Problem 48 - Fundamentals of Mathematics

Question

For the following problems, perform the subtractions. You may chedk each difference with a calculator. $$ \begin{array}{r} 42,041 \\ -15,355 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

26,686

Step 1: Align the Numbers

Make sure the numbers 42,041 and 15,355 are aligned properly, with each digit in a column corresponding to the same place value from right to left: units, tens, hundreds, thousands, tens of thousands.

Step 2: Subtract Each Column

Start subtracting from the rightmost column (the units column). If a digit in the minuend (top number) is smaller than the corresponding digit in the subtrahend (bottom number), borrow from the next column on the left.

Step 3: Units Column

Subtract the ones digits: 1 (from 42,041) - 5 (from 15,355). Since 1 is less than 5, borrow 10 from the tens column, making it 11. So, 11 - 5 = 6.

Step 4: Tens Column

Subtract the tens digits: 3 (from next column after borrowing) - 5. Since 3 is less than 5, borrow from the hundreds column, making it 13. So, 13 - 5 = 8.

Step 5: Hundreds Column

Subtract the hundreds digits: 0 (after borrowing) - 3. Since 0 is less than 3, borrow from the thousands column, making it 10. So, 10 - 3 = 7.

Step 6: Thousands Column

Subtract the thousands digits: 1 (from next column after borrowing) - 5. Again, 1 is less than 5, borrow from the ten thousands column, making it 11. So, 11 - 5 = 6.

Step 7: Ten Thousands Column

Subtract the ten thousands digits: 3 (after borrowing) - 1 = 2.

Step 8: Compile the Result

Combine each of the digits' results from right to left to get the final answer: 26,686.

Key Concepts

Borrowing in Subtraction

In subtraction, there are instances when the digit being subtracted (the subtrahend) is larger than the digit from which it is being subtracted (the minuend). This requires a process called "borrowing". Borrowing is crucial when the top digit in a column is smaller than the bottom digit and involves transferring value from the next left column.
For example, in the subtraction problem 42,041 - 15,355, while subtracting the units column (1-5), we notice that 1 is smaller than 5. So, we borrow 1 set of 10 from the tens column, making the 1 become 11. It's essential to understand that borrowing reduces the number in the column from which you borrow by 1.
Remember that borrowing affects not just individual digits but entire adjacent columns, which requires careful adjustment of each column's digit.

Place Value

Understanding place value is fundamental in subtraction, especially for multi-digit numbers. Each digit in a number has a specific place value; for instance, in the number 42,041:

• The rightmost digit (1) is in the units place.
• The next is in the tens place.
• The following is in the hundreds place.
• Then, the thousands place.
• The leftmost digit is in the ten thousands place.

Grasping place value ensures that digits are appropriately aligned by their magnitudes during subtraction. Each column corresponds to a place value, which helps in understanding when to borrow and adjust during the process.

Column Subtraction

Column subtraction is a method where you subtract numbers positively by their place value columns, rightmost to leftmost. It ensures accuracy and organization in subtraction tasks. Let's revisit our example:
When subtracting 42,041 from 15,355, each digit should be placed directly above or below its corresponding place value.

• Start with the units column: adjust if borrowing is necessary.
• Move on to the tens column: again, borrow if the top digit is smaller.
• Continue this process through hundreds, thousands, and ten thousands columns.

Column subtraction is practical for ensuring no confusion arises between different digit magnitudes and borrowed values.

Checking Subtraction with a Calculator

After solving a subtraction problem manually, it's always wise to verify the result using a calculator. This provides a quick way to confirm the accuracy of manual work.

• Input the original numbers into a calculator.
• Press the subtract function key ("-") located on most calculators.
• Compare the calculated result with your manual answer.

In our example, enter 42,041 and subtract 15,355 using the calculator. You should confirm that the result is 26,686, thus verifying the manual subtraction process was accurate. The calculator acts as a reliable double-check, ensuring your borrowing and place value adjustments were correctly applied.