Problem 59 - Prealgebra

Question

Perform the indicated operations. $$\begin{array}{r} 75 \\ -34 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The result of 75 - 34 is 41.

Step 1: Understand the Problem

The problem asks us to subtract two numbers: 75 and 34. This is a simple subtraction problem.

Step 2: Set Up the Subtraction

Write the numbers vertically with 75 above 34, ensuring the digits are correctly aligned according to place value (ones under ones, tens under tens).

Step 3: Subtract the Ones Column

Subtract the ones digits: 5 (from 75) minus 4 (from 34) equals 1. Write down the 1 in the ones column of the answer.

Step 4: Subtract the Tens Column

Subtract the tens digits: 7 (from 75) minus 3 (from 34) equals 4. Write down the 4 in the tens column of the answer.

Step 5: Write the Final Answer

Combine the results from the ones and tens columns to form the complete answer, which is 41.

Key Concepts

Place Value

In subtraction, the concept of place value plays a crucial role. Understanding place value means recognizing the position of each digit in a number. In the number 75, the digit '7' is in the tens place, which means it represents 70, while the '5' is in the ones place, representing just 5.

- **Tens Place**: This is the second digit from the right, and its value is ten times the digit itself. For instance, in 75, the '7' means seventy, not just seven.
- **Ones Place**: This is the rightmost digit. In 75, the '5' represents five ones, or simply 5.

By aligning the digits according to their place value, we ensure that we subtract tens from tens and ones from ones, maintaining the proper numerical order.

Vertical Subtraction

Vertical subtraction involves lining up numbers according to their place values and then subtracting corresponding digits from top to bottom. It is a straightforward approach once you see how it works.

1. **Aligning the Numbers**: Write the numbers vertically, ensuring that the ones digits are lined up, as well as the tens digits.
2. **Subtracting Step-by-Step**: Start from the rightmost column (the ones), move to the left column (tens), and subtract each pair of digits.

This method helps break down the subtraction process into manageable steps, making it easier to handle, especially for larger numbers. Vertical subtraction also minimizes errors by keeping related digits in clear view.

Borrowing in Subtraction

Sometimes, when subtracting in a vertical format, the top number in a column is smaller than the bottom number, requiring us to borrow from the next left column.

- **When to Borrow**: If you cannot subtract the bottom digit from the top digit in the same column, you need to borrow.
- **How to Borrow**: Move to the immediate left column, reduce its digit by 1, and add 10 to the current column. This effectively increases the top digit so you can complete the subtraction.

Borrowing is like a lifesaver for tough subtraction problems. It helps us adjust the numbers without changing the overall value or balance, ensuring accurate results.