Problem 26 - Prealgebra

Question

Write each decimal as a fraction or mixed number in simplest form. $$0 . \overline{2}$$

Step-by-Step Solution

Answer

The decimal \(0.\overline{2}\) is \(\frac{2}{9}\) in simplest fraction form.

Step 1: Understanding Recurring Decimals

The decimal \(0.\overline{2}\) is a recurring decimal where the digit 2 repeats indefinitely. It can be expressed as \(x = 0.2222\ldots\).

Step 2: Set Up the Equation

To convert this repeating decimal into a fraction, first set \(x = 0.\overline{2}\). Then, multiply both sides of the equation by 10 to shift the decimal point: \(10x = 2.2222\ldots\).

Step 3: Subtract and Simplify

Subtract the equation \(x = 0.2222\ldots\) from \(10x = 2.2222\ldots\) to eliminate the repeating part: \[10x - x = 2.2222\ldots - 0.2222\ldots\]\[9x = 2\]

Step 4: Solve for x

Divide both sides of the equation \(9x = 2\) by 9 to solve for \(x\): \[x = \frac{2}{9}\].

Step 5: Verify the Solution

Verify that \(\frac{2}{9}\) is in its simplest form and correctly represents the repeating decimal. The calculation checks out, confirming that the fraction is \(\frac{2}{9}\).

Key Concepts

Prealgebra

In mathematical studies, prealgebra acts as the foundational phase where students begin to explore basic mathematical concepts that prepare them for more advanced topics like algebra and beyond.
One core aspect of prealgebra is the understanding of numbers and their relationships, which includes:

• Understanding whole numbers, fractions, and decimals
• Grasping the concept of number lines
• Basic operations like addition, subtraction, multiplication, and division
• Comprehending factors and multiples
• Introduction to variables and simple equations

When dealing with decimals in prealgebra, students are often tasked with converting them to fractions. This foundational skill is crucial because it establishes a base for solving more complex problems. For example, knowing how to convert a repeating decimal such as 0.2̅ into a fraction not only improves computational fluency but also aids in understanding recurring patterns in numbers. It bridges the gap between many arithmetic concepts and algebraic functions.

Repeating Decimals

Repeating decimals are an interesting part of the number system where after a certain point, a digit or group of digits keeps repeating indefinitely.

This can be represented by placing a line (vinculum) over the repeating section. In our example, the decimal 0.2̅ repeats the digit 2. To convert repeating decimals to fractions:

• Identify the repeating digit(s).
• Set an equation where the repeating decimal is represented as a variable, say, x.
• Multiply this equation to shift the decimal point so the repeating section aligns.
• Subtract the original equation from the new equation to get a linear equation.
• Solve the equation to find the fraction equivalent.

These steps effectively translate an infinite repeating sequence into a manageable fraction, making computations and comparisons easier. Decoding repeating decimals into fractions unveils their exact numerical value, providing insight into the precision and recurring nature of numbers.

Fractions in Simplest Form

A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.

For example, the fraction \(\frac{4}{18}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). Here, the GCD of 4 and 18 is 2. Thus, reducing \(\frac{4}{18}\) results in \(\frac{2}{9}\).

Converting decimals, specifically repeating decimals like 0.2̅, into their simplest fractional form involves solving the derived equation and checking if the result can be further simplified.

• Check if the fraction obtained can be simplified.
• Divide both numerator and denominator by the greatest common divisor.
• Ensure the resulting fraction accurately represents the decimal in its entirety.

This simplification process not only makes the fraction easier to interpret but also sets a solid understanding for more complex fractions and operations in higher mathematics.