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If a is a positive number less than 10, is c greater than the average (arithmetic mean) of a and 10?
(1) On the number line, c is closer to 10 than it is to a. (2) 2c – 10 is greater than a.
Given: \(0<a<10\). Question: is \(c\) greater than the average (arithmetic mean) of \(a\) and 10? --> or is \(c>\frac{a+10}{2}=average\)? --> or is \(2c>a+10\)?
(1) On the number line, c is closer to 10 than it is to a.
Number line approach: a-----average-----10----- (average of a and 10 is halfway between a and 10). So the question ask whether c is either in the BLUE or GREEN area.
As, c is closer to 10 than it (c) is to a then this statement directly tells us that c is either in the BLUE or GREEN area. Sufficient.
Algebraic approach: c is closer to 10 than it is to a, means that the distance between c and 10 is less than the distance between c and a. So, \(|10-c|<|c-a|\). Now, as c is closer to 10 than it is to a, then c>a, so \(|c-a|=c-a\) --> two cases for 10-z:
A. \(c\leq{10}\) --> \(|10-c|=10-c\) --> \(|10-c|<|c-a|\) becomes: \(10-c<c-a\) --> \(2c>10+a\). Answer to the question YES.
B. \(c>{10}\) --> in this case \(2c>20\) and as \(a<10\), then \(a+10<20\), hence \(2c>10+a\). Answer to the question YES.
(2) 2c – 10 is greater than a --> \(2c-10>a\) --> \(c>\frac{a+10}{2}=average\), again directly tells us that c is greater than the average of a and 10. Sufficient.
q is c > (a+10)/2 from statement 1 c is closer to 10 than a let us take a=9.4 c= 9.5 in this case c < (a+10)/2 if a=7 c=9 then c > (a+10)/2
so statement 1 not sufficient can some one explain is there any wrong in this
Statement (1) says: on the number line, c is closer to 10 than it is to a --> means that the distance between c and 10 is less than the distance between c and a.
Now, your example \(a=9.4\) and \(c=9.5\) is not valid as in this case \(c\) is obviously closer to \(a\) than to 10 (c-a=0.1 and 10-c=0.6).
There are 2 different approaches in my previous post shoving why is this statement sufficient.
Re: If a is a positive number less than 10, is c greater than [#permalink]
23 Jan 2014, 23:28
I did it with numbers.
From (1) I know that C is closer to 10 than to a. Pluggin numbers gives me e.g. c= 9, a = 7 average = 8,5 so true....continuing I figured that since c is ALWAYS closer (even if you take 9.99995) to 10, it will be always greater than the average. SUFF.
(2) 2c -10 > a --> c > 10 +a --> c > (10+a)/2 which is the average. SUFF. Also: Since a is < 10, C is at least 10. which would give us the same answer.
Re: If a is a positive number less than 10, is c greater than [#permalink]
14 Jan 2016, 03:51
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Re: If a is a positive number less than 10, is c greater than [#permalink]
15 Jan 2016, 12:39
Hi Bunuel. Actually, i understand your logic in algebraic approach in statement 1 while i can not as you mentioned that such statement tells us directly that c (either) in green area or red area. Isn't "either" in a DS Q. means that it has two solutions and is insufficient ?
Re: If a is a positive number less than 10, is c greater than [#permalink]
15 Jan 2016, 12:48
hatemnag wrote:
Hi Bunuel. Actually, i understand your logic in algebraic approach in statement 1 while i can not as you mentioned that such statement tells us directly that c (either) in green area or red area. Isn't "either" in a DS Q. means that it has two solutions and is insufficient ?
You are confusing getting "either" in the form of 2 differnet answers for the same statements and have 2 cases for the same statement that give you the same answer.
Example, if the question is " what is the value of x?
Statement 1 tells you that x is either 1 or 2, then in this case the statement is NOT sufficient.
BUT
if the question asks " is x>0?"
Statement 1 tells you that x is either 1 or 2, then in this case the statement is sufficient as for x=1, you get "YES" for the question asked, similar to the case when x=2. FYI, if you ended with different values of negative values of 'x' , even then this statement sould have been SUFFICIENT, as you would have obtained a "NO" for all possible values of 'x'.
Thus, a statement or a combination of statements is SUFFICIENT if and only if you get 1 UNIQUE/UNAMBIGUOUS answer.
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