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Given a positive number N, when N is rounded by a certain method (for

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Given a positive number N, when N is rounded by a certain method (for  [#permalink]

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New post 21 Sep 2019, 19:14
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Given a positive number N, when N is rounded by a certain method (for convenience, call it Method Y), the result is \(10^n\) if and only if n is an integer and \(5*10^{n − 1} ≤ N < 5*10^n\). In a certain gas sample, there are, when rounded by Method Y, \(10^{21}\) molecules of \(H_2\) and also \(10^{21}\) molecules of \(O_2\). When rounded by Method Y, what is the combined number of \(H_2\) and \(O_2\) molecules in the gas sample?

(1) The number of \(H_2\) molecules and the number of \(O_2\) molecules are each less than \(3*10^{21}\).
(2) The number of \(H_2\) molecules is more than twice the number of \(O_2\) molecules.



DS36141.01
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Re: Given a positive number N, when N is rounded by a certain method (for  [#permalink]

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New post 22 Sep 2019, 07:05
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1
Given a positive number N, when N is rounded by a certain method (for convenience, call it Method Y), the result is \(10^n\) if and only if n is an integer and \(5*10^{n − 1} ≤ N < 5*10^n\). In a certain gas sample, there are, when rounded by Method Y, \(10^{21}\) molecules of \(H_2\) and also \(10^{21}\) molecules of \(O_2\). When rounded by Method Y, what is the combined number of \(H_2\) and \(O_2\) molecules in the gas sample?

This means RANGE of both \(H_2\) and \(O_2\) will be \(5*10{21-1}<10^{21}<5*10^{21}\)

(1) The number of \(H_2\) molecules and the number of \(O_2\) molecules are each less than \(3*10^{21}\).
We know the range of both as \(5*10{21-1}<10^{21}<5*10^{21}\).
If both are less than \(2.5*10^{21}\), answer will be \(10^{21}\) as the TOTAL will always be less than \(5*{21}\).
But if both are, say, \(2.6*10^{21}\), then total is \(5.2^{21}\), which will get rounded off to \(10^{22}\).
Insufficient

(2) The number of \(H_2\) molecules is more than twice the number of \(O_2\) molecules.
Again various possibilities..
If \(H_2\) molecules = \(2.6*10^{21}\) and \(O_2\) molecules=\(1.3^{21}\)....total will become \(3.9^{21}\), rounded off to \(10^{21}\).
But if \(H_2\) molecules = \(4.8*10^{21}\) and \(O_2\) molecules=\(2.4^{21}\)....total will become \(7.2^{21}\), rounded off to \(10^{22}\).
Insuff

Combined..
Let us take the MAX value of \(H_2\) ~ or just less than \(3*10^{21}\), so max value of \(O_2\) will be just less than \(1.5*10^{21}\).
Total = \(4.5*10^{21}\), rounded off to 10^21..
Thus answer will always be \(10^{21}\)

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Re: Given a positive number N, when N is rounded by a certain method (for  [#permalink]

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New post 18 Oct 2019, 15:16
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\(5*10^{20} ≤ N_h < 5*10^{21}\)
\(5*10^{20} ≤ N_o < 5*10^{21}\)

Statement 1-
\(5*10^{20} ≤ N_h < 3*10^{21}\)......(1)

\(5*10^{20} ≤ N_o < 3*10^{21}\)......(2)

Combining (1) and (2), wen can get the range of \(N_h\)+\(N_o\).

\(10^{21} ≤ N_h+N_o < 6*10^{21}\)

Case 1- If \(10^{21} ≤ N_h+N_o < 5*10^{21}\)
\(N_h\)+\(N_o\)is rounded off to \(10^{21}\)

Case 2- \(5*10^{21} ≤ N_h+N_o < 6*10^{21}\)
\(N_h\)+\(N_o\) is rounded off to \(10^{22}\)

Insufficient

Statement 2-
\(10^{21} ≤ N_h < 5*10^{21}\)......(1)

\(0.5*10^{21} ≤ N_o < 2.5*10^{21}\)......(2)

Hence,

\(1.5*10^{21} ≤ N_h+N_o < 7.5*10^{21}\)

Case 1- If \(1.5*10^{21} ≤ N_h+N_o < 5*10^{21}\)
\(N_h\)+\(N_o\) is rounded off to \(10^{21}\)

Case 2- \(5*10^{21} ≤ N_h+N_o < 7.5*10^{21}\)
\(N_h\)+\(N_o\) is rounded off to \(10^{22}\)

Insufficient

Combining both statements

\(10^{21} ≤ N_h < 3*10^{21}\)......(1)

\(0.5*10^{21} ≤ N_o < 1.5*10^{21}\)......(2)


\(1.5*10^{21} ≤ N_h+N_o < 4.5*10^{21}\)

Hence, \(N_h\)+\(N_o\) rounded off to \(10^{21}\)

Sufficient

gmatt1476 wrote:
Given a positive number N, when N is rounded by a certain method (for convenience, call it Method Y), the result is \(10^n\) if and only if n is an integer and \(5*10^{n − 1} ≤ N < 5*10^n\). In a certain gas sample, there are, when rounded by Method Y, \(10^{21}\) molecules of \(H_2\) and also \(10^{21}\) molecules of \(O_2\). When rounded by Method Y, what is the combined number of \(H_2\) and \(O_2\) molecules in the gas sample?

(1) The number of \(H_2\) molecules and the number of \(O_2\) molecules are each less than \(3*10^{21}\).
(2) The number of \(H_2\) molecules is more than twice the number of \(O_2\) molecules.



DS36141.01
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Re: Given a positive number N, when N is rounded by a certain method (for   [#permalink] 18 Oct 2019, 15:16
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