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# If a is a positive number less than 10, is c greater than

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

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02 Dec 2010, 03:27
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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.
[Reveal] Spoiler: OA

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Re: DS Algebra [#permalink]

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02 Dec 2010, 03:43
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rxs0005 wrote:
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.

Similar questions:
number-line-problem-22709.html
x-is-a-positive-number-less-than-86563.html
600-level-question-95138.html?hilit=halfway%20between%20greater%20closer

Hope it helps.
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Re: DS Algebra [#permalink]

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05 Dec 2010, 07:08
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
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Re: DS Algebra [#permalink]

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05 Dec 2010, 07:23
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anilnandyala wrote:
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.

Hope it's clear.
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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.
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Re: If a is a positive number less than 10, is c greater than [#permalink]

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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 ?
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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, 13:48
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.

Hope this helps.
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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, 16:10
yes. Now it is clear. great thanks Engr2012.
Re: If a is a positive number less than 10, is c greater than   [#permalink] 15 Jan 2016, 16:10
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# If a is a positive number less than 10, is c greater than

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