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Curly brackets {} around the last digits of a decimal fraction signify

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Curly brackets {} around the last digits of a decimal fraction signify  [#permalink]

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New post 27 Nov 2016, 01:57
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Question Stats:

57% (00:51) correct 43% (01:10) wrong based on 94 sessions

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Curly brackets {} around the last digits of a decimal fraction signify that these digits recur an infinite number of times. For example 0.1{5} is a shorthand for 0.1555555… Similarly, 0.{15} is a shorthand for 0.15151515…. 10/99 is which of the following?

A. 0.{11}
B. 0.{101}
C. 0.{10}
D. 0.{09}
E. 0.{01}

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Curly brackets {} around the last digits of a decimal fraction signify  [#permalink]

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New post 27 Nov 2016, 03:12
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Bunuel wrote:
Curly brackets {} around the last digits of a decimal fraction signify that these digits recur an infinite number of times. For example 0.1{5} is a shorthand for 0.1555555… Similarly, 0.{15} is a shorthand for 0.15151515…. 10/99 is which of the following?

A. 0.{11}
B. 0.{101}
C. 0.{10}
D. 0.{09}
E. 0.{01}


\(10/99=0.1010101010...=0.\{10\}\)

Also, there is a way to convert recurring decimal into a fraction.

Given \(a_1,a_2,...\) are digits

\(\overline{0.\{a_1\}}=\frac{a_1}{9}\)
\(\overline{0.\{a_1 a_2\}}=\frac{\overline{a_1 a_2}}{99}\)
\(\overline{0.\{a_1 a_2 ... a_n\}}=\frac{\overline{a_1 a_2 ... a_n}}{99...9}=\frac{\overline{a_1 a_2 ... a_n}}{10^n-1}\)

For example, \(0.\{1\} = \frac{1}{9}\)
\(0.\{3\}=\frac{3}{9}=\frac{1}{3}\)
\(0.\{10\}=\frac{10}{99}\)
\(0.\{12345\}=\frac{12345}{99999}\)

\(0.\{11\}=\frac{11}{99}=\frac{1}{9}\). Actually \(0.\{11\}=0.\{1\}\)
\(0.\{101\}=\frac{101}{999}\)
\(0.\{09\}=\frac{9}{99}=\frac{1}{11}\)
\(0.\{01\}=\frac{1}{99}\)

What about the number like \(0.12\{3\}\)?

Simply, \(0.12\{3\}=10^{-2} \times 12.\{3\}=10^{-2} \times 12\frac{3}{9}=10^{-2} \times 12\frac{1}{3}=10^{-2} \times \frac{37}{3}=\frac{37}{300}\)
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Re: Curly brackets {} around the last digits of a decimal fraction signify  [#permalink]

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New post 27 Nov 2016, 04:28
Bunuel wrote:
Curly brackets {} around the last digits of a decimal fraction signify that these digits recur an infinite number of times. For example 0.1{5} is a shorthand for 0.1555555… Similarly, 0.{15} is a shorthand for 0.15151515…. 10/99 is which of the following?

A. 0.{11}
B. 0.{101}
C. 0.{10}
D. 0.{09}
E. 0.{01}



IMO C

0.101010=x
10.1010= 100x
99x= 10
x=10/99
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Re: Curly brackets {} around the last digits of a decimal fraction signify  [#permalink]

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New post 22 Feb 2018, 09:10
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Re: Curly brackets {} around the last digits of a decimal fraction signify   [#permalink] 22 Feb 2018, 09:10
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