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M27-17
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16 Sep 2014, 01:27
16 Sep 2014, 01:27
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A
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C
D
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Difficulty:
85%
(hard)
Question Stats:
51% (02:13) correct
49%
(02:11)
wrong
based on 158
sessions
If \(x=0.abcd\), where \(a\), \(b\), \(c\), and \(d\) are digits from 0 to 9, inclusive, is \(x \gt \frac{7}{9}\)?
(1) \(a+b \gt 14\)
(2) \(a-c \gt 6\)
(1) \(a+b \gt 14\)
(2) \(a-c \gt 6\)
Re M27-17
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16 Sep 2014, 01:27
16 Sep 2014, 01:27
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Expert Reply
Official Solution:
If \(x=0.abcd\), where \(a\), \(b\), \(c\), and \(d\) are digits from 0 to 9, inclusive, is \(x \gt \frac{7}{9}\)?
First of all, \(\frac{7}{9}\) is a recurring decimal and can be expressed as \(0.\overline{7}\) (a bar over a sequence of digits in a decimal indicates that the sequence repeats indefinitely, so \(0.\overline{7}=0.777...\)).
(1) \(a+b \gt 14\).
The least value of \(a\) that satisfies this condition is 6 (since \(6+9=15 \gt 14\)), so in this case, \(x=0.69d \lt 0.\overline{7}\). However, \(a=7\) and \(b=9\) is also possible, and in that case \(x=0.79d \gt 0.\overline{7}\). Not sufficient.
(2) \(a-c \gt 6\).
The least value of \(a\) that satisfies this condition is 7 (since \(7-0=7 \gt 6\)), but we don't know the value of \(b\). Not sufficient.
(1)+(2) The least value of \(a\) that satisfies both conditions is 7, and in this case, from (1), the least value of \(b\) is 8 (since \(7+8=15 \gt 14\)). Hence, the least value of \(x\) is \(0.78d \gt 0.\overline{7}\). Sufficient.
Answer: C
If \(x=0.abcd\), where \(a\), \(b\), \(c\), and \(d\) are digits from 0 to 9, inclusive, is \(x \gt \frac{7}{9}\)?
First of all, \(\frac{7}{9}\) is a recurring decimal and can be expressed as \(0.\overline{7}\) (a bar over a sequence of digits in a decimal indicates that the sequence repeats indefinitely, so \(0.\overline{7}=0.777...\)).
(1) \(a+b \gt 14\).
The least value of \(a\) that satisfies this condition is 6 (since \(6+9=15 \gt 14\)), so in this case, \(x=0.69d \lt 0.\overline{7}\). However, \(a=7\) and \(b=9\) is also possible, and in that case \(x=0.79d \gt 0.\overline{7}\). Not sufficient.
(2) \(a-c \gt 6\).
The least value of \(a\) that satisfies this condition is 7 (since \(7-0=7 \gt 6\)), but we don't know the value of \(b\). Not sufficient.
(1)+(2) The least value of \(a\) that satisfies both conditions is 7, and in this case, from (1), the least value of \(b\) is 8 (since \(7+8=15 \gt 14\)). Hence, the least value of \(x\) is \(0.78d \gt 0.\overline{7}\). Sufficient.
Answer: C
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