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16 Sep 2014, 01:17
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If \(a=0.xyz\), where \(x\), \(y\), and \(z\) are digits from 0 to 9, inclusive, is \(a \gt \frac{2}{3}\)?
(1) \(x + y \gt 13\)
(2) \(x + z \gt 14\)
(1) \(x + y \gt 13\)
(2) \(x + z \gt 14\)
16 Sep 2014, 01:17
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Official Solution:
If \(a=0.xyz\), where \(x\), \(y\), and \(z\) are digits from 0 to 9, inclusive, is \(a \gt \frac{2}{3}\)?
First of all, \(\frac{2}{3}\) is a recurring decimal and can be expressed as \(0.\overline{6}\) (a bar over a sequence of digits in a decimal indicates that the sequence repeats indefinitely, so \(\frac{2}{3}=0.666...=0.\overline{6}\)).
(1) \(x+y \gt 13\).
The least value of \(x\) that satisfies this condition is 5 (since \(5+9=14 \gt 13\)), so in this case, \(a=0.59z \lt 0.\overline{6}\). However, \(x=7\) and \(y=9\) is also possible, and in that case \(a=0.79d \gt 0.\overline{6}\). Not sufficient.
(2) \(x+z \gt 14\).
The least value of \(x\) that satisfies this condition is 6 (since \(6+9=15 \gt 14\)), but we don't know the value of \(y\). Not sufficient.
(1)+(2) The least value of \(x\) that satisfies both conditions is 6, and in this case, from (1), the least value of \(y\) is 8 (since \(6+8=14 \gt 13\)). Hence, the least value of \(a\) is \(0.68z \gt 0.\overline{6}\). Sufficient.
Answer: C
If \(a=0.xyz\), where \(x\), \(y\), and \(z\) are digits from 0 to 9, inclusive, is \(a \gt \frac{2}{3}\)?
First of all, \(\frac{2}{3}\) is a recurring decimal and can be expressed as \(0.\overline{6}\) (a bar over a sequence of digits in a decimal indicates that the sequence repeats indefinitely, so \(\frac{2}{3}=0.666...=0.\overline{6}\)).
(1) \(x+y \gt 13\).
The least value of \(x\) that satisfies this condition is 5 (since \(5+9=14 \gt 13\)), so in this case, \(a=0.59z \lt 0.\overline{6}\). However, \(x=7\) and \(y=9\) is also possible, and in that case \(a=0.79d \gt 0.\overline{6}\). Not sufficient.
(2) \(x+z \gt 14\).
The least value of \(x\) that satisfies this condition is 6 (since \(6+9=15 \gt 14\)), but we don't know the value of \(y\). Not sufficient.
(1)+(2) The least value of \(x\) that satisfies both conditions is 6, and in this case, from (1), the least value of \(y\) is 8 (since \(6+8=14 \gt 13\)). Hence, the least value of \(a\) is \(0.68z \gt 0.\overline{6}\). Sufficient.
Answer: C
14 Aug 2023, 07:41
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
