Problem 8 - Prealgebra

Question

Convert each of the following fractions to a decimal. $$\frac{1}{2}$$

Step-by-Step Solution

Answer

\(\frac{1}{2}\) as a decimal is \(0.5\).

Step 1: Understanding the Fraction

The fraction given is \( \frac{1}{2} \). Here, \(1\) is the numerator and \(2\) is the denominator. The task is to convert this fraction into a decimal.

Step 2: Dividing the Numerator by the Denominator

To convert the fraction into a decimal, divide the numerator by the denominator: \( 1 \div 2 \).

Step 3: Performing the Division

Perform the division \( 1 \div 2 \). Since \(2\) does not go into \(1\), consider it as \(10\) and place a decimal point. \(2\) goes into \(10\) five times. So \( 1 \div 2 = 0.5 \).

Step 4: Concluding the Conversion

The decimal representation of the fraction \( \frac{1}{2} \) is \(0.5\).

Key Concepts

Understanding Numerator and Denominator

A fraction is a mathematical expression that represents the division of two numbers. It consists of two parts: the numerator and the denominator. The numerator is the top number, indicating how many parts of the whole we have. In our example, the fraction is \( \frac{1}{2} \), where the numerator is \(1\). This tells us we have one part.
The denominator is the bottom number and it shows the number of equal parts the whole is divided into. For \( \frac{1}{2} \), the denominator is \(2\), meaning the whole is divided into two equal parts. Understanding these two components helps us realize what the fraction is representing in a real-world or mathematical context.

• Numerator: the part that is taken or considered.
• Denominator: the total number of equal parts.

Recognizing the roles of the numerator and denominator is the first step in converting fractions into decimals.

Grasping the Division Process

Converting a fraction to a decimal involves an important arithmetic method: division. The division process is the key to changing a fraction form to a decimal. Here, the operation is to divide the numerator by the denominator. In our example, you execute \( 1 \div 2 \).
Initially, notice that \(2\) does not go into the smaller number \(1\) neatly. This is where the need for decimals arises. We convert and express the numerator as \(10\) by imagining it as \(1.0\), allowing us to undertake the division properly.
To perform this division:

• Extend the numerator with a point, making it \(10\).
• \(2\) goes into \(10\) a total of \(5\) times perfectly.

Therefore, the division process gives us the decimal digit by digit, equipping us to handle even more complex divisions.

Achieving Decimal Representation

Decimal representation is the outcome we seek when converting fractions. It represents numbers as parts of whole numbers in a straightforward form. In our case, expressing \( \frac{1}{2} \) in terms of decimals results in \(0.5\).
This format is more intuitive when dealing with everyday math problems, offering a better grasp of value when precision is vital. When \(1\) is divided by \(2\), as found through the division, we reach the decimal \(0.5\).

• Decimal representation is practical in calculations like percentages, measurements, and money.
• Decimals offer a great means for comparing fractional values.

By getting comfortable with decimal representation, we can interpret and manipulate numbers in versatile ways, aiding both academic pursuits and daily activities.