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Chapter 8: Problem 21

Write the repeating decimal as a fraction in simplest form. \(0.161616 \ldots\)

Short Answer

Expert verified
The repeating decimal \(0.161616... \) as a fraction in simplest form is \( \frac{16}{99} \)

Step by step solution

01

Applying Algebra

Let \(x = 0.161616...\) This equation represents the decimal expansion of \(x\). Multiply this equation by 100 to shift the decimal point two places to the right. This yields \(100x = 16.161616...\)
02

Subtracting the Initial Equation

By subtracting the initial equation from the one obtained after multiplication, repetitive decimal parts will cancel out. The equation is \(100x - x = 16.161616... - 0.161616...\). This simplifies to \(99x = 16.\)
03

Solving for x

To solve for \(x\), divide both sides of the equation by 99. This results in \(x = \frac{16}{99}\), which is a fraction equivalent to the repeating decimal.

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Key Concepts

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

Algebraic Manipulation
Understanding algebraic manipulation is crucial to converting repeating decimals into fractions. When faced with a repeating decimal like \(0.161616\ldots\), we can utilize variable assignments and strategically apply algebraic operations, such as multiplication and subtraction, to identify the fractional equivalent.
  • Start by assigning a variable to the repeating decimal. For instance, let \(x = 0.161616\ldots\).
  • Multiply the variable by an appropriate power of ten. This shifts the decimal point to aid in alignment for subsequent elimination of repetition. In this case, multiplying by 100 gives \(100x = 16.161616\ldots\).
  • Next, subtract the original equation from the multiplied equation. This leverages the repetition, resulting in elimination of the repeating series on both sides of the equation.
This step-by-step procedure simplifies our workbars. Ultimately, this will help us isolate \(x\) for an easy-to-handle fraction.
Simplifying Fractions
Once a repeating decimal has been converted into a fraction via algebra, the next task is to ensure our answer is in its simplest form. Simplifying fractions involves reducing the numerator and denominator to their smallest values while retaining equivalent values.
  • After deriving \(x = \frac{16}{99}\), we must check if there's a common factor between the numerator and the denominator that could further reduce the fraction.
  • Conduct a factorization of both 16 and 99. As neither number shares a common factor aside from 1, \(\frac{16}{99}\) is already in simplest form.
Simplifying fractions ensures clarity and consistency in mathematical expressions, making them easier to understand and utilize for further calculations.
Decimal Representation
Understanding decimal representation clarifies why we sometimes end up with repeating decimals. Repeating decimals occur when a fraction's denominator has a prime factor that does not divide evenly into the fraction's base after division.
  • The decimal \(0.161616\ldots\) results from an infinite expansion, indicating it stems from a fraction that doesn鈥檛 resolve neatly in base ten.
  • Recognizing these characteristics reveals that repeating decimals are a natural outcome of our numerical system. They also provide insights into the properties of numbers, helping us convert these sequences back to manageable fractions.
By acknowledging how decimals relate back to fractions, we gain a fundamental understanding of fractional calculations and can better handle operations like division or multiplication of repeating decimals.

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Most popular questions from this chapter

On January 1, you deposit $$\$ 2000$$ in a retirement account that pays \(5 \%\) annual interest. You make this deposit each January 1 for the next 30 years. How much money do you have in your account immediately after you make your last deposit?

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write a rule for the nth term of the arithmetic sequence. \(a_7=58, a_{11}=94\)

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