Problem 32 - Fundamentals of Mathematics

Question

For the following 18 problems, perform each subtraction. Use a calcula tor to cherk each result. $$ 67-38 $$

Step-by-Step Solution

Answer

The subtraction of 67 and 38 is 29.

Step 1: Set Up the Problem

Write the subtraction problem vertically, aligning the numbers by their place values: 67 and 38. This will help us perform the subtraction operation easily.

Step 2: Perform the Subtraction

Start by subtracting the digits in the units place. Subtract 8 from 7. Since 8 is larger than 7, we borrow 1 from the tens digit of 67 (making it 6), converting 7 to 17. Subtract 8 from 17 to get 9. Then in the tens place, subtract 3 from 6, which gives 3.

Step 3: Check with a Calculator

Enter the numbers into a calculator: 67 - 38, and verify that the result is 29. This confirms our subtraction is correct.

Key Concepts

Place Value Alignment

When we subtract numbers, it is crucial to align them correctly by their place values. This means arranging numbers in columns according to their units, tens, hundreds, and so on. For instance, in the problem \(67 - 38\), we need to line up the digits such that their corresponding places are directly above or below each other.
This step ensures each digit is in the right place.

• Units Place: Align the '7' and '8' under the units' column.
• Tens Place: Align the '6' and '3' under the tens column.

Starting the subtraction process becomes straightforward once the numbers are aligned. Without proper alignment, the calculation can become messy and it's easy to make mistakes. By keeping the numbers in their correct columns, we can focus on one place value at a time, leading to more accurate results.

Borrowing in Subtraction

Borrowing is a technique used in subtraction when a digit in the minuend (the number being subtracted from) is smaller than the digit in the same column of the subtrahend (the number to be subtracted). This involves borrowing '1' from the next higher place value. For the problem \(67 - 38\), let's break down the steps:- **Subtract Units Place:** Attempt to subtract \(8\) from \(7\). Here, \(7\) is smaller than \(8\). - **Borrow from Tens Place:** We borrow \(1\) from the '6' in the tens column, making it a '5'. This adds \(10\) to the '7', transforming it into '17'. - **Perform the Subtraction:** Now, subtract \(8\) from \(17\) to get \(9\).Next, move to the tens place. Subtract \(3\) from the adjusted tens position, \(5\), which results in \(2\). Borrowing helps when dealing with situations where subtraction within a place value column isn't straightforward. Recognizing when and how to borrow is a fundamental aspect of performing accurate subtraction.

Checking Subtraction with a Calculator

After manually performing subtraction, it's a good idea to confirm your results with a calculator. This helps ensure that there are no errors in your manual calculation. Here's how you can use a calculator for verification:

• Enter the minuend: Type in the first number, in this case, \(67\).
• Subtract the subtrahend: Press the minus sign, then enter \(38\).
• Check the result: Complete the operation to verify that the result is \(29\).

Using a calculator helps by providing a quick and reliable check against manual errors. While manual calculations are important for learning and understanding math concepts, double-checking ensures accuracy, especially in more complex problems or when you're just starting out with subtraction skills. This method builds confidence in your computational abilities.