If a b c 0, is c

Question

If a > b > c > 0, is c < 3 ?

(1) 1/a > 1/3 
(2) 1/a + 1/b + 1/c = 1

Bunuel · Accepted Answer

If a > b > c > 0, is c < 3 ?

(1) 1/a > 1/3 --> cross-multiply (notice that we can safely do this since we know that a > 0): a < 3. As c < a, then c < a < 3. Sufficient.

(2) 1/a + 1/b + 1/c = 1. If c is more than or equal to 3, then 1/c is less than or equal to 1/3 (for example, if c is 3, then 1/c is 1/3 and if c is 4, then 1/c is 1/4). In this case both 1/b and 1/a would be less than 1/3. So, 1/a + 1/b + 1/c = (less than 1/3) + (less than 1/3) + (less than or equal to 1/3) = (less than 1), which contradicts the given statement. Therefore our assumption that c could be more than or equal to 3 was wrong, which implies that c < 3. Sufficient.

Answer: D.

BrentGMATPrepNow · Answer

Here's a slightly different approach . . . Target question: Is c < 3? Given: 0 < c < b < a Statement 1 : 1/a > 1/3 Since we can be certain that a is positive, it's safe to take the inequality 1/a > 1/3 and multiply both sides by a to get: 1 > a/3 Likewise, we can take 1 > a/3 and multiply both sides by 3 to get: 3 > a If 3 > a and c < a , then we can conclude that c < 3 Since we can answer the target question with certainty, statement 1 is SUFFICIENT Statement 2 : (1/a) + (1/b) + (1/c) = 1 IMPORTANT: If 0 < c < a , we can conclude that 1/a < 1/c Likewise, since 0 < c < b , we can conclude that 1/b < 1/c In other words, 1/c is BIGGER than both 1/a and 1/b So, if we take the equation (1/a) + (1/b) + (1/c) = 1 and replace both 1/a and 1/b with 1/c, the resulting sum will be BIGGER than 1 That is, (1/c) + (1/c) + (1/c) > 1 Simplify to get: 3/c > 1 Since we know that c is positive, it's safe to take the inequality and multiply both sides by c to get: 3 > c Since we can answer the target question with certainty, statement 2 is SUFFICIENT Answer = D Cheers, Brent

scarish · Answer

IMO....D

(1) 1/a>1/3
Solution: 0 < a < 3
Since a > b > c > 0....and a < 3, then c must be < 3.
Sufficient.

2) 1/a+1/b+1/c=1

1 = 1/3 + 1/3 + 1/3
if C=3 then 1/a < 1/b < 1/3
Hence, the sum of the fraction can never reach 1....

Therefore, c has to be < 3 in order to get the sum of fractions reach 1
Sufficient.

GMAT TIGER · Answer

Thats D. good job for good question. 8-)

tejal777 · Answer

Stmt 2 is not crystal clear..

We have here,a+b+c=abc which is only true when a,b,c=3.However,from the stem we know that a>b>c..hmmm..getting there but missing something...

ichha148 · Answer

humm , for stmt 2 , is there any other way to solve the problem other than one mentioned by scarish

ab+bc+ab = abc , now can i solve from this ?

ashutoshbarawkar · Answer

Can some one explain Stm 2 better than the above explanations

VenoMfTw · Answer

Statement 2 : 1/a + 1/b + 1/c = 1

given a>b>c and all are positive thus == >1/c > 1/b > 1/a --(i)

let me assume values

1/a 1/b 1/c C
0.1 0.2 0.7 10/7 
0.2 0.3 0.5 10/5
0.3 0.4 0.4 ---> This violates the condition--(i)

Thus C value will always be <3 ..

mcwoodhill · Answer

definitely you can get c<3 from the condition: a>b>c>0 so 1/a<1/b<1/c because 1/a+1/b+1/c=1 if you replace 1/a with 1/c, and 1/b with 1/c, you will get a sum bigger than 1: 1/c+1/c+1/c>1/a+1/b+1/c=1 so 3/c>1 so c<3

deepak268 · Answer

a>b>c>0 is c>3? 1) 1/a >1/3 since a is positive we can multiply both sides by a . therefore a <3 33 ...sufficient 2) 1/a+1/b+1/c =1 we have to prove c>3 ... also means 1/c >1/3 ...also means 1/c>33.33% ( since 1/3 is 33.33% for getting rid of fraction, i have a phobia of fractions

) also 1/c>1/b>1/a ( rephrasing given data, it says 1/c should be greatest) now back to st. 2 we can write this as 1/a + 1/b +1/c =100% as we have to prove 1/c >33.33% take a case where 1/c is 30% .. then 35+35+30 =100 ( cannot happen since 1/c should be greatest) 1/c has to greater than equal to 40% in this case (21+39+40) so 1/c >33.33 % sufficient. ans is D i hope i am right

hudhudaa · Answer

Hi,

is it possible to do the following for the 2nd statement:

= 1

a+b+c = abc

this forces us to use numbers for a, b, and c, that will both add and multiply to give the same answer. Is this a viable approach?

Basshead · Answer

If a > b > c > 0 , is c < 3 ? (1) \frac{1}{a} > \frac{1}{3} a < 3. If a < 3, then c < 3. SUFFICIENT. (2) \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1 We are told a > b > c . Thus \frac{1}{c} > \frac{1}{b} > \frac{1}{a} . Suppose we had 3 of c: \frac{3}{c} \frac{3}{c} > 1 3 > c SUFFICIENT. Answer is D.

Kimberly77 · Answer

Great and detailed explanation BrentGMATPrepNow. To clarify St 2, given condition is (1/a) + (1/b) + (1/c) = 1 but calculation result is 3/c > 1. Doesn't this contradict as is not =1 ? Not sure what did I miss? Thanks Brent

BrentGMATPrepNow · Answer

We took the given information, a > b > c > 0, and made two conclusions: 1/a < 1/c and 1/b < 1/c So, when took statement two, 1/a + 1/b + 1/c = 1, and we replaced 1/a with 1/c, and replaced 1/b with 1/c, we were able to say that (1/c) + (1/c) + (1/c) > 1 bigger values ]

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If a > b > c > 0, is c < 3 ?

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