1 from the hundreds place, making it 10 - 6 = 4.

Step 4: Borrow from the thousands place

Since you borrowed 1 in the previous step, the hundreds place becomes 5. Subtract the bottom digit from the top in the hundreds place: 5 - 6 = -1. Borrow 1 from the thousands place to make it 15 - 6 = 9.

Step 5: Subtract the thousands place

In the thousands place, after borrowing, you have 5 (original 6 minus 1 borrowed) and need to subtract 1: 5 - 1 = 4.

Step 6: Subtract the ten thousands place

Finally, subtract the remaining place. There's no borrowing here, so it's straightforward: 1 (ten thousands) - 0 = 1.

Step 7: Write down the result

Combine all the results from the subtraction steps: 16,608 - 1,660 = 14,948.

Key Concepts

Place Value

Understanding place value is vital when dealing with subtraction, especially with multi-digit numbers. Each digit in a number has a specific place value which determines its actual worth. For example, in the number 16,608, each digit is positioned in a different place:

• 1 is in the ten thousands place
• 6 is in the thousands place
• 6 is in the hundreds place
• 0 is in the tens place
• 8 is in the units place

Recognizing the place for each digit helps you line up numbers correctly when setting up for subtraction.
This alignment ensures you subtract digits assigned to the same place value, which is critical for keeping your calculations accurate. If numbers aren't aligned by place value, it becomes easy to make mistakes.

Borrowing in Subtraction

Borrowing, sometimes called 'regrouping', is a technique used in subtraction when a smaller number is subtracted from a larger number, but there isn't enough value in the top digit. For instance, when you try to subtract 6 from 0 in the tens place, you can’t simply take it away.
Instead, you borrow from the next column to the left, increasing the digit at your current column:

• Borrow 1 from the hundreds place (making it decrease by one) so the tens place now reads 10.
• The subtraction then becomes: 10 - 6, which equals 4.

It's pivotal to adjust the numbers properly as you go along, not just where you're borrowing from, but also remembering to decrease the digit from where you borrowed. This ensures the final result remains correct.

Arithmetic Operations

Subtraction is one of the core operations in arithmetic and involves taking one number away from another. In the equation given (16,608 - 1,660), the process involves solving each column from right to left. Begin from the units place first, moving sequentially to the left.
Unlike addition, subtraction sometimes involves borrowing, as explained earlier, which requires careful attention to detail:

• Understand whether your subtraction needs borrowing at each place value.
• Subtract digits within the same place value column.
• Write down your result as you progress through each subtraction step.

Mastering arithmetic operations like subtraction is critical because they form the foundation for solving more complex mathematical problems later on. Practicing these basic steps will improve efficiency and accuracy in calculations.