Problem 29 - Prealgebra

Question

Find each of the following differences. (Subtract.) $$45-0.067$$

Step-by-Step Solution

Answer

The difference is 44.933.

Step 1: Align the Numbers

Write the numbers vertically, aligning the decimal points. This helps ensure that each digit is in the correct place value when subtracting.\[\begin{array}{c}45.000 \-0.067 \\hline\end{array}\]

Step 2: Subtract the Hundredths

Start with the rightmost digit, subtracting the digits in the hundredths place: 0 minus 7. Since 0 is less than 7, borrow from the tenths column across the other columns, effectively turning the 45 into 44.900.\[\begin{array}{c}45.000 \-0.067 \\hline44.933\end{array}\]

Step 3: Subtract the Tenths

Move to the tenths place, subtract 6 from 9 (after the borrowing from previous step). This is straightforward: 9 - 6 = 3. Write down the 3 below the subtraction bar.

Step 4: Subtract the Units

Finally, subtract in the units column, 45 minus 0 remains 45, since we had borrowed to make it 44, it now becomes 44.933.

Key Concepts

Decimal Subtraction

Decimal subtraction is the process of subtracting numbers with decimal points. The key difference between whole numbers and decimals is the decimal point, which ensures we align numbers based on their place values.

When subtracting decimals, it's crucial to:

• Align the numbers vertically, making sure the decimal points line up.
• Use zeros as placeholders to ensure each column is filled.
• Once aligned, subtract each column starting from the rightmost digit.

Following these guidelines will help avoid mistakes and make decimal subtraction straightforward.

Place Value

Understanding place value is foundational to math. It refers to the value of a digit depending on its position within a number.

For example, in the number 45.067:

• The '4' is in the tens place, giving it a value of 40.
• The '5' is in the ones place, representing just 5.
• To the right of the decimal, '0', '6', and '7' are in the tenths, hundredths, and thousandths places, respectively.

Understanding place value helps you know the magnitude of each digit and guides you during subtraction, especially when borrowing is involved.

Borrowing in Subtraction

Borrowing in subtraction is used when a digit in one place is too small to subtract from. Think of it as rearranging value from a higher place to help a lower place do its job.

Here’s how borrowing works:

• If a digit can't be subtracted (e.g., 0 minus 7), borrow from the next higher place value.
• Reduce the borrowed place by 1 and add 10 to the place needing help.
• Repeat the process from right to left as needed.

In our example, we borrow from the tens place to fill up the tenths and hundredths places. This borrowing converts the original 45 into 44.933, allowing us to perform subtraction smoothly.