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

[gmattesttaker2]

Hello,

Can you please assist with this:

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

1) 1/a > 1/3

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

OA: D


I tried to solve as follows:

(1/a) > (1/3) => 3 > a
Since a > b > c > 0
=> 3 > c

However, I was not able to solve Statement 2.

Can you please assist?

Thanks,
Sri

[Uva@90]

Hi Sri,

To find:c<3?
As you know statement 1 is sufficient.
Lets take Statement 2: (1/a) + (1/b) + (1/c) = 1
when all the numbers are 3 then,
1/3+1/3+1/3 = 3
So if C > 3 and a>b>c then, 1/c + 1/b + 1/c cant add up to give 1.
hence c<3
Hence Answer is D

Regards,
Uva.

[gmattesttaker2]

Hi Sri,

To find:c<3?
As you know statement 1 is sufficient.
Lets take Statement 2: (1/a) + (1/b) + (1/c) = 1
when all the numbers are 3 then,
1/3+1/3+1/3 = 3
So if C > 3 and a>b>c then, 1/c + 1/b + 1/c cant add up to give 1.
hence c<3
Hence Answer is D

Regards,
Uva.

Hello Uva,

Thanks for your reply. I had a question here:

when all the numbers are 3 then,
1/3+1/3+1/3 = 3

I was just wondering if we can take a = b = c = 3 since the question says a > b > c ?

Thanks for your help.

Best Regards,
Sri

[Uva@90]

Hi Sri,

To find:c<3?
As you know statement 1 is sufficient.
Lets take Statement 2: (1/a) + (1/b) + (1/c) = 1
when all the numbers are 3 then,
1/3+1/3+1/3 = 3
So if C > 3 and a>b>c then, 1/c + 1/b + 1/c cant add up to give 1.
hence c<3
Hence Answer is D

Regards,
Uva.

Hello Uva,

Thanks for your reply. I had a question here:

when all the numbers are 3 then,
1/3+1/3+1/3 = 3

I was just wondering if we can take a = b = c = 3 since the question says a > b > c ?

Thanks for your help.

Best Regards,
Sri

Sri,
Yes you are right, we can't take a=b=c=3.
What I am trying to say in above post is that,

Take 3 cases,
where c=3,c>3 and c<3

When C=3
Yes it is possible to add up 1 only when a=3 and b =3 which is does not satisfy our criteria(a>b>c)
Hence not sufficient.
when c>3
if c > 3 and a>b>c. we can't achieve 1/a+1/b+1/c = 1
Hence not sufficient.
when c<3
it is possible.
Hence C should <3

Regards,
Uva.

[gmattesttaker2]

Hi Sri,

To find:c<3?
As you know statement 1 is sufficient.
Lets take Statement 2: (1/a) + (1/b) + (1/c) = 1
when all the numbers are 3 then,
1/3+1/3+1/3 = 3
So if C > 3 and a>b>c then, 1/c + 1/b + 1/c cant add up to give 1.
hence c<3
Hence Answer is D

Regards,
Uva.

Hello Uva,

Thanks for your reply. I had a question here:

when all the numbers are 3 then,
1/3+1/3+1/3 = 3

I was just wondering if we can take a = b = c = 3 since the question says a > b > c ?

Thanks for your help.

Best Regards,
Sri

Sri,
Yes you are right, we can't take a=b=c=3.
What I am trying to say in above post is that,

Take 3 cases,
where c=3,c>3 and c<3

When C=3
Yes it is possible to add up 1 only when a=3 and b =3 which is does not satisfy our criteria(a>b>c)
Hence not sufficient.
when c>3
if c > 3 and a>b>c. we can't achieve 1/a+1/b+1/c = 1
Hence not sufficient.
when c<3
it is possible.
Hence C should <3

Regards,
Uva.

Hello Uva,

Thank you very much for your detailed and excellent explanation. It is clear now.

Best Regards,
Sri

[Brent@GMATPrepNow]

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

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

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

[gmattesttaker2]

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

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

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

Hello Brent,

I was just wondering what would be an example for values of a, b and c that satisfies Statement 2.

