Problem 12 - Prealgebra

Question

Convert the given fraction to a terminating decimal. \(\frac{4}{8}\)

Step-by-Step Solution

Answer

The fraction \( \frac{4}{8} \) simplifies to \( 0.5 \), a terminating decimal.

Step 1: Simplify the Fraction

To convert a fraction to a decimal, first check if it can be simplified. The given fraction is \( \frac{4}{8} \). Both the numerator and the denominator can be divided by their greatest common divisor, which is 4. Thus, \( \frac{4}{8} \) simplifies to \( \frac{1}{2} \).

Step 2: Divide the Numerator by the Denominator

After simplifying, use division to convert the fraction to a decimal. Divide the numerator (1) by the denominator (2). Perform the division: \( 1 \div 2 = 0.5 \).

Step 3: Verify the Decimal is Terminating

A terminating decimal has a finite number of digits after the decimal point. The result from the division is \( 0.5 \), which has one digit after the decimal point, confirming it is a terminating decimal.

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This means adjusting the numerator and the denominator until they no longer have any common factors other than 1. For instance, the fraction \( \frac{4}{8} \) can be simplified. Both the numbers 4 and 8 share a common factor of 4.

• Divide the numerator (4) and the denominator (8) by their greatest common divisor (GCD), which is 4.
• The simplified fraction is \( \frac{1}{2} \). This form is easier to work with and can make future calculations simpler.
• Recognizing the GCD is a key skill in simplifying fractions; with practice, this becomes second nature.

By simplifying fractions, you lay the groundwork for converting fractions to decimals more effectively.

Terminating Decimals

Terminating decimals are decimals that have a set number of digits after the decimal point. They do not go on forever, unlike repeating decimals. When converting fractions like \( \frac{1}{2} \) to decimals, you often end up with a terminating decimal.

• The decimal equivalent of \( \frac{1}{2} \) is 0.5, which is a terminating decimal because it only has one digit after the decimal point.
• A terminating decimal is an indicator that the fraction has been divided wholly and finitely.
• To identify if a decimal is terminating, you could look for factors of the denominator. If after simplification, the denominator consists solely of the prime factors 2 and/or 5, the decimal will terminate.

Terminating decimals are straightforward and clear, enhancing comprehension of the specific value without ambiguity.

Division Method for Fractions

The division method for converting fractions to decimals is both practical and straightforward. In this process, you divide the numerator by the denominator. Let's explore how this works using our example of \( \frac{1}{2} \).

• Place the numerator (1) inside the division bracket and the denominator (2) outside.
• Divide 1 by 2. Since 1 is smaller than 2, it enters into the bracket as 0, and you add a decimal point.
• Create a zero in the tenths place, making it 10, and then divide again: 10 divided by 2 gives a quotient of 5.
• The result is 0.5, confirming \( \frac{1}{2} \) equals 0.5 as a decimal.

Using the division method for fractions is reliable for converting both simple and complex fractions into decimals and ensures accuracy every step of the way.