Problem 46 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ 67-38 $$

Step-by-Step Solution

Answer

Answer: The result of subtracting 38 from 67 is 29.

Step 1: Write Down the Numbers

Firstly, write the numbers vertically, aligning each digit to its respective place value (ones and tens): ``` 67 - 38 ```

Step 2: Subtract the Ones Place

Subtract the digits in the ones place, proceeding from right to left: ``` 67 - 38 ------ 9 ``` Since 7 (ones place in 67) is greater than 3 (ones place in 38), subtract directly: 7 - 3 = 4.

Step 3: Subtract the Tens Place

Next, subtract the digits in the tens place: ``` 67 - 38 ------ 29 ``` Since 6 (tens place in 67) is greater than 3 (tens place in 38), subtract directly: 6 - 3 = 3. So, the final answer is 29.

Key Concepts

Understanding Arithmetic Operations

Arithmetic operations are the basic building blocks of mathematics. They include addition, subtraction, multiplication, and division. In this exercise, the focus is on subtraction. Subtraction is an operation that involves taking one number away from another. It helps us determine the difference between numbers. When we perform subtraction, we often use it to measure how much one value is less than another. To execute subtraction correctly, it's important to follow a few steps:

• Line up the numbers vertically according to their place values.
• Start subtracting from the rightmost digits, usually referred to as the ones place.
• Proceed to the next left place value (like tens, hundreds, etc.), subtracting each following the ones.

Understanding these steps makes subtraction easier and ensures accuracy. Always begin with aligned numbers to prevent mistakes. This method helps maintain clarity, especially with larger digits.

Decoding Place Value

Place value is crucial in understanding numbers and performing arithmetic operations. Each digit in a number has a position or 'place,' and each place has a value. For instance, in the number 67, 7 is in the ones place, and 6 is in the tens place. Recognizing place value is essential because it determines the value of each digit depending on where it is.

• Ones place: The rightmost digit represents units.
• Tens place: The second digit from the right represents groups of ten.

When subtracting, place value helps us subtract corresponding digits correctly. Aligning digits according to their place value ensures that we handle each part of the number map properly. It prevents errors in subtraction, especially in cases where one requires borrowing.

The Concept of Borrowing in Subtraction

Borrowing in subtraction is a technique used when the digit in the minuend is smaller than the corresponding digit in the subtrahend. In such cases, we cannot directly subtract; we need to "borrow" from the next higher place value. For example, in the number 67 and 38, if we had a situation where we needed to subtract 8 from 7, borrowing would be required:

• Find the nearest higher place value from which to borrow. In this case, it would be the tens place.
• Reduce the tens digit by one (in 67, changing 6 to 5).
• Add ten to the ones place digit (increasing 7 to 17, making the subtraction possible).

Understanding borrowing is vital for accurately performing subtraction with larger numbers. It ensures that even when direct subtraction is not possible, we can adjust numbers systematically to carry out the operation.