What is the Numerical Value of 1/a + 1/b + 1/c ? GMAT Data Sufficiency

Question: What is the numerical value of 1/a + 1/b + 1/c ?

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

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: E

Solution and Explanation
Approach Solution 1:

Any solution based solely on counting equations and unknowns is mathematical not possible. We are unable to count equations and unknowns for three reasons:
One of the equations in this section is non-linear. There are no simple rules for determining the number of solutions to nonlinear equation systems.
There are three unknowns. When there are more than two unknowns, you can't simply count equations and unknowns because you don't know when your equations will be independent.
Because the question does not require us to solve for any of the unknowns, it is unclear what counting equations tell us. We might be able to figure out what 1/a + 1/b + 1/c is without knowing the value of any of our unknowns. If one of the statements said something like ab + ac + bc = abc, that would suffice (since we can rewrite it to get 1/a + 1/b + 1/c = 1).
In this question, counting equations and unknowns tells you almost nothing.

Approach Solution 2:
There is another way to derive that neither of the statement can be used to find the equation-
1/a + 1/b + 1/c is given.
To each term, combine to a single fraction and multiply by abc/abc:
(bc + ac+ ab) / abc = 1/a + 1/b + 1/c
There is no way to get bc / ac / ab from any of the following:

a + b + c = 1
abc = 1

Therefore In this question, solving equations tells you almost nothing.

Approach Solution 3
We have:

a + b + c = 1 ... (i)
abc = 1 ... (ii)

There are 3 unknowns and 2 equations => We cannot solve for a, b and c => We cannot determine the value of (1/a + 1/b + 1/c)
Often, we don't really need to determine the values of a, b, c and still can determine the value of an expression. For example, had the question been to determine the value of (1/ab + 1/bc + 1/ac), we can easily get the answer by combining the statements.

Hence, we need to do some analysis:

Assuming a, b, c are positive and using AM ≥ GM for (i):
(a + b + c)/3 ≥ (a * b * c)^(1/3)
=> (a + b + c)/3 ≥ (a * b * c)^(1/3)
=> (a * b * c)^(1/3) ≤ 1/3
=> abc ≤ 1/27

Clearly, this contradicts with the second statement: abc = 1
This leads to 2 possibilities:

Two of the terms of a, b, c are negative (note - we cannot have only one negative)
OR
a, b, c are complex numbers (beyond the scope of GMAT)
Considering the first case: We can at best simplify as follows:
a = 1 - b - c
Thus, abc = 1 implies: (1 - b - c) * bc = 1
=> (b^2) * c + b * (c^2 - c) + 1 = 0
Which is a quadratic in 'b' and CAN have multiple real solutions => there are many such values of a, b, c possible.
A couple of such values I obtained:
Say a = 2 => b + c = 1 - a = -1 and bc = 1/a = 1/2
=> 1/b + 1/c = (b + c)/bc = -2
=> 1/a + 1/b + 1/c = 1/2 - 2 = -3/2

Say a = 3 => b + c = 1 - a = -2 and bc = 1/a = 1/3
=> 1/b + 1/c = (b + c)/bc = -6
=> 1/a + 1/b + 1/c = 1/3 - 6 = -17/3
Thus, the value of 1/a + 1/b + 1/c is NOT unique - Not sufficient

“What is the numerical value of 1/a + 1/b + 1/c ?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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What is the Numerical Value of 1/a + 1/b + 1/c ? GMAT Data Sufficiency

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