Problem 49 - Prealgebra

Question

Simplify the given expression by first converting the fraction into a terminating decimal. \(\frac{9}{10}-8.61\)

Step-by-Step Solution

Answer

The simplified expression is -7.71.

Step 1: Identify the Fraction

The given fraction is \( \frac{9}{10} \). This is our first term in the expression.

Step 2: Convert Fraction to Decimal

To convert the fraction \( \frac{9}{10} \) into a decimal, we divide the numerator by the denominator: \[ \frac{9}{10} = 0.9 \] Thus, \( \frac{9}{10} \) as a decimal is 0.9.

Step 3: Write the Expression with Decimals

Now replace the fraction with its decimal equivalent in the original expression. The expression now reads \[ 0.9 - 8.61 \]

Step 4: Perform the Subtraction

Subtract 8.61 from 0.9. To perform the subtraction correctly, align the decimal points: \[ 0.90 - 8.61 \underline{\phantom{-}-------} \] To subtract, we start from the rightmost digit and borrow if necessary: 0.90 minus 8.61 gives us -7.71.

Key Concepts

Fraction to Decimal Conversion

Converting a fraction to a decimal might sound challenging at first, but it’s quite straightforward once you get the hang of it. A fraction is simply another way to represent division. The fraction \(\frac{9}{10}\) means "9 divided by 10." When you perform that division, you get the decimal 0.9. This is called a terminating decimal because it has a finite number of digits. To convert any fraction:

• Divide the numerator by the denominator.
• Ensure the decimal is simplified; reduce fractions if necessary before conversion.

Understanding this process is important, as decimals are often easier to work with in mathematical operations. Remember, not all fractions will convert into such neat decimals. Some might result in repeating decimals, but our focus here is on fractions like \(\frac{9}{10}\) that convert neatly to a finite decimal result.

Subtraction of Decimals

Subtracting decimals requires careful alignment of the decimal points. The key is precision and keeping everything lined up properly. Let’s take the example of subtracting 8.61 from 0.9:

• First, ensure the decimal places are aligned vertically, which may mean adding trailing zeros to the decimal with fewer digits. So, 0.9 becomes 0.90.
• Begin subtracting from the rightmost digit. If necessary, borrow from the digit to the left in cases where the top digit is smaller than the bottom one.
• Continue the subtraction process across all the decimal places.

In our detailed example, we ended up with -7.71. This result indicates that the minuend (0.9) was smaller than the subtrahend (8.61), resulting in a negative difference—a crucial point to understand when working with real-world decimal operations.

Place Value in Decimals

Understanding place value in decimals is essential for accurate calculations. Each position in a decimal number represents a different value based on its position relative to the decimal point. Let's break this down using the number 8.61:

• The digit 8 is in the ones place, contributing 8 to the total value.
• The digit 6 is in the tenths place, adding 0.6.
• The digit 1 is in the hundredths place, contributing 0.01.

Recognizing the roles of these positions helps you perform operations like subtraction with confidence and accuracy. When converting fractions or performing operations with decimals, always double-check each digit's place value. This ensures clarity and precision, making math a much smoother process.