Problem 31 - Prealgebra

Question

Subtract. $$65 .002-24.003$$

Step-by-Step Solution

Answer

The result of the subtraction is 40.999.

Step 1: Align Decimal Points

First, make sure to write the numbers vertically and neatly align the decimal points. This helps avoid mistakes with the decimal subtraction. Write down 65.002 and below it, write 24.003 ensuring the decimal points are one below the other.

Step 2: Subtract Digit by Digit

Subtract the digits in each column from right to left (rightmost digits to leftmost). Starting at the rightmost digit: - The last digit is 2 - 3. Since 2 is smaller than 3, borrow 1 from the next column. The 2 becomes 12. - Now, 12 - 3 = 9. - Move to the next left column, subtract the digits: 0 - 0 = 0. - Then, in the hundredths, it becomes 9 (borrowed) - 0 = 9. - Proceed to the tenths: 4 - 0 = 4. - Finally, subtract the remaining whole numbers separately from left to right: 65 - 24. This is split further into: - 4 - 4 = 0 (since we already borrowed from the column on the right earlier, so it's effectively 4). - And 5 - 2 = 3.

Step 3: Write the Result

After completing the subtraction in all columns, write down the remaining number putting the decimal point in the appropriate place. The entire number should read 40.999.

Key Concepts

Understanding Decimal Borrowing

Decimal borrowing may feel a bit confusing at first, but it's quite similar to borrowing in whole number subtraction. When you encounter a situation like subtracting 3 from 2 in decimals, and the top number lacks enough to subtract directly, borrowing is your friend.
Here's how it works:

• Look at the digit in the next left column. It's like asking your neighbor for sugar: "May I borrow 1?"
• In the case where you have a zero or another number that isn't enough, you continue the borrowing process until you reach a value that can lend 1.
• Decrease the lending digit by 1 and give that extra ten to your needy column.

With decimals, this borrowed '1' becomes '10' due to place value, like changing dimes into pennies. Now, you have enough to complete your subtraction smoothly.
For example, if you need to subtract 3 from 2, you borrow 1 from the next column, turning 2 into 12. Easy, right?

Place Value Alignment Crucial for Decimals

When subtracting decimals, ensuring that the numbers are correctly lined up according to their place value is essential. This means the decimal points of the numbers must align vertically.
Proper alignment simplifies the subtraction process because:

• The digits in the same columns represent the same place value (ones, tenths, hundredths, etc.).
• Misalignment could result in adding or subtracting values from the wrong place, leading to inaccurate results.
• Alignment automates the borrowing mechanism, making it visually clear where borrowing might be needed.

Always remember to carefully align your decimal points before starting your subtraction. This step, though small, makes a significant difference and ensures your work is error-free.
Think of it as ensuring your ducks are in a row before you begin the calculations!

The Simple Art of Subtracting Decimals

While subtracting decimals might seem daunting at first, following a clear process step-by-step makes it straightforward.
Here’s how you can simplify it:

• Align the decimal points as your starting line. This setup ensures you’re dealing with like values vertically.
• After ensuring proper alignment, start subtracting from the rightmost digit and move leftwards. This traditional approach helps manage borrowing effectively.
• If borrowing is necessary, seamlessly incorporate it into your process (as discussed before), creating a smooth transition as you move across columns.

Once you’ve completed calculating each column, don’t forget to place the decimal in its rightful position in your final answer. This seemingly tiny step ensures that everything makes sense from a value perspective.
And there you have it: a complete and accurate subtraction, with your decimals tidy and correctly handled!