If abc ≠ 0 is (a/b)/c = a/(b/c) GMAT Data Sufficiency

Question: If abc ≠ 0, 

\(\)\(\frac{a}{\frac{b}{c}}=\frac{a}{\frac{b}{c}} \)

(1) a = 1

(2) c = 1

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

4. EACH statement ALONE is sufficient.

5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: B

Solution and Explanation 
Approach Solution 1:

It's easy to answer the following equation and the algebraic method can also be used in this situation. This can be explored in the following manner:

abc ≠ 0, is \(\frac{a}{\frac{b}{c}}=\frac{a}{\frac{b}{c}} \)

\(\frac{a}{\frac{b}{c}}= \frac{a}{b}x\frac{1}{c}=\frac{a}{bc} \)

\(\frac{a}{\frac{b}{c}}= a x \frac{c}{b}=\frac{ac}{b} \)

Now let's simplify the equation, so now the question becomes

\(\frac{a}{bc}=\frac{ac}{b} \)

Now, lets reduce this above equation even further by \(\frac{a}{b} \)

The question becomes,

\(\frac{1}{c}=c \)

Further simplifying the question becomes
\(c^2=1\)

a = 1. Not sufficient.
c = 1. Sufficient.

Approach Solution 2:

Let's try to answer the following equation by Cross multiplying method. This can be explored in the following manner:

We need to see if \(\frac{a}{\frac{b}{c}}=\frac{a}{\frac{b}{c}} \)
On Cross multiplying, the above equation we get

\(\frac{a}{b}x\frac{b}{c}=axc \)

\(\frac{a}{c}=c \)

\(\frac{1}{c}=c \)

we need to know only the value of c to determine whether \(\frac{1}{c}=c \)

Now looking at statement 1
Since the value of c is unknown, statement one alone is insufficient.
looking at statement 2
Since its known c = 1,

we have \(\frac{1}{c}=c \).

Statement two alone is sufficient.

Approach Solution 3:
If abc≠0, is a/b/c=a/b/c?

a/b/c=a/b∗1/c= a/bc

a/bc=a∗c/b=ac/b

So, the question becomes: is a/bc=ac/b? --> reduce by a/b: is 1/c=c? --> is c^2=1.

(1) a = 1. Not sufficient.
(2) c = 1. Sufficient.

“If abc ≠ 0 is (a/b)/c = a/(b/c)”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "The Official Guide for GMAT Quantitive 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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