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Chapter 11: Problem 52

Find fraction notation. $$0.2222 \ldots, \text { or } 0 . \overline{2}$$

Short Answer

Expert verified
\( \frac{2}{9} \)

Step by step solution

01

- Let x be the repeating decimal

Define the repeating decimal as a variable: Let \( x = 0.2222 \ldots \).
02

- Multiply by a power of 10

To isolate the repeating part, multiply both sides of the equation by 10: \( 10x = 2.2222 \ldots \).
03

- Set up an equation to subtract

Subtract the original equation from this new equation to eliminate the repeating part: \( 10x - x = 2.2222 \ldots - 0.2222 \ldots \).
04

- Simplify the equation

This will simplify to: \( 9x = 2 \).
05

- Solve for x

Divide both sides by 9 to solve for \( x \): \( x = \frac {2}{9} \).
06

- Conclusion

The fraction form of the repeating decimal \( 0.2222 \ldots \) is \( \frac{2}{9} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Repeating Decimals
Repeating decimals are numbers that have one or more digits after the decimal point repeating infinitely. For example, in the decimal 0.2222..., the digit 2 repeats forever. We can represent repeating decimals using a bar notation, for example, 0.2虆, where the bar above the 2 indicates that 2 is the repeating digit.
Converting Decimals to Fractions
To convert a repeating decimal to a fraction, follow these steps:

  • Assign the repeating decimal to a variable. Let's call it x.
  • Multiply x by a power of 10 so that the decimal part aligns.
  • Subtract the original x from the result to eliminate the repeating part.
  • Solve for x to find the fraction form.


For example, let's convert 0.2222... to a fraction:

  • Let x = 0.2222...
  • Multiply by 10: 10x = 2.2222...
  • Subtract the original x from 10x: 10x - x = 2.2222... - 0.2222...
  • This simplifies to: 9x = 2
  • Solve for x: x = 2/9

Thus, the fraction form of 0.2222... is 2/9.
Algebraic Manipulation
Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions. When dealing with repeating decimals and converting them to fractions, we often use simple algebraic techniques. Here's a breakdown of what happens during this process:

  • Define the decimal as a variable: Let x = 0.2222...
  • Multiply by a power of 10: This step helps in aligning the repeating part of the decimal. For our example, we multiply by 10: 10x = 2.2222...
  • Set up an equation to subtract: By subtracting the original variable from the multiplied one, we eliminate the repeating portion: 10x - x = 2.2222... - 0.2222...
  • Simplify the equation: The subtraction simplifies to 9x = 2
  • Solve for the variable: Finally, divide both sides by 9 to isolate x, resulting in x = 2/9

These steps use basic algebra to transform an infinite repeating decimal into a precise fraction form.

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Most popular questions from this chapter

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