/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Find the fractions equal to the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 19: Problem 20

Find the fractions equal to the given decimals. $$0.070707 \ldots$$

Short Answer

Expert verified
0.070707... equals the fraction \( \frac{7}{99} \).

Step by step solution

01

Identify the Repeating Decimal

The decimal is given as 0.070707... The repeating part of the decimal is 07. This means the decimal 0.070707... is a repeating decimal with block 07.
02

Set Up the Equation for the Repeating Decimal

Let the repeating decimal be represented as a variable, say \( x = 0.070707\ldots \). The repeating part has two digits (07), hence we will multiply \( x \) by 100, because 10 squared is 100.
03

Form the Equation by Subtracting

Calculate \( 100x = 7.070707\ldots \). Now subtract the original \( x = 0.070707\ldots \) from this equation to eliminate the repeating part: \( 100x - x = 7.070707\ldots - 0.070707\ldots \).
04

Simplify the Equation

The subtraction yields: \( 99x = 7 \).
05

Solve for x

Divide both sides of \( 99x = 7 \) by 99 to isolate \( x \): \( x = \frac{7}{99} \).
06

Conclusion

The decimal 0.070707... can be expressed as the fraction \( \frac{7}{99} \). This fraction is already in simplest form as 7 is a prime number and does not divide 99 evenly.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Fraction Conversion
When faced with a repeating decimal, the goal is to convert it into a simple fraction. This allows for a clearer representation of the number. It might seem a bit tricky at first, but it's quite systematic once you understand the steps.
  • Start by identifying the repeating sequence. For example, in the decimal 0.070707..., the repeating part is '07'.
  • Next, assign a variable like \( x \) to represent the decimal. Here, \( x = 0.070707... \).
  • Decide how many digits are in the repeating block. If there are two digits, you will multiply \( x \) by 100 (because \( 10^2 = 100 \)), effectively shifting the decimal to remove the repeats temporarily.
  • Use basic algebra to set up an equation where the repeating parts cancel out, allowing you to solve for \( x \).
With this understanding, the transformation from a repeating decimal to a fraction can be achieved systematically.
Decimal Representation
Decimals have two major types: terminating and repeating. A repeating decimal shows a sequence that endlessly recurs, such as 0.070707..., where '07' is the repeating part.
  • Repeating decimals can often be identified by an overline, such as \( 0.07\overline{07} \) or by simply analyzing the sequence presented.
  • The block of repeating numbers sets the stage for how we shift and manipulate the decimal during conversion to a fraction.
Decimals like these may seem complex, but grasping the concept of repetition can simplify understanding and ultimately make it easier to handle in mathematical equations.
Simplifying Fractions
Once you've converted a repeating decimal into a fraction, you might need to simplify it. Simplifying is the process of reducing the fraction to its smallest numerator and denominator without changing its value.
  • For example, after solving the equation from a repeating decimal, you may find \( \frac{7}{99} \). To check if it's in simplest form, find the greatest common divisor (GCD) of the numerator and the denominator.
  • If the GCD is 1 (meaning the numbers are coprime), the fraction is already simplified.
  • If not, divide both the numerator and the denominator by their GCD.
In our case, since 7 is a prime number and doesn't divide 99, \( \frac{7}{99} \) is automatically in its simplest form. Simplifying makes fractions more manageable and easier to integrate into further calculations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the indicated quantities for the appropriate arithmetic sequence. Show that the sum of the first \(n\) positive odd integers is \(n^{2}\)

Find the first four terms of the indicated expansions by use of the binomial series. $$(1-3 x)^{-2}$$

Find the indicated quantities.A professional baseball player is offered a contract for an annual salary of \(\$ 5,000,000 dollars for six years. Also offered is a bonus (based on performance) of either \)\$ 400,000\( each year, or a \)5.00 \%$ increase in salary each year. Which bonus option pays more over the term of the contract, and how much more?

Find the indicated quantities for the appropriate arithmetic sequence. At a logging camp, 15 layers of logs are so piled that there are 20 logs in the bottom layer, and each layer has 1 less log than the layer below it. How many logs are in the pile?

Find the indicated quantities for the appropriate arithmetic sequence. If \(x, 5,\) and \(x+8\) are the first three terms of an arithmetic sequence, find the sum of the first 20 terms.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.