If abc ≠ 0 is (a/b)/c = a/(b/c)

SOLUTION

If \(abc\neq{0}\), is \(\frac{\frac{a}{b}}{c}=\frac{a}{\frac{b}{c}}\)?

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


\(\frac{a}{\frac{b}{c}}=a*\frac{c}{b}=\frac{ac}{b}\).

So, the question becomes: is \(\frac{a}{bc}=\frac{ac}{b}\)? --> reduce by a/b: is \(\frac{1}{c}=c\)? --> is \(c^2=1\).

(1) a = 1. Not sufficient.

(2) c = 1. Sufficient.

Answer: B.
_________________

I reduced the equation but did not cancel variables in the end, instead I put a=1, so I got this:

a/bc = ac/b

if a=1 (given) we get,

1/bc = c/b
which can never be equal to the question stem, hence sufficient?

I guess I didnt interpret the question correctly then

Solution:

Question Stem Analysis:

We need to determine whether (a/b)/c = a/(b/c). Cross multiplying, we have:

(a/b) * (b/c) = a * c

a/c = ac

1/c = c

That is, we need to determine whether 1/c = c. In other words, we need to know only the value of c to determine whether 1/c = c.

Statement One Alone:

Since we don’t know the value of c, statement one alone is not sufficient.

Statement Two Alone:

Since c = 1, we have 1/c = c. Statement two alone is sufficient.

Answer: B
_________________

I found this confusing so I just used some numbers to see the relationship
Say a=1, b=2, c=3

\(\frac{\frac{1}{2}}{3}=\frac{1}{\frac{2}{3}}\)?

\(\frac{\frac{1}{2}}{3}=\frac{1}{2}*\frac{1}{3}=\frac{1}{6}\);

\(\frac{1}{\frac{2}{3}}=1*\frac{3}{2}=\frac{3}{2}\).

(1) a = 1
So the above gives a NO
while if we make a,b,c = 1 then it gives a YES
Not sufficient.

(2) c = 1
a=3, b=2, c=1

\(\frac{\frac{3}{2}}{1}=\frac{3}{2}\);

\(\frac{3}{\frac{2}{1}}=3*\frac{1}{2}=\frac{3}{2}\).

This will hold true for any fraction.

An easier way to see it is to convert 3/2 to 1.5 ... so you get 1.5/1 both ways.

Hi Bunuel,
I solved it very easily. I just have one confusion in data sufficiency i.e. the answer should be consistent or same for whatever value we use to check the main question.
So in first part when a=1 the answer will always be NO so why this info is not sufficient?

Posted from my mobile device

Hi all,
Can we use if: a/b = c/d then ad = bc ?

So in this case we will get:
a*c = \(\frac{a}{b}\) * \(\frac{b}{c}\)
a*c = \(\frac{a}{c}\)

and nowif we know from (1):
a=1 then let's take c=3 as an example:
1*3 ≠ 1/3

(2) c=1
any a we take it always holds true:
4*1 = 4/1

Could anyone explain the logic of this question? Clearly, it asks whether two equations with same variables in same positions are equal to each other. How, then, could they be different?
Cheers

Posted from my mobile device

They are not in the same position. On the left part you are dividing fraction a/b by number c, and on the right you are dividing number a by fraction b/c

The question is asking if c= 1 or -1 because if it is then both sides become equivalent fractions or integer values. If it's not, then you get different fractions, I picked numbers above to show this.

If you look at Bunuel's solution, the actual values of a,b don't matter as they cancel out.

Hope it helps.

As shown in my solution above, a gets reduced (in fact we reduced by a/b). After that the question simplified to: is c^2 = 1? That means that the values of a and b does not matter. How did you get that if a = 1, the answer to the question is always NO?
_________________

For statement 1 after you substitute a=1 the question then infact becomes "Is 1/bc= c/b" and this cannot be answered without knowing the values of b and c. If say the values are b=c=1 then the answer is YES and if b=c=2 then the answer is NO. Hence, statement 1 is not sufficient.

This question wants you to use your basic understanding of the process of simplification when dealing with fractions and fraction division and multiplication in particular.
On GMAT DS questions always try to utilize information and make logical inferences before you move to statements!
In the statements, analyze if there is an easy kill and you can spot a statement that can be easily proved sufficient or insufficient. Eliminate the corresponding answer choices and move to the next statement.

Some GMAT basics for fraction multiplication

Fraction Multiplication
When multiplying fractions, multiply the numerators (value of p in a fraction p/q) together, and multiply the denominators (value of q in a fraction p/q) together. This gives you the answer in lowest terms:
1. Multiply the numerators (top numbers) together, and put that result over the denominator (bottom number) of the original fractions.
2. Reduce (or simplify) the fraction by dividing both the numerator and denominator by their greatest common factor.
Ex: \(\frac{2}{3 }\)* \(\frac{9}{4}\) = \(\frac{18 }{ 12}\) = \(\frac{3}{2}\)

Let’s get to the question now.

So how would a LOGICAL & SMART GMAT aspirant think through this question?

GMAT Track of Thought 1

What do I have from the Question stem?
abc ≠ 0 and hence we know that none of the fractions on LHS and RHS are equal to 0.

I am asked – Is \(\frac{a}{b / c}\) =\(\frac{ a }{b/c}\) ?

GMAT Tip: Avoid rushing through questions like this and note carefully the Numerator and Denominator. You are a SMART thinker and this question and questions like this have to be nailed by you avoiding ALL ERRORS OF FOCUS !

Let me simplify the LHS first which is (a/b) / c

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

Let me simplify the RHS now which is a /(b/c)

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

So I am asked is \(\frac{a}{bc} \) = \(\frac{ac}{b }\)?

Cancelling the common terms ‘a’ & ‘b’ from the Numerator and Denominator respectively, I have

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

Is \(c^2 \)= 1? (On cross multiplying)

From Is \(\frac{a}{b / c}\) = \(\frac{a }{(b/c)}\) ? , I have reduced and simplified the question stem to Is \(c^2\) = 1 ?

It’s a YES/NO type of DS question and I must have a definite YES or a Definite NO as an answer to this.
I need to know the value of c or \(c^2 \)or any relation between the variables to lead me to it.

GMAT Track of Thought 2

Which of the statements is an easy kill?
Scanning the statements quickly, I can see c =1 in statement 2.
That alone is sufficient since it implies \(c^2\) =1 (upon squaring).
This answers my simplified question stem with a definite YES and I can eliminate A,C, E right away!

GMAT Track of Thought 3

Now let me look at statement 1.
It says a = 1. I am NOT provided with any value of c and nor can I connect a & c with any relationship from the statement and/or the Q. Stem.
Eliminate Statement 1 and hence option A

Mark answer option B as the correct answer choice for this question.

You can break any complex looking statement with a stepwise approach and simplify it. On doing so you would sometimes be happily surprised on looking at the statements that may provide you with a direct answer in GMAT DS questions.

Hope this post of mine has lent clarity and added value to your learning.

Devmitra Sen
GMAT Mentor
_________________

a/b/(c) = a/(b*c)

a/(b/c) = (a*c)/b

a/(b*c) = (a*c)/b => 1/c = c => c^2 = 1

1. Not sufficient do not provide any info on c
2. c = 1 => 1^2 = 1 sufficient => B