Problem 30 - Prealgebra

Question

Find each of the following differences. (Subtract.) $$48-0.075$$

Step-by-Step Solution

Answer

The difference is 47.925.

Step 1: Understand the Problem

We need to find the difference between two numbers: 48 and 0.075. This means we will subtract 0.075 from 48.

Step 2: Align the Numbers for Subtraction

First, write the number 48 with three decimal places to match the decimal places of 0.075. This can be written as 48.000 - 0.075.

Step 3: Subtract Decimal Numbers

Subtract 0.075 from 48.000. Line up the decimal points:\[\begin{array}{r} 48.000 \- 0.075 \\hline x.\text{xxx}\end{array}\]Start subtracting from the rightmost digit. Since 0 is less than 5, borrow from the next digit in 48, making it:\[\begin{array}{r} 47.\color{red}{9}10 \ - 0.075 \\hline x.\text{xxx}\end{array}\]Now subtract:9 - 7 = 2,\9 - 0 = 9,\'7' becomes '47'.So, the final subtraction is 47.925.

Key Concepts

Understanding Place Value in Subtraction

Place value is a fundamental concept in mathematics that helps us understand the worth of each digit in a number. When subtracting decimal numbers, recognizing place value is crucial.

• In 48.000, the '4' is in the tens place, the '8' is in the ones place, followed by zeroes in the tenths, hundredths, and thousandths places.
• In 0.075, the '7' is in the hundredths place, and the '5' is in the thousandths place.

When we write equivalent decimals, such as converting 48 to 48.000, we are essentially acknowledging these unused places. This ensures each digit is appropriately lined up according to its respective place in value, allowing proper subtraction to occur without confusion or error.

Borrowing in Subtraction

Borrowing can often seem tricky, but it's all about redistributing value to perform subtraction when a digit is smaller than the one being subtracted from it. When you attempt to subtract 5 from 0, which is impossible, you need to "borrow" from the nearest left place value that has a higher number.

• In our example, the thousandth place is reduced to a 10 from borrowing 1 from the hundredths place, changing it to 9.
• Next, borrow from the tenths place, turning it into a 9, and reducing the ones place to a 7.

This sequential borrowing ensures that each place value can provide enough value to complete each subtraction step correctly. This way, the subtraction becomes feasible, and you can proceed effectively toward the correct answer.

Decimal Alignment

Aligning decimals is an essential step when subtracting decimal numbers because it guarantees that each place value is correctly matched. Without this alignment, subtraction would lead to incorrect results. First, ensure both numbers have the same number of decimal places by padding with zeros if necessary.

• Write 48 as 48.000, matching it to the three decimal places of 0.075.
• Align the decimals vertically, making it clear which digits you are subtracting from.

This clear layout allows you to perform borrowing correctly, as seen with the thousandths and hundredths places in the example. Proper alignment of the decimal points ensures that each digit is subtracted from the right corresponding place value, avoiding potential errors in calculations.