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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]

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24 Jan 2014, 00:28

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]

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14 Jan 2016, 04:51

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Re: If a is a positive number less than 10, is c greater than [#permalink]

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15 Jan 2016, 13:39

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]

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15 Jan 2016, 13:48

Expert's post

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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