I tried picking some numbers for a that is less than 3 (from Statement 1). But I couldn't find anything that satisfies both a > b > c > 0 and 1/a + 1/b + 1/c = 1

I tried the following as well:

1/a + 1/b + 1/c = 1
=> ( bc + ac + ab )/abc = 1
=> ab + bc + ac = abc

So I was just wondering what would be an example here? Thanks a lot for your help.

Best Regards,
Sri

[Brent@GMATPrepNow]

Hello Brent,

I was just wondering what would be an example for values of a, b and c that satisfies Statement 2.

I tried picking some numbers for a that is less than 3 (from Statement 1). But I couldn't find anything that satisfies both a > b > c > 0 and 1/a + 1/b + 1/c = 1

How about a = 6, b = 3 and c = 2 (satisfies the condition that 0 < c < b < a)
We get 1/6 + 1/3 + 1/2 = 1 (satisfies the condition in statement 2)

orrr..

How about a = 10, b = 5/2 and c = 2 (satisfies the condition that 0 < c < b < a)
We get 1/10 + 2/5 + 1/2 = 1 (satisfies the condition in statement 2)

Cheers,
Brent

[gmattesttaker2]

Hello Brent,

I was just wondering what would be an example for values of a, b and c that satisfies Statement 2.

I tried picking some numbers for a that is less than 3 (from Statement 1). But I couldn't find anything that satisfies both a > b > c > 0 and 1/a + 1/b + 1/c = 1

How about a = 6, b = 3 and c = 2 (satisfies the condition that 0 < c < b < a)
We get 1/6 + 1/3 + 1/2 = 1 (satisfies the condition in statement 2)

orrr..

How about a = 10, b = 5/2 and c = 2 (satisfies the condition that 0 < c < b < a)
We get 1/10 + 2/5 + 1/2 = 1 (satisfies the condition in statement 2)

Cheers,
Brent

Hello Brent,

Thanks for the examples. I was also just wondering if there is any value of a which is less than 3 that will satisfy Statement 2. I was trying to pick a value for a which is less than 3 since Statement 1 says that a < 3. Thanks a lot for all your help.

Best Regards,
Sri

[GMATGuruNY]

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

1) 1/a > 1/3

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

Statement 1: 1/a > 1/3.
Since a>0, we can cross-multiply:
3*1 > a*1
3 > a.
Thus, 3 > a > b > c, implying that c<3.
SUFFICIENT.

Statement 2: 1/a + 1/b + 1/c = 1.
If c≥3, then a > b > c ≥ 3.
Implication:
1/a + 1/b + 1/c = (less than 1/3) + (less than 1/3) + (1/3 or less)
1/a + 1/b + 1/c = less than 1.
Since statement 2 requires that 1/a + 1/b + 1/c = 1, it is not possible that c≥3.
Thus, c<3.
SUFFICIENT.

The correct answer is D.

[Brent@GMATPrepNow]

Hello Brent,

Thanks for the examples. I was also just wondering if there is any value of a which is less than 3 that will satisfy Statement 2. I was trying to pick a value for a which is less than 3 since Statement 1 says that a < 3. Thanks a lot for all your help.

Best Regards,
Sri

Hi Sri,

I see the problem now. The question is not a true GMAT question, because the two given statements contradict each other. The statements in a true Data Sufficiency question will never contradict each other. (for more about this, see our free video: https://www.gmatprepnow.com/module/gmat- ... cy?id=1104)

Here's what I mean:

Statement 1: 1/a > 1/3
This means that a < 3
Since we're told that a > b > c > 0, we can conclude that b < 3 and c < 3
If b and c are both less than 3 (and greater than zero), we can conclude that 1/b > 1/3 and 1/c > 1/3

Now comes the contradiction.
If 1/a > 1/3, 1/b > 1/3 and 1/c > 1/3, then it's IMPOSSIBLE for 1/a + 1/b + 1/c to equal 1 (statement 2).
It's impossible because, if we take three fractions (each of which is GREATER THAN 1/3), the sum cannot equal 1. The sum will be GREATER THAN 1.

This explains why we cannot find values for a, b and c that satisfy both statements.

Cheers,
Brent