/*! 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 19 Express $$ 0.23232323 \ldots... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 19

Express $$ 0.23232323 \ldots $$ as a fraction; here the digits 23 repeat forever.

Short Answer

Expert verified
The fraction representation of the repeating decimal \(0.232323 \ldots\) is \(\frac{23}{99}\).

Step by step solution

01

Define the variable x

Let x be the repeating decimal \(0.232323 \ldots \), which we want to express as a fraction.
02

Multiply by 100

To isolate the repeating part of the decimal, multiply x by 100. This will give us a new equation: \(100x = 23.2323 \ldots \)
03

Subtract the original number

Subtract the original equation (step 1) from the new equation (step 2) to eliminate the repeating decimal portion: $$ (100x - x) = (23.2323 \ldots - 0.2323 \ldots) \\ 99x = 23 $$
04

Solve for x

Divide both sides of the equation by 99 to solve for x: $$ x = \frac{23}{99} $$ Therefore, the fraction representation of the repeating decimal \(0.232323 \ldots\) is \(\frac{23}{99}\).

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Most popular questions from this chapter

Find all infinite sequences that are both arithmetic and geometric sequences.

Define a sequence recursively by \(a_{1}=3 \quad\) and \(\quad a_{n+1}=\frac{1}{2}\left(\frac{7}{a_{n}}+a_{n}\right)\) for \(n \geq 1 .\) Find the smallest value of \(n\) such that \(a_{n}\) agrees with \(\sqrt{7}\) for at least six digits after the decimal point.

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Evaluate \(\lim _{n \rightarrow \infty} \frac{2 n^{2}+5 n+1}{5 n^{2}-6 n+3}\).

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