Problem 26 - Prealgebra

Question

Find each of the following differences. (Subtract.) $$7.5-3.04$$

Step-by-Step Solution

Answer

The difference is 4.46.

Step 1: Align the Numbers

First, align the numbers by their decimal points to ensure correct subtractions. Write 7.5 as 7.50 so that it has the same number of decimal places as 3.04. Now, we have 7.50 and 3.04.

Step 2: Perform the Subtraction

Starting from the rightmost digit, subtract the bottom number from the top number, column by column. If necessary, borrow from the next left column. Start with the hundredths place: 0 - 4. Since 0 is less than 4, we need to borrow 1 from the tenths place, turning the 5 into 4, and the 0 into 10. Now, subtract 10 - 4 = 6.

Step 3: Continue Subtraction for Tenths Place

Next, with the updated tenths place, subtract 4 (from borrowing) - 0 = 4.

Step 4: Subtraction for the Units Place

Now, move to the units place: subtract 7 - 3 = 4.

Step 5: Write Down the Result

Combine all results: the difference in each column gives us a final result of 4.46 for the expression 7.50 - 3.04.

Key Concepts

Aligning Decimal Numbers

When subtracting decimal numbers, the first crucial step is to align them correctly. This alignment is done by ensuring that each decimal point lines up vertically with the others. This might sound simple, but it's a critical step that sets the stage for accurate subtraction. Imagine you want to subtract \(7.5 - 3.04\). To align them properly, you can rewrite \(7.5\) as \(7.50\). This way, both numbers have the same number of decimal places, making it much easier to manage each column of numbers.

• Ensure the decimal points are in a vertical line.
• Fill in trailing zeros where necessary to match the decimal places.

By aligning the numbers, you are organizing them into neat columns of units, tenths, and hundredths. This allows you to focus on subtracting one place value at a time, just like you would with whole numbers.

Borrowing in Subtraction

Borrowing is like a lifesaver in subtraction when you encounter a situation where you can't directly subtract the smaller digit from the larger digit in the same place value. In the example of \(7.50 - 3.04\), look at the hundredths place: \(0 - 4\). Since \(0\) is smaller than \(4\), you cannot subtract it directly. So, you "borrow" 1 from the next column on the left, the tenths place, to make the subtraction possible.

• Subtracting within a column sometimes requires borrowing from the column immediately to the left.
• This borrowed 1 turns the \(0\) into \(10\) in the hundredths place.
• After borrowing, you subtract \(10 - 4\) to get \(6\).

Borrowing ensures you can still perform the operation correctly even when numbers in the subtrahend are larger than those in the minuend.

Place Value in Subtraction

Understanding place value is essential not just for subtraction, but for all arithmetic operations with numbers. Each digit in a decimal number has a specific value depending on its position: units, tenths, hundredths, and so on. When you subtract numbers, especially decimals, you subtract each place value separately.

• The units place in \(7.50\) and \(3.04\) is straightforward as \(7 - 3 = 4\).
• The tenths place becomes \(4\) after borrowing, allowing you to subtract \(4 - 0\) directly.
• The hundredths place, as discussed, requires borrowing, leading to \(10 - 4 = 6\).

Understanding place values ensures you subtract corresponding parts of the numbers and helps avoid mistakes. By recognizing and respecting each place's role in the overall value of the number, you maintain accuracy throughout the calculation. This systematic approach not only simplifies arithmetic but also builds a strong foundational skill for more complex math concepts.