Converting Periodic Fractions to Regular Fractions

This page provides a comprehensive guide on how to convert periodic fractions to regular fractions. The process is explained through three detailed examples, each demonstrating a different scenario.

The general method for converting a periodic fraction to a regular fraction involves the following steps:

1. Represent the periodic fraction as a variable (usually x).
2. Set up an equation using this variable.
3. Multiply the equation by an appropriate power of 10 to align the decimal points.
4. Subtract the original equation from the multiplied equation.
5. Solve for x to obtain the regular fraction.

Vocabulary: A periodic fraction, also known as a repeating decimal, is a decimal number where a digit or group of digits repeats indefinitely after the decimal point.

Example 1: Converting 0.(5) to a regular fraction

1. Let x = 0.5555...
2. Multiply both sides by 10: 10x = 5.5555...
3. Subtract the original equation: 9x = 5
4. Solve for x: x = 5/9

Example: 0.(5) = 5/9

Example 2: Converting 0.(17) to a regular fraction

1. Let x = 0.171717...
2. Multiply both sides by 100: 100x = 17.1717...
3. Subtract the original equation: 99x = 17
4. Solve for x: x = 17/99

Example: 0.(17) = 17/99

Example 3: Converting 0.5(8) to a regular fraction

1. Let x = 0.58888...
2. Multiply by 10: 10x = 5.8888...
3. Multiply by 100: 100x = 58.8888...
4. Subtract the equations: 90x = 53
5. Solve for x: x = 53/90

Example: 0.5(8) = 53/90

Highlight: The key to converting periodic fractions is identifying the repeating pattern and setting up the appropriate equations to isolate the fraction.

These examples demonstrate how to apply the ułamek okresowy na zwykły kalkulator (periodic fraction to regular fraction calculator) method manually. Understanding this process is crucial for students learning about zamiana ułamków okresowych na zwykłe (converting periodic fractions to regular fractions).