Textbook Question
In Exercises 49–52, a single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 2 the second time.
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 9, Problem 50
Express each repeating decimal as a fraction in lowest terms. 0.6 (repeating 6)
Verified step by step guidance1
Let x = 0.666... (where the digit 6 repeats indefinitely).
Multiply both sides of the equation by 10 to shift the decimal point one place to the right: 10x = 6.666....
Subtract the original equation (x = 0.666...) from the new equation (10x = 6.666...): 10x - x = 6.666... - 0.666....
Simplify the subtraction: 9x = 6.
Solve for x by dividing both sides of the equation by 9: x = 6/9. Simplify the fraction to its lowest terms by dividing the numerator and denominator by their greatest common divisor (GCD), which is 3, resulting in x = 2/3.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Repeating Decimals
Repeating decimals are decimal numbers in which a digit or a group of digits repeats infinitely. For example, 0.666... can be expressed as 0.6 with a bar over the 6 to indicate that it repeats. Understanding how to identify and represent repeating decimals is crucial for converting them into fractions.
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Conversion of Decimals to Fractions
To convert a repeating decimal to a fraction, one typically sets the decimal equal to a variable, multiplies by a power of 10 to shift the decimal point, and then subtracts the original equation from this new equation. This process eliminates the repeating part, allowing for the fraction to be derived. Simplifying the resulting fraction to its lowest terms is also essential.
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Lowest Terms
A fraction is in lowest terms when the numerator and denominator have no common factors other than 1. To simplify a fraction, one must divide both the numerator and denominator by their greatest common divisor (GCD). This concept is important to ensure that the final answer is presented in its simplest form, making it easier to understand and use.
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Related Practice
Textbook Question
Use the Binomial Theorem to expand each expression and write the result in simplified form. (x1/3 +x-1/3)3
Textbook Question
Use the formula for nCr to solve Exercises 49–56. An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?
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Textbook Question
Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1/2+2/3+3/4+⋯+ 14/(14+1)
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Textbook Question
Use the formula for nCr to solve Exercises 49–56. Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?
Textbook Question
In Exercises 49–52, a single die is rolled twice. Find the probability of rolling a 2 the first time and a 3 the second time.
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