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07 Jan 2024, 09:33
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A
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Difficulty:
65%
(hard)
Question Stats:
63% (01:41) correct
37%
(01:42)
wrong
based on 1293
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What is the 100th digit to the right of the decimal point in decimal expression of 1/27?
(A) 0
(B) 2
(C) 3
(D) 6
(E) 7
(A) 0
(B) 2
(C) 3
(D) 6
(E) 7
02 May 2024, 01:10
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sayan640
That's because it's not 0.111 there, it’s \(0.\overline{111}\), or simply \(0.\overline{1}\), which means the 1's there repeat indefinitely: 0.111...
What is the 100th digit to the right of the decimal point in decimal expression of 1/27?
(A) 0
(B) 2
(C) 3
(D) 6
(E) 7
\(\frac{1}{27}=\)
\(=\frac{1}{9}*\frac{1}{3}=\)
\(=0.\overline{111}*\frac{1}{3}=\)
\(=0.\overline{037}\)
(Alternatively, if you recognize that \(\frac{1}{27} = \frac{37}{999}\), you could directly write: \(\frac{1}{27} = \frac{37}{999} = 0.\overline{037}\))
The digits after the decimal repeat in blocks of three (037 - 037 - 037 - ...). Thus, the 99th digit to the right of the decimal point will be 7, and the 100th digit will be 0.
Answer: A.
THEORY:
If you have a fraction where the denominator is 9, 99, 999, etc., the decimal equivalent will have a repeating pattern in the decimal part. The repeating part is just the numerator of the fraction, with enough leading zeroes added to match the number of 9s in the denominator.
Examples:
• \(\frac{2}{9} = 0.2222...\)
• \(\frac{3}{99} = 0.030303...\)
• \(\frac{45}{999} = 0.045045045...\)
Thus, for any fraction \(\frac{n}{999...}\), the decimal representation will be \(0.nnn...\), with the number \(n\) (padded with leading zeroes if necessary) recurring as many times as there are 9s in the denominator.
General Discussion
07 Jan 2024, 13:39
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AnkurGMAT20
\(\frac{1}{27 }= 0.\overline{0370}\)
Therefore to find the \(100^{\text{th}}\) digit to the right of the decimal point 0370 must repeat 25 times. Hence, the \(100^{\text{th}}\) digit to the right of the decimal point is equal to \(0\)
Option A
