Problem 39 - Fundamentals of Mathematics

Question

For the following problems, perform the subtractions. You may chedk each difference with a calculator. $$ \begin{array}{r} 46,400 \\ -\quad 2,012 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

44,388.

Step 1: Arrange the Numbers Vertically

Write down the numbers so that their digits are aligned by place value. \[\begin{array}{r} 46,400 \- \quad 2,012 \\hline\end{array}\] Start from the units digit on the right to the left.

Step 2: Subtract the Units Column

Subtract the units digits: 0 - 2. Since 0 is less than 2, you need to borrow 1 from the tens place. After borrowing, change the 0 to 10. So, 10 - 2 = 8.

Step 3: Subtract the Tens Column

After borrowing, the tens digit is changed from 0 to 9. So, subtract 1 from the rest of the existing number: \(9 - 1 = 8\).

Step 4: Subtract the Hundreds Column

Subtract the hundreds digits: \(4 - 0 = 4\). There is no need to borrow here.

Step 5: Subtract the Thousands Column

Subtract the thousands digits: \(6 - 2 = 4\). Again, no borrowing is necessary.

Step 6: Subtract the Ten Thousands Column

Subtract the ten thousands digits: \(4 - 0 = 4\). No borrowing is required.

Step 7: Compile the Answer

Put the digits together from each column to get the final answer: 44,388.

Key Concepts

Borrowing in Subtraction

When you're subtracting numbers, sometimes the number on top is smaller than the number on the bottom in a particular column. This requires borrowing, a crucial subtraction concept that can trip up students. To borrow means taking a value from the next higher place (like tens, hundreds) to make the subtraction possible. For instance, in subtracting 0 from 2, since 0 is less than 2, we borrow 1 from the tens place, making it 10.

• Start with the smaller digit.
• If it's less than the digit you're subtracting, borrow from the next column to the left.
• Adjust the column you're borrowing from by decreasing its value by 1.

After borrowing, you can then complete the subtraction by subtracting the numbers as they now appear. Borrowing makes it possible to handle cases where direct subtraction isn't possible due to smaller digits involved.

Place Value Alignment

Place value alignment is incredibly important when solving subtraction problems. Each digit has a position that defines its value, such as units, tens, hundreds, and so on. Ensuring these are properly aligned takes the complexity out of subtraction as it prevents errors like subtracting a hundreds digit from a tens digit. Here's how to align correctly:

• Write each number in a vertical column, corresponding each digit to its place value.
• The far-right column should always be the smallest place, usually units.
• Place one digit directly underneath its corresponding digit from the other number.

Proper place value alignment is foundational in multi-digit subtraction to ensure that each operation makes logical and mathematical sense.

Step-by-Step Subtraction

Step-by-step subtraction offers a clear path forward when solving subtraction problems. Breaking down each digit and solving column-by-column simplifies the process and reduces the chance of mistakes. Here's a typical sequence you can follow:

• Step 1: Align digits by place value.
• Step 2: Start with the units, borrow if necessary, and subtract.
• Step 3: Move to the tens, subtract, borrow if needed.
• Step 4: Continue to higher place values, ensuring correct borrowing.
• Step 5: Compile all digits into the final answer.

This systematic approach helps to organize thoughts and make each subtraction step clearer, ensuring that numbers do not get mixed up and the correct solution is reached.