Problem 40 - Fundamentals of Mathematics

Question

For the following problems, perform the subtractions. You may chedk each difference with a calculator. $$ \begin{array}{r} 77,893 \\ -\quad 421 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

77,472

Step 1: Set Up the Operation

Write the numbers with the larger number on top and aligned by place value. Ensure the digits of 77,893 are written above the corresponding digits of 421 with proper alignment for thousands, hundreds, tens, and units.

Step 2: Subtract the Units Column

Start with the units column on the right. Subtract 1 from 3 to get 2. Write 2 under the line in the units column.

Step 3: Subtract the Tens Column

In the tens column, subtract 2 from 9, which gives you 7. Write 7 under the line in the tens column.

Step 4: Subtract the Hundreds Column

Move to the hundreds column and subtract 4 from 8. Place the result, which is 4, under the line in the hundreds column.

Step 5: Handle the Thousands Column

Since there is a 0 in the thousands column of 421, subtract 0 from 7 in the thousands column of 77,893, resulting in 7. Place 7 under the line in this column.

Step 6: Result Verification

The result, written under the horizontal line, is 77,472. Optionally, use a calculator to verify the subtraction accuracy.

Key Concepts

Place Value

Place value is a fundamental concept in arithmetic, particularly important when performing operations like subtraction. Each digit in a number has a value depending on its position. For example, in the number 77,893:

• The digit 7 in the ten-thousands place represents 70,000.
• The next digit 7 in the thousands place represents 7,000.
• The 8 in the hundreds place represents 800.
• The 9 in the tens place represents 90.
• The 3 in the units place represents 3.

Understanding place value helps us align numbers correctly to make sure we are subtracting each column accurately. Each column corresponds to a particular place value. Aligning numbers by place value ensures digits are subtracted from their corresponding positions, which simplifies the operation.

Borrowing In Subtraction

Borrowing in subtraction is a method used when the top digit is smaller than the bottom digit in a column. Let's say we're subtracting the numbers 677 and 489. In the units column, 7 is larger than 9, so we need to borrow from the tens column to perform this operation.

• Borrow from the tens column. In 677, the tens digit is 7, but we make it 6 and add 10 to the units place.
• Now we have 17 in the units place, from which we can subtract 9.
• We do the same in the tens column if needed, borrowing from the next higher place value.

This borrowing process continues across the columns to ensure that you can subtract each digit. It's about redistributing the place values to make the subtraction possible rather than changing the numbers themselves.

Arithmetic Operations

Arithmetic operations are the basic operations of mathematics, which include addition, subtraction, multiplication, and division. Subtraction, the focus of this exercise, is essentially about finding the difference between numbers. It is one of the first operations taught in mathematics and forms a foundation for understanding more complex concepts.
When subtracting:

• Ensure that the numbers are properly aligned by their place value, starting from the right-most digit.
• Subtract each column starting from the units and moving left to the higher place values, borrowing if necessary.
• This operation reduces the value of the minuend by the value of the subtrahend.

Subtraction is applied frequently in daily life, from simple tasks like budgeting money to more complex calculations in various fields. Mastering subtraction through exercises develops problem-solving skills that are essential for advanced math learning.