Problem 6 - Fundamentals of Mathematics

Question

Perform the following subtractions. $$ \begin{array}{r} 95,628 \\ -34,510 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result of the subtraction is 61,118.

Step 1: Align the Numbers

Write the numbers in a column, making sure that each digit is aligned according to its place value (ones under ones, tens under tens, etc.). \[\begin{array}{r} 95,628\- 34,510\\hline\end{array}\]

Step 2: Subtract the Units Place

Start by subtracting the digits in the ones place: 8 - 0 = 8. Write 8 below the line in the ones place.

Step 3: Subtract the Tens Place

Move to the tens place. Subtract 1 from 2: 2 - 1 = 1. Write 1 below the line in the tens place.

Step 4: Subtract the Hundreds Place

Next, subtract the hundreds place. Subtract 5 from 6: 6 - 5 = 1. Write 1 below the line in the hundreds place.

Step 5: Subtract the Thousands Place

Subtract the digits in the thousands place. Subtract 4 from 5: 5 - 4 = 1. Write 1 below the line in the thousands place.

Step 6: Subtract the Ten Thousands Place

Finally, subtract the digits in the ten thousands place. Subtract 3 from 9: 9 - 3 = 6. Write 6 below the line in the ten thousands place.

Step 7: Present the Final Result

Combine all the subtracted digits together to get the final result of the subtraction:\[\begin{array}{r} 95,628\- 34,510\\hline 61,118\end{array}\]

Key Concepts

Place Value

When performing subtraction, understanding place value is essential for accurate calculations. Each digit in a number has a different value depending on its position. The value increments by ten times as you move left in the number:

• Ones
• Tens
• Hundreds
• Thousands
• Ten thousands

For example, in the number 95,628, the digit 9 is in the ten thousands place, representing 90,000, whereas 5 represents 5,000 in the thousands place. It's crucial to align digits correctly according to their place values when performing any arithmetic operations. This ensures that each digit is properly aligned with its counterpart in the other number, making calculations straightforward.

Column Subtraction

Column subtraction is a method of solving subtraction problems with clarity and ease. It involves writing numbers vertically, one beneath the other, ensuring digits are aligned in columns. Begin with the larger number on top, and place the smaller number directly beneath it, aligning all digits according to place value. Here’s how we proceed:

• Start from the rightmost column, which is the ones place, and work leftwards.
• Subtract each digit of the bottom number from the corresponding digit of the top number.
• If a digit in the top number is smaller than the corresponding bottom digit, borrow 10 from the next left column.

This method is intuitive, visually clear, and effective for manual calculations, especially for younger learners. It reinforces the concept of borrowing, making it an excellent way to double-check your work for accuracy.

Multi-Digit Subtraction

Multi-digit subtraction takes the process of subtraction up a notch by involving larger, more complex numbers. Here's a simplified breakdown:

• Ensure that each digit has its place correctly aligned before starting the subtraction.
• Begin subtracting from the ones place and move consecutively through to the highest place value present, such as the ten thousands in our example.
• Utilize borrowing where necessary, particularly when a digit in the minuend (top number) is less than the digit below it.

For instance, consider subtracting 34,510 from 95,628. Follow the subtraction column-wise:

• Ones: 8 - 0 = 8
• Tens: 2 - 1 = 1
• Hundreds: 6 - 5 = 1
• Thousands: 5 - 4 = 1
• Ten Thousands: 9 - 3 = 6

The clarity and precision in each step allow errors to be caught easily and corrections made swiftly, teaching students the importance of careful calculations.