Problem 11 - Fundamentals of Mathematics

Question

Perform the following subtractions. Show the expanded form for the first three problems. $$ \begin{array}{r} 872 \\ -565 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The answer is 307.

Step 1: Write the Numbers in Expanded Form

First, we need to break down the numbers 872 and 565 into their expanded forms. 872 can be written as 800 + 70 + 2, and 565 can be written as 500 + 60 + 5.

Step 2: Subtract the Hundreds

Subtract the hundreds from each number. That is 800 - 500 = 300.

Step 3: Subtract the Tens

Next, subtract the tens. We take 70 from 872 and 60 from 565. Perform the subtraction: 70 - 60 = 10.

Step 4: Subtract the Ones

Finally, subtract the ones. The numbers are 2 and 5. Since 2 is smaller than 5, we need to borrow 1 ten from the tens column, changing the 10 to a 0 and adding 10 to the 2, making it 12. Then subtract: 12 - 5 = 7.

Step 5: Add the Differences Together

Now, add up the differences obtained from each column: 300 + 10 + 7 = 317.

Key Concepts

Expanded Form

Expanded form is a way of breaking down numbers to show the value of each individual digit based on its position. For instance, let's explore this with the number 872.

• The "8" is in the hundreds place, so it represents 800.
• The "7" is in the tens place, representing 70.
• The "2" is in the ones place, showing 2.

Putting it all together, 872 can be expressed as 800 + 70 + 2. Likewise, for the number 565:

• "5" in the hundreds place equals 500.
• "6" in the tens place equals 60.
• "5" in the ones place equals 5.

Thus, 565 expands to 500 + 60 + 5. Expanded form helps us see and understand the value of each digit clearly, which becomes crucial when performing operations like subtraction.

Place Value

Place value tells us the position of a digit in a number and the value it holds. Each position represents a power of ten in our base-10 number system.

• Starting from the right, the first position is the ones place.
• Next is the tens place, followed by the hundreds place, and so on.

In the number 872:

• The "2" is in the ones place and represents 2.
• The "7" is in the tens place, representing 70 (7 times 10).
• The "8" is in the hundreds place, meaning 800 (8 times 100).

Place value is essential for understanding how numbers work in operations such as subtraction. It helps us to align digits correctly when subtracting, ensuring each value is accurately accounted for according to its position.

Borrowing in Subtraction

Borrowing in subtraction, sometimes called regrouping, is a crucial technique when a smaller digit needs to be subtracted from a larger one. Let's see this in action with our example involving 872 and 565:

• When subtracting ones, we encounter 2 - 5.
• Direct subtraction isn't possible as 2 is less than 5, so we borrow from the tens column.

By borrowing 10 from the "7" in 70 (thus making it 60), we add it to the 2. This makes 12, allowing us to perform the subtraction 12 - 5, which equals 7.

In brief:

• Borrowing means "reducing" a digit in the next left column by 1, while increasing the current column's number by 10.
• This only applies when subtracting a larger number from a smaller number in any given column.

Mastering borrowing is vital for effectively solving subtraction problems, particularly when dealing with numbers where the digits in the minuend (top number) are smaller than the corresponding digits in the subtrahend (bottom number).