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

Question

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.

Bunuel · Accepted Answer

Given: 0 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 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. Answer: D. Similar questions to practice: https://gmatclub.com/forum/if-x-is-a-po ... 67734.html (OG) https://gmatclub.com/forum/if-x-is-a-po ... 28086.html https://gmatclub.com/forum/if-y-is-a-ne ... 95138.html https://gmatclub.com/forum/if-x-is-a-po ... 31322.html https://gmatclub.com/forum/if-x-and-z-a ... 55651.html Hope it helps.

anilnandyala · Answer

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

Bunuel · Answer

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.

unceldolan · Answer

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.

hatemnag · Answer

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 ?

ENGRTOMBA2018 · Answer

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.

hatemnag · Answer

yes. Now it is clear. great thanks Engr2012.

varunsan97 · Answer

It's pretty much straight forward to look at it this way Easier and faster approach 1) If C is less than a it is c ---- a ----- avg --- 10 which gives us a result of c< avg(a,10) YES If C is slightly greater than a ( since it is closer to a ) on number line it'll be a--c----avg---10 still we have Ca => c>a+10/2 c is greater than average YES 2 is sufficient Hence option D

MathRevolution · Answer

Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

DS question with 2 variables and 1 Equation: Let the original condition in a DS question contain 2 variables and 1 Equation. Now, 2 variables and 1 Equation would generally require 1 more equation to give us the value of the variables.

We know that each condition would usually give us an equation, and since we need 1 equation to match the number of variables and equations in the original condition, the equal number of equations and variables should logically lead to answer D.

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find whether 'c > \(\frac{(a + 10) }{ 2}\)'.

=> Given that 'a' is a positive number less than '10' => 0 < a < 10

Second and the third step of Variable Approach: From the original condition, we have 2 variables (a and c) and 1 Equation (0 < a < 10). To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Let’s take a look at each condition.

Condition(1) tells us that on the number line, c is closer to 10 than it is to a.

=> Let a = 9 then c = (9 + 10) / 2 = 9.5 - Is c > \(\frac{(a + 10) }{ 2}\) - YES

=> Let a = 5 then c = (5 + 10) / 2 = 7.5- Is c > \(\frac{(a + 10) }{ 2}\) - YES

Since the answer is unique YES, the condition is sufficient by CMT 1.

Condition(2) tells us that 2c -10 > a.

=> 2c > a + 10

=> c > \(\frac{(a + 10) }{ 2}\) - Is c > \(\frac{(a + 10) }{ 2}\) - YES

Since the answer is unique YES, the condition is sufficient by CMT 1.

Both conditions (1) and (2) alone are sufficient.

So, D is the correct answer.

Answer: D

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

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