Problem 69 - Fundamentals of Mathematics

Question

Find the difference between 7,214 and 2,049 .

Step-by-Step Solution

Answer

The difference is 5,165.

Step 1: Set up the subtraction

Write down the numbers one below the other, making sure to line up the digits by place value. Like this:\[ \begin{array}{c} \,7214 \- \,2049\end{array} \]

Step 2: Subtract the units place

Subtract the digits in the rightmost column, the units place: \(4 - 9\). Because 4 is smaller than 9, you need to borrow 1 from the tens place. This makes the tens digit in 7,214 reduce by 1, turning 1 into 0, and turns 4 into 14. Now, subtract: \(14 - 9 = 5\).

Step 3: Subtract the tens place

Move to the tens place column and subtract: \(0 - 4\). Since you can't subtract 4 from 0, borrow 1 from the hundreds place. Make the hundreds digit in 7,214 reduce by 1, turning 2 into 1. Now, the tens digit becomes 10. Subtract: \(10 - 4 = 6\).

Step 4: Subtract the hundreds place

With the hundreds place column, subtract: \(1 - 0 = 1\).

Step 5: Subtract the thousands place

Finally, in the thousands place column, subtract: \(7 - 2 = 5\).

Step 6: Write the result

Write down the numbers calculated from each column. The difference between 7,214 and 2,049 is 5,165. The completed subtraction looks like this:\[ \begin{array}{c}\, 7214\\underline{-2049} \\, 5165\end{array} \]

Key Concepts

Borrowing in Subtraction

Borrowing in subtraction, often called "regrouping," is an essential technique used to manage subtraction problems where the top digit is smaller than the bottom digit within the same column. This process occurs frequently in multi-digit subtraction problems. Here's a simplified breakdown of how and why we borrow:

• When subtracting digits, if the upper digit is smaller than the lower digit, borrowing becomes necessary.
• You "borrow" from the next left column to make the current column's top number larger, allowing the subtraction to occur.
• For example, in subtracting 4 from 9, you take 1 from the tens column, making the tens column decrease by one and turning the current column's top number into 14 instead of 4.

After borrowing, the subtraction can carry on smoothly. Practicing borrowing is key to becoming proficient in two-digit or greater subtraction problems. Remember, borrowing can happen in any place value column, moving neatly from right to left.

Place Value

Place value is a crucial concept in mathematics that helps you understand the position and value of each digit in a number.

• The rightmost digit in a number is the 'units' or 'ones' place.
• The next digit to the left is the 'tens' place, then 'hundreds,' and so on.
• The value of a digit increases tenfold as you move one place to the left.

In subtraction, aligning numbers by their place values is vital for accuracy. This ensures that you subtract hundreds from hundreds, tens from tens, and so forth. As seen in the example given:
- First line up 4 with 9, 1 with 4, 2 with 0, and 7 with 2.
This alignment allows for proper borrowing and accurate subtraction. Without placing numbers correctly, borrowing might be incorrectly executed or skipped.

Arithmetic Operations

Arithmetic operations are the building blocks of mathematics, with subtraction being one of the four essential operations.

• Addition brings numbers together, making sums larger.
• Subtraction, in contrast, reduces numbers, finding their difference.
• Multiplication focuses on adding groups of numbers, and division on distributing a number into equal parts.

Subtraction requires careful attention to the numbers' order and their placement. This attention ensures each digit is correctly subtracted from its corresponding digit. Here’s an example from our solution:
- First, ensure the numbers are lined up correctly.
- Then, subtract each column carefully, considering borrowing when necessary.
Understanding these operations is crucial for moving to more complex math problems, making it necessary to practice thoroughly. By valuing each operation, like subtraction, you lay a stronger foundation for tackling more challenging mathematical tasks in the future.