Problem 44 - Algebra 2

Question

Write each repeating decimal as a fraction 0.2\(\overline{31}\)

Step-by-Step Solution

Answer

0.2\(\overline{31}\) is \(\frac{229}{990}\) as a fraction.

Step 1: Understand the Notation

The notation 0.2\(\overline{31}\) means that the digits 31 repeat indefinitely after the decimal point. This number can be expressed as 0.231313131... where 31 repeats continuously.

Step 2: Identify the Parts to Separate

Separate the non-repeating part from the repeating part. In 0.231313131..., '2' is the non-repeating part and '31' is the repeating part.

Step 3: Formulate the Equations

Let \(x = 0.231313131...\). Multiply by a power of 10 to eliminate the non-repeating part. Multiply \(x\) by 10 to get \(10x = 2.31313131...\). Then multiply \(x\) by 1000 to move the repeating part: \(1000x = 231.31313131...\).

Step 4: Subtract the Equations

Subtract the first equation from the second: \[1000x - 10x = 231.31313131... - 2.31313131...\]This results in \(990x = 229\).

Step 5: Solve for x as a Fraction

Solve \(990x = 229\) for \(x\) by dividing both sides by 990:\[x = \frac{229}{990}\]

Step 6: Simplify the Fraction

Both 229 and 990 are already in simplest form since they have no common factors other than 1. So, the fraction \(\frac{229}{990}\) is the simplest form.

Key Concepts

Fraction Simplification

Fractions represent parts of a whole, and simplifying them makes them easier to understand and work with. The simplest form of a fraction is achieved when the numerator and denominator have no common factors other than 1. To simplify, one can:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by this GCD.

In our exercise, the fraction \(\frac{229}{990}\) is already in simplest form because 229 and 990 share a greatest common divisor of 1. Always check this step to ensure the fraction doesn't break down further, which keeps the solution elegant and efficient. For practice, try simplifying fractions manually, checking if smaller numbers can be divided by any common factors.

Infinite Series

An infinite series arises when a sequence of numbers is added, extending endlessly. In our problem, the decimal 0.231313131... illustrates this via repeating decimals. This means the digits 31 continue indefinitely. Such sequences demand specific methods to convert into fractions.

• The repeating part of the decimal forms a geometric series.
• Mathematically, this series allows representation via algebraic equations.

Understanding infinite series paves the way for converting complex repeating decimals into fractions. It's about seeing patterns and using algebra to express these infinite behaviors in a finite way, making them much more manageable.

Algebraic Equation

Algebraic equations are essential tools for solving problems involving unknowns. In the context of repeating decimals like 0.2\(\overline{31}\), they help transform a complex repeating number into a simple fraction.
By letting \(x = 0.231313131...\), we set an equation that models the decimal. Multiplying by powers of 10 (like \(10x\) and \(1000x\)) moves the repeating sequence to create a solvable system of equations:

• Write multiple forms of the decimal using these multipliers.
• Subtract them to remove the repeating sequence.

The algebra simplifies the process to one clear step - divide, revealing that the equation simplifies into a tidy fraction. So, using minimal yet impactful algebra, repeating decimals become straightforward fractions.

Decimal Representation

Understanding decimal representation is key to converting repeating decimals into fractions. Decimals express numbers that aren't whole, often pinpointed to after a decimal point. A repeating decimal adds complexity, as parts of it extend indefinitely.

• Identify the non-repeating section, which in 0.2\(\overline{31}\) is '2'.
• Spot the repeating section which is '31'.

Grasping these components allows separating the number accurately, achieving proper fraction representation. Decimals serve a vital role in mathematics by offering precise representation of numbers that would otherwise be cumbersome in fractional form alone. Converting them to fractions ensures they are practical for most calculations, while acknowledging their repeating nature gives insights into their infinite behavior.