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Chapter 11: Problem 44

\(41-46\) Express the number as a ratio of integers. $$6.2 \overline{54}=6.2545454 \ldots$$

Short Answer

Expert verified
The number is \( \frac{344}{55} \).

Step by step solution

01

Express the repeating decimal in fractional form

Let \( x = 6.2545454\ldots \). To manage the repeating part, multiply by 1000 to shift the repeating part: \( 1000x = 6254.5454\ldots \).
02

Eliminate the repeating part

Subtract the equation \( 10x = 62.5454\ldots \) from \( 1000x = 6254.5454\ldots \) to eliminate the repeating decimals. This yields:\[ 1000x - 10x = 6254.5454\ldots - 62.5454\ldots \]\[ 990x = 6192 \]
03

Solve for x

Solve for \( x \) by dividing both sides of the equation by 990:\[ x = \frac{6192}{990} \]
04

Simplify the fraction

Simplify \( \frac{6192}{990} \) by finding the greatest common divisor (GCD) of 6192 and 990. The GCD is 18, so divide both numerator and denominator by 18:\[ \frac{6192}{990} = \frac{344}{55} \]Thus, \( 6.2\overline{54} = \frac{344}{55} \).

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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 a sequence of digits after the decimal point that repeats infinitely. In our example, the repeating decimal is represented as \(6.2\overline{54}\), which means \(6.2545454\ldots\). Here, \(54\) is the repeating part. This concept is vital because repeating decimals can often be expressed as fractions.When dealing with repeating decimals:
  • Identify the repeating sequence of digits.
  • Multiply the number to shift the decimal and align the repeating part.
For instance, in our example, multiplying by \(1000\) helps in aligning the repeating digits for subtraction, allowing us to form an equation without an infinite decimal part.
Fractions
Fractions represent a part of a whole and are expressed as a ratio: a numerator over a denominator. Turning repeating decimals into fractions is a key skill, as it allows us to represent decimals precisely.To convert a repeating decimal into a fraction:
  • Assign a variable to the repeating decimal; for instance, let \(x = 6.2545454\ldots\).
  • Shift the decimal to the right by multiplying by a power of 10, such that the entire repeating sequence is aligned just after the decimal point.
  • Align the decimal parts such that subtraction can eliminate the repeating section. This helps in turning the decimal into an algebraic equation which can then be solved to find the fraction.
After setting up the equation and subtracting, you will get a fraction of the form \(\frac{a}{b}\), where both \(a\) and \(b\) are integers.
Simplifying Fractions
Simplifying fractions is about reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This step is crucial for clarity and efficiency in mathematical expressions.To simplify a fraction:
  • Find the greatest common divisor (GCD) of the numerator and denominator.
  • Divide both the numerator and denominator by the GCD.
In our example, the fraction \(\frac{6192}{990}\) was simplified by dividing the numerator and denominator by 18, the GCD, resulting in \(\frac{344}{55}\).Simplification helps in making calculations easier and the fraction easier to understand.

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Most popular questions from this chapter

Use a computer algebra system to find the Taylor polynomials \(T_{n}\) centered at \(a\) for \(n=2,3,4,5 .\) Then graph these polynomials and \(f\) on the same screen. $$f(x)=\sqrt[3]{1+x^{2}}, \quad a=0$$

A car is moving with speed 20 \(\mathrm{m} / \mathrm{s}\) and acceleration 2 \(\mathrm{m} / \mathrm{s}^{2}\) a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled during the next minute?

Test the series for convergence or divergence. $$\sum_{n=1}^{\infty} \frac{n !}{e^{n^{2}}}$$

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. \(y=e^{x} \ln (1-x)\)

(a) Let \(a_{1}=a, a_{2}=f(a), a_{3}=f\left(a_{2}\right)=f(f(a)), \ldots\) \(a_{n+1}=f\left(a_{n}\right),\) where \(f\) is a continuous function. If \(\lim _{n \rightarrow \infty} a_{n}=L,\) show that \(f(L)=L\) (b) Illustrate part (a) by taking \(f(x)=\cos x, a=1,\) and estimating the value of \(L\) to five decimal places.
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