M01-36

Question

Is the product of  a  and  b  equal to 1?

(1)  a^2b = a 
 
(2)  ab^2 = b

Bunuel · Accepted Answer

Official Solution:

Is the product of  a  and  b  equal to 1? 
 
Question: is  ab=1 ?
 
(1)  a^2b=a 
 
 a^2b-a=0 ;
 
 a(ab-1)=0 . This implies either  a=0  (and  b  can be any value, including zero) resulting in  ab=0
eq 1   OR   ab=1 . Since there are two different outcomes, this statement is not sufficient.
 
It is important to note that we cannot reduce  a^2b = a  by  a , because  a  could be equal to 0, and division by 0 is not allowed. By reducing the equation this way, we lose a solution, namely  a = 0 .
 
Always be cautious when simplifying an equation involving a variable (or an expression containing a variable). Do not divide by the variable (or the expression with the variable) unless you are certain that it does not equal zero, as division by zero is not allowed.
 
(2)  ab^2=b 
 
 ab^2-b=0 ;
 
 b(ab-1)=0 . This implies either  b=0  (and  a  can be any value, including zero) resulting in  ab=0 
eq 1   OR   ab=1 . Since there are two different outcomes, this statement is not sufficient.
 
(1)+(2) The combined statements lead to two possibilities: either  a=b=0 , which results in  ab=0 
eq 1  and the answer to the question is NO, OR  ab=1  and the answer to the question is YES. As there are still two different outcomes, the combined statements are not sufficient.

Answer: E

mvictor · Answer

why can't we conclude that:
a^2*b=a=>b=1/a
in this case a*1/a = 1
and statement 1 and 2 are both sufficient
hm..

Bunuel · Answer

Please re-read the solution above. Does not a = 0 satisfy the equation? You cannot write b=1/a from a^2*b=a because a could be 0, and we cannot reduce or divide by 0.

svijayaug12 · Answer

Hi Bunuel,

I have a question based on the solution that you have mentioned below:

Let's say a question in DS is as follows?

If x is a positive integer, is x=4?

Stmt -1) X is a root of the quadratic equation - x^2 - 12x+12=0

Stmt -2) X is a root of the quadratic equation - x^2 - 9x+20=0

Now solving each statement alone,

Stmt 1 -> Allows x value to be either x= 3,4
Stmt 2 -> Allows x value to be either x= 4,5

Now for such a Data sufficiency question, don't we find that x=4 is a common solution in both statements together and hence mark the answer as Answer choice (C) - Both statements together are sufficient but neither alone.

Now based on this understanding, for the current question,

Both statement (1) and statement (2) when taken together show that ab = 1.
Can we not select option choice (c), based on this?

Please correct me if my understanding is wrong some where?

Vijay

Bunuel · Answer

When we consider two statements together we should take the values which satisfy both statements. For this question  ab=1  satisfies both statement, but  a=b=0  also satisfies both statements. So what you call "common solution" for this question is:  ab=1  OR  ab=0
eq{1} .

mbsingh · Answer

I am confused,

Why cant we say 
a*b*a = a 
a*b = a/a
a*b = 1

What am i missing  Bunuel

Bunuel · Answer

You cannot reduce a*b*a = a by a, because a can be 0, and we cannot divide by 0. By doing so you are loosing a root, namely a = 0.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.

patto · Answer

I think this is a  high-quality  question and I agree with explanation.

AndrewN · Answer

An alternative approach : Just use easy numbers and test the statements one by one. -1, 0, and 1 will do just fine.

We can see that whether  a  is positive or negative, squaring it on the left-hand side will give us a positive quantity. Let a = -1. For this equation to be true,  b  would also have to equal -1:

(-1)*(-1)*(-1)=(-1)

The answer to the question would then be  yes . But what about using a = 0 instead? It is pretty clear that  ab  could also equal 0, and the answer to the question would be  no . Conflicting information leads to the conclusion that Statement (1) ALONE is NOT sufficient. Strike choices (A) and (D).

We can play the same game here, using the same easy numbers. Let a = -1. Once again,  b  would also have to equal -1:

(-1)*(-1)*(-1)=(-1)

The answer to the question would again be  yes . But there is nothing prohibiting us from using 0 for either variable, at which point the product on the left-hand side would also be 0, and we would get 0 = 0. The answer to the question would be  no . Thus, with conflicting information in hand, Statement (2) ALONE is NOT sufficient.

Since combining the statements would do nothing in the way of reducing our potential number pool, the answer must be (E).

Good luck with your studies.

- Andrew

anilkkhyalia · Answer

I think this is a  high-quality  question and I agree with explanation.

mSKR · Answer

Hi  Bunuel

can you please post some question that has:
I. x= 4 or 5
II x= 4 or 6

I+ II = should be 4 ( can you please post such a question where we need to take intersection of both options)

I+II = can be 4 or 5 or 6( this one is current question)

It would help me to understand the difference in scenarios of both types .

Please suggest @Bunel  IanStewart

Thanks!

IanStewart · Answer

I don't think I understand what you mean when you talk about "both types" of scenarios. If you have a DS question that asks for the value of x, and from Statement 1 you know x = 4 or x = 5, and from Statement 2 you know x = 4 or x = 6, then when you combine the two Statements, x absolutely must equal 4. There is no situation in DS where you would conclude, combining these two statements, that x = 4 or x = 5 or x = 6. When you combine the information from two Statements, you are always looking only at where the solutions overlap.

The kind of question you're referring to might look like this (not a high-level question, but it produces the exact numerical solutions you list above) -

What is the value of x?
1. x^2 - 9x + 20 = 0
2. x^2 - 10x + 24 = 0

The answer is C, because x = 4 is the only value that will work in both equations. You should never conclude, when you combine these two Statements, that x can be 4, 5 or 6.

Hope that answers your question!

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

varad98 · Answer

Hi,

I subtracted (1) from (2)

a2b-ab2=a-b

ab(a-b)=a-b

ab=1

As this was unique solution, I selected c as option.

Bunuel · Answer

You cannot reduce ab(a - b) = a - b by a - b, because it can be 0 and division by 0 is not allowed. By doing so, you exclude a possible case, namely the one where a - b = 0.

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