Problem 14 - Calculate With Confidence

Question

Perform the indicated operations. Give exact answers. $$8.17-3.05=$$

Step-by-Step Solution

Answer

The result is 5.12.

Step 1: Align the Numbers

To subtract the numbers, write them one underneath the other, aligning the decimal points. Arrange it like so:\[\begin{array}{c}8.17\-3.05\\hline?\end{array}\]

Step 2: Subtract the Hundredths Place

Start from the rightmost decimal place (hundredths). Subtract the digits in the hundredths place:\[7 - 5 = 2\]

Step 3: Subtract the Tenths Place

Move to the tenths place and subtract those digits:\[1 - 0 = 1\]

Step 4: Subtract the Units Place

Finally, subtract the numbers in the units place:\[8 - 3 = 5\]

Step 5: Combine Results

Combine the results from the previous steps. Write the final result:\[8.17 - 3.05 = 5.12\]

Key Concepts

Decimal Subtraction

Decimal subtraction is a type of arithmetic where you take away one decimal number from another. Like ordinary subtraction, decimal subtraction involves aligning numbers, but it has the added challenge of dealing with decimal points. One of the key things to remember is to line up the decimal points of both numbers. This ensures all the place values are properly aligned.

While subtracting, the digits are processed from the right to the left, just like in whole number subtraction. This starts from the rightmost digit, known as the hundredths place for decimals. Once you have subtracted the rightmost digits, you continue moving left through the tenths place and then the units place.

To keep things simple, consider the whole number subtraction rules you already know and apply them to each decimal place. Remember, each place value must be considered separately.

Place Value

Place value is an essential concept not just for decimal subtraction but for all mathematical operations involving numbers. In decimal numbers, place value refers to the position of a digit in relation to the decimal point, and each position represents a power of ten.

Here's a quick breakdown:

• The first digit to the right of the decimal is in the tenths place.
• The next digit is in the hundredths place.
• The digit just left of the decimal is in the units place.

Understanding and identifying these place values is critical when performing operations because subtraction must account for each digit's position value. When you align the decimal points in a subtraction problem, you ensure that each digit is in its proper place value column, and you subtract according to those values.

Mathematical Operations

In mathematics, operations refer to the basic processes we perform with numbers. Subtraction is one of the essential operations, alongside addition, multiplication, and division. Decimal subtraction, specifically, requires you to use these operations in a coordinated way, focusing on details such as decimal alignment and place value.

When performing subtraction:

• Align numbers based on their decimal points so that you work with the correct place values.
• Start with the smallest place value (the rightmost digit) and work your way to the left.
• If any digit on the minuend (the number being subtracted from) is smaller than the corresponding digit in the subtrahend, you may need to borrow from the next higher digit.

These carefully applied steps ensure the subtraction is precise, as each positional value is correctly processed.