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07 Jan 2024, 09:33
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
63% (01:41) correct
37%
(01:42)
wrong
based on 1040
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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
chetan2u , Can you please help me understand how you derived that highlighted portion ? 111/3 = 37 only. How did yo arrive at .037..that extra 0..Simply by dividing or any other trick ?
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...chetan2u
There has to be a pattern otherwise finding 100th digit would be a herculean task.
If I were to do it, I would try to think about the expression in terms of some more friendly ones.
So, \(\frac{1}{27}=\frac{1}{3}\frac{1}{3}\frac{1}{3}=0.3333333*\frac{1}{3}*\frac{1}{3}\)
\(0.111111*\frac{1}{3}\)
Now 111 add up to 3 when digits are added, so should be divisible by 3.
\(\frac{0.(111)(111)(111)...}{3}=0.037037037....\)
Thus 100th digit will come after 33 sets of these three digits 037.
Hence, 0 is the answer.
If I were to do it, I would try to think about the expression in terms of some more friendly ones.
So, \(\frac{1}{27}=\frac{1}{3}\frac{1}{3}\frac{1}{3}=0.3333333*\frac{1}{3}*\frac{1}{3}\)
\(0.111111*\frac{1}{3}\)
Now 111 add up to 3 when digits are added, so should be divisible by 3.
\(\frac{0.(111)(111)(111)...}{3}=0.037037037....\)
Thus 100th digit will come after 33 sets of these three digits 037.
Hence, 0 is the answer.
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
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
\(\frac{1}{27 }= 0.\overline{0370}\)
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
