Problem 20 - Prealgebra

Question

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$1 \frac{1}{2}$$

Step-by-Step Solution

Answer

The decimal of \(1 \frac{1}{2}\) is 1.5.

Step 1: Convert the Mixed Number to an Improper Fraction

First, convert the mixed number \(1 \frac{1}{2}\) into an improper fraction. Multiply the whole number by the denominator of the fraction and add the numerator. This results in: \(1 \times 2 + 1 = 3\). So, the mixed number as an improper fraction is \(\frac{3}{2}\).

Step 2: Divide to Convert to Decimal

Next, divide the numerator by the denominator to convert \(\frac{3}{2}\) into a decimal. Perform the division: \(3 \div 2 = 1.5\).

Step 3: Check for Repeating Decimal

Check if the decimal \(1.5\) has a repeating part. Since it terminates after the tenths place and doesn’t repeat, there's no need for a bar.

Key Concepts

Understanding Mixed Numbers

A mixed number is a combination of a whole number and a fraction. Mixed numbers like \(1 \frac{1}{2}\) often appear in everyday situations, such as measuring ingredients in cooking. Understanding how to convert mixed numbers into improper fractions and then to decimals is crucial in mathematics.

Mixed numbers can be tricky at first, but following a simple process makes them easier to handle:

• The whole number part is straightforward.
• The fraction consists of a numerator (top number) and a denominator (bottom number).

To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator of the fraction.
• Add the numerator to the product from the previous step.
• Place this result over the original denominator.

For example, with \(1 \frac{1}{2}\):

• 1 (whole number) \(\times\) 2 (denominator) = 2
• Add 1 (numerator) = 3
• The improper fraction is \(\frac{3}{2}\)

This conversion is essential for solving equations and simplifying expressions in mathematics.

Improper Fractions Explained

Improper fractions have numerators that are equal to or greater than their denominators. For example, \(\frac{3}{2}\) is an improper fraction because the numerator (3) is larger than the denominator (2). These fractions are common in algebra and calculus.

To understand improper fractions, consider:

• Improper fractions can be greater than 1 or less than -1.
• They are easy to manipulate in calculations, especially when converting to decimals.

Converting improper fractions to decimals involves straightforward division:

• Divide the numerator by the denominator.
• The quotient can be a whole number or a decimal.
• In our example, \(3 \div 2 = 1.5\)

Understanding improper fractions helps in performing arithmetic operations and in the context of algebraic expressions. They make calculations precise and much easier to handle.

The Basics of Division in Decimal Conversion

Division is the mathematical process used to divide a whole into parts. It helps convert fractions into decimals. Converting fractions like \(\frac{3}{2}\) into decimal form relies heavily on division.

Steps for dividing fractions to get a decimal:

• Place the numerator inside the division symbol (or calculator, if using one).
• The denominator acts as the divisor outside.
• Divide the two to find the decimal equivalent.

As seen in the exercise, dividing 3 by 2 gives us 1.5. This division tells us how many times 2 fits into 3 and what remains, if any.

Keep in mind:

• A result with no remainder (like 1.5 in our example) is a terminating decimal.
• If the remainder repeats, it would be a repeating decimal, often shown with a bar over the digits that repeat.

Understanding division is vital for converting fractions to decimals accurately and efficiently. This skill is used frequently in everyday math and more complex mathematics.