Problem 176 - Prealgebra

Question

In the following exercises, subtract and then check by adding. $$ 542-288 $$

Step-by-Step Solution

Answer

542 - 288 = 254. Checked by adding 254 + 288 = 542.

Step 1: Write the Subtraction

Set up the numbers into a subtraction format: \[ 542 - 288 \]

Step 2: Subtract the Units Place

Subtract the digits in the units place: \[ 2 - 8 \] Since 2 is less than 8, you need to borrow from the tens place.

Step 3: Borrow from Tens Place

Borrow 1 from the tens place (4 becomes 3) and add 10 to the units place (2 becomes 12). Now, subtract: \[ 12 - 8 = 4 \] Write 4 in the units place of the result.

Step 4: Subtract the Tens Place

Subtract the digits in the tens place: \[ 3 - 8 \] Borrow 1 from the hundreds place (5 becomes 4). Add 10 to the tens place (3 becomes 13). Now subtract: \[ 13 - 8 = 5 \] Write 5 in the tens place of the result.

Step 5: Subtract the Hundreds Place

Subtract the digits in the hundreds place: \[ 4 - 2 = 2 \] Write 2 in the hundreds place of the result.

Step 6: Combine the Results

Combine the results from each place value to get the final subtraction result: \[ 542 - 288 = 254 \]

Step 7: Check by Adding

To ensure the subtraction is correct, add the result (254) to the subtracted number (288). The sum should equal the original number (542): \[ 254 + 288 \] Add the units place: \[ 4 + 8 = 12 \] Write 2 and carry 1.

Step 8: Add the Tens Place

Add the digits in the tens place including the carryover: \[ 5 + 8 + 1 = 14 \] Write 4 and carry 1.

Step 9: Add the Hundreds Place

Add the digits in the hundreds place including the carryover: \[ 2 + 2 + 1 = 5 \] Write 5.

Step 10: Verify the Sum

The sum matches the original number: \[ 542 = 542 \]. The subtraction is correct.

Key Concepts

Basic Arithmetic

Basic arithmetic is the foundation of all math calculations. It involves addition, subtraction, multiplication, and division. When solving problems like subtraction with borrowing, understanding these basic operations makes everything easier.

• **Addition**: Combining two or more numbers to get a total. For example, 2 + 3 = 5.
• **Subtraction**: Taking one number away from another. For instance, 5 - 3 = 2.
• **Multiplication**: Repeated addition. For instance, 4 x 3 means adding 4 three times (4 + 4 + 4 = 12).
• **Division**: Splitting a number into equal parts. For instance, 8 ÷ 4 = 2.

In our exercise, we worked through the subtraction of 542 - 288 using these basic arithmetic principles. We especially focused on borrowing, which is an essential concept in subtraction when the top number is smaller than the bottom one in a particular place value. This basic understanding is crucial to solving more complex problems.

Checking Subtraction by Addition

After subtracting, it's important to check our work. One effective way to do this is by using addition. If we add the result of our subtraction problem to the subtracted number, the sum should equal the original number. This helps us verify if our subtraction was done correctly.
In the exercise:

• **Subtract 542 - 288**: The result is 254.
• **Check with addition**: Add 254 to 288.
• **Step by step**: 4 + 8 = 12 (write 2, carry 1); 5 + 8 + 1 = 14 (write 4, carry 1); 2 + 2 + 1 = 5.

The sum, 542, matches the original number, confirming that our subtraction is correct. Checking subtraction by addition helps catch errors and builds confidence in your calculations. Always remember: confirming your work helps you learn from mistakes and ensures accuracy.

Place Value Operations

Understanding place value is critical in arithmetic operations. It refers to the value of each digit in a number based on its position.
For example, in the number 542:

• **Hundreds place**: The '5' represents 500.
• **Tens place**: The '4' represents 40.
• **Units place**: The '2' represents 2.

Place value becomes especially important when borrowing in subtraction. In our exercise, when we subtracted:

• Units place: We borrowed 1 from the tens place to make 12 - 8 = 4.
• Tens place: We borrowed 1 from the hundreds place to make 13 - 8 = 5.
• Hundreds place: We then subtracted 4 - 2 = 2.

Borrowing adjusts the place values temporarily to make subtraction possible. Mastering place value operations makes borrowing and other arithmetic processes much simpler and more intuitive.