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Chapter 11: Problem 55

Find fraction notation for each repeating decimal. $$0.5555 \ldots$$

Short Answer

Expert verified
\( x = \frac{5}{9} \)

Step by step solution

01

- Identify the repeating part

The repeating decimal is given as 0.5555... Here, the repeating part is 5.
02

- Assign a variable

Let's assign the repeating decimal to a variable: \( x = 0.5555 ... \)
03

- Multiply by a power of 10

Multiply both sides of the equation by 10 to shift the decimal point one place to the right: \( 10x = 5.5555 ... \)
04

- Set up a subtraction equation

To eliminate the repeating part, subtract the original equation from this new equation: \[ 10x - x = 5.5555 ... - 0.5555 ... \] This simplifies to: \[ 9x = 5 \]
05

- Solve for x

Divide both sides by 9 to solve for x: \( x = \frac{5}{9} \)

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Key Concepts

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

fraction notation
Fraction notation represents numbers in the form of a ratio between two integers, like \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. This notation is essential because it helps to convert repeating decimals, like \(0.5555\ldots\), into fractions that are simpler to understand and use. Starting with a repeating decimal, we assign a variable \(x\) to this number, making it easier to manipulate and eventually solve.
repeating decimal
Repeating decimals are decimals where a digit or group of digits repeat infinitely. For example, in \(0.5555\ldots\), the digit 5 repeats. Recognizing the repeating part is crucial for converting it to fraction notation. By assigning a variable to the repeating decimal, multiplying by a power of 10 to shift the decimal, and then subtracting the original equation, we can isolate the repeating part and solve for the fraction.
solving linear equations
Solving linear equations involves finding the value of a variable that makes the equation true. In our example with repeating decimals, once we've assigned \(x\) and manipulated the equation to isolate the repeating part, we get a clean algebraic equation: \(9x = 5\). Solving for \(x\) involves simple algebraic steps like subtracting and dividing, leading us to the fraction \(\frac{5}{9}\), which is the simplest form representing the repeating decimal.

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