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# M27-17

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 00:27
1
9
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Difficulty:

85% (hard)

Question Stats:

46% (02:10) correct 54% (01:45) wrong based on 178 sessions

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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) $$a-c \gt 6$$

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16 Sep 2014, 00:27
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Official Solution:

First of all, $$\frac{7}{9}$$ is a recurring decimal $$=0.77(7)$$.

(1) $$a+b \gt 14$$. The least value of $$a$$ is 6 ($$6+9=15 \gt 14$$), so in this case, $$x=0.69d \lt 0.77(7)$$, but $$a=7$$ and $$b=9$$ is also possible, and in this case $$x=0.79d \gt 0.77(7)$$. Not sufficient.

(2) $$a-c \gt 6$$. The least value of $$a$$ is 7 ($$7-0=7 \gt 6$$), but we don't know the value of $$b$$. Not sufficient.

(1)+(2) The least value of $$a$$ is 7 and in this case from (1) the least value of $$b$$ is 8 ($$7+8=15 \gt 14$$), hence the least value of $$x=0.78d \gt 0.77(7)$$. Sufficient.

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03 Oct 2017, 00:27
Hi Bunuel

Thank you for the explanation.

I am not able to follow when we combine the two statements.

For S1 a+b>14 , Can we not have 9+6=15 and in this case the least value of b will be 6 and we get 0.76 which is less than 0.77 and answer is E.

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03 Oct 2017, 00:36
KB04 wrote:
Hi Bunuel

Thank you for the explanation.

I am not able to follow when we combine the two statements.

For S1 a+b>14 , Can we not have 9+6=15 and in this case the least value of b will be 6 and we get 0.76 which is less than 0.77 and answer is E.

When we combine the statements we get that the least value of a is 7, so it can be 7, 8 or 9. If a = 9, as it is in your example, then x = 0.9... not 0.76.., so x > 0.777...
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12 Jul 2018, 17:17
Here is my approach

Question is x > 7/9 which is 0.777..

(1). Given a+b>14

Only possible values are (Please note the max value is 9 for any digit)
9+6=15>14 --> 0.96.. (Greater than 0.77.., Yes)
8+7=15>14 --> 0.87..(greater than 0.77.., yes)
7+8=15>14 --> 0.78..(greater than 0.77.., Yes)
6+9=15>14 --> 0.69..(Lesser than 0.77.., NO)
Thus Statement 1 is not sufficient.

(2). Given a-b >6
similar way
9-2=7>6 --> 0.9b2d (greater than 0.7...,Yes)
8-1=7>6 --> 0.8b1d (greater than 0.7...,Yes)
7-0=7>6 --> 0.7b0d (lesser than 0.7777, NO)
Thus statement 2 is not sufficient.

(1) and (2) >> Given a+b>14 and a-c>6
Overlapping both, we can take a=6, as it will not satisfy (1) and (2) together.

a=7 and then b must be 8 from above.

so x=0.78cd >0.7777 Thus (1) and (2) are sufficient.

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Joined: 07 Feb 2017
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12 Jul 2018, 18:51
Is x>0.77777777 ?

(1) a=7; b=8
x=0.78 yes
a=6; b=9
x=0.69 no
Insufficient

(2) a>c+6
c=0; a=7
x=0.700d no
x=0.780d yes
Insufficient

(1) and (2) 7 is min value of a
x=0.7b0d
then b>=8
yes

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Joined: 12 Mar 2018
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Location: Iran (Islamic Republic of)
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01 Oct 2018, 08:06
Very intersting... thank you Bunuel
Re: M27-17 &nbs [#permalink] 01 Oct 2018, 08:06
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# M27-17

Moderators: chetan2u, Bunuel

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