91Ó°ÊÓ

When facing a problem involving decimal subtraction, such as the exercise \(345.213-27.35\), the first step is to align the decimal points. This means writing the numbers vertically, one under the other, with their decimal points directly in the same column. This is crucial because it allows us to accurately match each digit according to its place value. For instance, in our example, aligning them would look like this:

\[\begin{align*}&\phantom{0}345.213\- &\phantom{00}27.35\end{align*}\]
The numbers are arranged so the hundreds, tens, ones, tenths, hundredths, and thousandths are all in their respective columns. Neglecting to align the decimal points correctly could result in erroneous calculations, as the value of each digit is tied to its position relative to the decimal.

With our decimal points aligned, we move on to column subtraction. This process starts from the farthest column on the right, which are the smallest place values, and moves to the left.

Begin subtracting each column individually. If you can’t subtract because the top number is smaller than the bottom one, you will need to use the process of borrowing. For each subtraction, make sure you bring down the decimal point directly below the others in the result so it remains aligned.

Working through our example, we subtract from right to left, taking the hundredths, then tenths, and so on, until we've subtracted every column. If you confront a situation where there are missing digits in one of the decimal places, like the '0' in the hundredths place for 27.35, remember to treat it as zero. This ensures that you maintain the correct place value during subtraction.

Borrowing, also known as regrouping, is a pivotal concept in subtraction whenever a larger digit is subtracted from a smaller one. It involves 'borrowing' one unit from the next higher place value to make the subtraction possible.

Let's illustrate this with our exercise, specifically looking at the tenths place. When we need to subtract 3 tenths from 1 tenth, we borrow 1 whole (which is equivalent to 10 tenths) from the ones column. This turns the 1 tenth into 11 tenths. After borrowing, we can then perform the subtraction. It's important to remember to decrease the digit in the column you borrowed from by one.

Repeat this process for each column as necessary. Through effective borrowing, numbers that seemed impossible to subtract become manageable and the subtraction can proceed smoothly. It's beneficial to practice this step by adding zeros to any empty decimal places. By mastering borrowing, you will vastly improve your confidence and accuracy with decimal subtraction.