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Magazine Chapter 13: Problem 24
Express each repeating decimal as a fraction. $$0 . \overline{47}$$
Short Answer
Expert verified
The repeating decimal \(0.\overline{47}\) as a fraction is \(\frac{47}{99}\).
Step by step solution
01
Understand the Problem
We need to convert the repeating decimal \(0.\overline{47}\) into a fraction. The number \(0.\overline{47}\) means that \(47\) is repeating indefinitely.
02
Assign a Variable
Assign a variable to the repeating decimal. Let \(x = 0.\overline{47}\). This means \(x = 0.474747\ldots\).
03
Eliminate the Repetition
Multiply both sides of the equation by 100 (since the repeated block has two digits) to shift the decimal point two places to the right: \[100x = 47.474747\ldots\]
04
Set Up the Subtraction
Subtract the original equation from this new equation to eliminate the repeating part: \[100x - x = 47.474747\ldots - 0.474747\ldots\]
05
Solve for x
The subtraction simplifies to: \[99x = 47\]Now, solve for \(x\) by dividing both sides by 99:\[x = \frac{47}{99}\]
06
Simplify the Fraction
Check if the fraction \(\frac{47}{99}\) can be simplified. Since 47 is a prime number and not a factor of 99, \(\frac{47}{99}\) is already in its simplest form.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fraction Conversion
Converting a repeating decimal into a fraction might seem tricky, but it's all about seeing the pattern and using some simple algebra. When you see a repeating decimal, you're looking at a situation where the decimal part repeats over and over again. The key is to express this repeating decimal as a fraction, which is a much neater form and often easier to work with.
Here's a simple breakdown:
Here's a simple breakdown:
- Identify the repeating part: In the decimal \(0.\overline{47}\), the repeating part is \(47\).
- Assign this repeating decimal to a variable, like \(x\).
- Use a method to eliminate the repeating part so that you can express the number as a straightforward fraction.
Decimal Representation
Decimals are numbers expressed with a point separating the whole number from the fractional part. A repeating decimal is one where digits after the decimal point continue infinitely. Repeating decimals have a pattern or sequence of digits that repeat endlessly. For example, in \(0.\overline{47}\), the sequence \(47\) repeats forever.
Understanding decimal representation allows you to move between different forms of numbers:
Understanding decimal representation allows you to move between different forms of numbers:
- Non-terminating: Decimals that go on forever without ending, but with a repeating pattern.
- Terminating: Decimals that come to an end after a certain number of digits.
Step-by-Step Solution
Solving problems with repeating decimals involves clear and precise steps. Each step helps in transforming the repeating decimal into a fraction form. Let's walk through this:
- Assign a Variable: Start by setting your repeating decimal to a variable, such as \(x\). This turns your problem into an algebraic equation and makes it easier to manage.
- Multiply to Shift Decimal: Since the repeating sequence in \(0.\overline{47}\) is two digits, multiply both sides of your equation by 100. This moves the decimal point two places, lining up the repeating sections.
- Subtract to Eliminate Repeats: With the aligned decimals, subtract the original equation from the shifted equation. This operation clears out the repeating decimals from one side of the equation.
- Solve for \(x\): Once the repeating part is gone, you're left with a simple equation like \(99x = 47\). Solve for \(x\) by isolating it on one side: \(x = \frac{47}{99}\).
