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Chapter 16: Problem 26

Express the given nonterminating repeating decimal as a common fraction.\(0.272727 \ldots\)

Short Answer

Expert verified
The repeating decimal \( 0.272727 \text{...} \) as a fraction is \( \frac{3}{11} \).

Step by step solution

01

- Assign a variable

Let the repeating decimal be represented by the variable, say, x. So, let \( x = 0.272727 \text{...} \).
02

- Multiply by the power of 10

Multiply both sides by 100 to shift the decimal point two places to the right: \( 100x = 27.272727 \text{...} \).
03

- Set up an equation

Now, notice the repeating part: \( x = 0.272727 \text{...} \) and \( 100x = 27.272727 \text{...} \). Create an equation by subtracting \( x \) from \( 100x \): \( 100x - x = 27.272727 \text{...} - 0.272727 \text{...} \).
04

- Solve the equation

Perform the subtraction on both sides: \( 100x - x = 27.272727 \text{...} - 0.272727 \text{...} \) gives \( 99x = 27 \). Next, solve for x: \( x = \frac{27}{99} \).
05

- Simplify the fraction

Simplify the fraction by finding the greatest common divisor (GCD) of 27 and 99. Since the GCD of 27 and 99 is 9, divide both the numerator and the denominator by 9 to get \( \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \). Therefore, \( x = \frac{3}{11} \).

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Key Concepts

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

Nonterminating Decimal
A nonterminating decimal is a decimal number that continues indefinitely without ending. These numbers cannot be represented exactly with a finite number of digits.
We often see nonterminating decimals in two forms: repeating and non-repeating.
For example:
- Repeating: The decimal 0.272727... is repeating because the sequence '27' repeats.
- Non-repeating: The decimal 0.1010010001... does not have a repeating sequence.
In this exercise, we analyze a repeating nonterminating decimal. By recognizing and using its repeating nature, we can convert it into a simple fraction.
Common Fraction
A common fraction, also simply called a fraction, is a way to represent a part of a whole number using two integers, the numerator and the denominator. The numerator is the top part, which shows the number of equal parts we have. The denominator is the bottom part, which shows the total number of equal parts the whole is divided into.
For example, in the fraction \(\frac{3}{11}\), the 3 is the numerator, and the 11 is the denominator.
When we transformed the repeating decimal \(0.272727...\) to the fraction \(\frac{27}{99}\), we simplified it to \(\frac{3}{11}\).
This step is vital because it helps in comparing, adding, and subtracting fractions more straightforwardly. Simplifying fractions makes them easier to work with and understand.
Greatest Common Divisor
To simplify a fraction, we use the greatest common divisor (GCD). The GCD of two numbers is the largest number that divides both of them without leaving a remainder.
Let's look at our example: \( \frac{27}{99} \). Here, the GCD of 27 and 99 is 9. This means both 27 and 99 can be divided by 9 perfectly:
\[ \text{GCD of 27 and 99 is 9} \]
Simplifying the fraction then becomes easier:
\[ \frac{27 \text{ divided by } 9}{99 \text{ divided by } 9} = \frac{3}{11} \]
Understanding the GCD is crucial because it helps reduce fractions to their simplest form. This makes calculations simpler and helps in understanding the relationships between different numbers.
So, always check for the GCD to simplify fractions efficiently.

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Most popular questions from this chapter

Determine if the given alternating series is convergent or divergent.\(\sum_{n=1}^{+\infty}(-1)^{n+1} \sin \frac{\pi}{n}\)

A function \(f\) is defined by a power series. In each exercise do the following: (a) Find the radius of convergence of the given power series and the domain of \(f\); (b) write the power series which defines the function \(f^{\prime}\) and find its radius of convergence by using methods of Sec. \(16.7\) (thus verifying Theorem 16.8.1); (c) find the domain of \(f^{\prime}\).\(f(x)=\sum_{n=1}^{+\infty} \frac{x^{n}}{\sqrt{n}}\)

Find a power-series representation for the given function at the number \(a\) and determine its radius of convergence.\(f(x)=2^{x} ; a=0\)

Find the sum of the given infinite series, accurate to three decimal places.\(\sum_{n=1}^{+\infty}(-1)^{n+1} \frac{1}{n 2^{n}}\)

Determine if the given series is absolutely convergent, conditionally convergent, or divergent. Prove your answer.\(\sum_{n=1}^{+\infty}(-1)^{n+1} \frac{1}{(2 n-1) !}\)
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