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Math Expert V
Joined: 02 Sep 2009
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If pqrst = 4, then is p = 1/q ?  [#permalink]

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12 00:00

Difficulty:   85% (hard)

Question Stats: 44% (02:04) correct 56% (01:59) wrong based on 160 sessions

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If pqrst = 4, then is p = 1/q ?

(1) r = s = t

(2) Three of p, q, r, s, t are integers

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Re: If pqrst = 4, then is p = 1/q ?  [#permalink]

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1) Case 1: r = s = t = 1 then pq = 4
Case 2: pq = 1 and r = s = t --> rst=4 --> r=2^2/3
Not sufficient

2) Not sufficient

1+2) since three of p, q, r, s, t are integers Case 2 cannot take place. Answer C.
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Math Expert V
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Re: If pqrst = 4, then is p = 1/q ?  [#permalink]

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Bunuel wrote:
If pqrst = 4, then is p = 1/q ?

(1) r = s = t

(2) Three of p, q, r, s, t are integers

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Re: If pqrst = 4, then is p = 1/q ?  [#permalink]

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gmat6nplus1 wrote:
1) Case 1: r = s = t = 1 then pq = 4
Case 2: pq = 1 and r = s = t --> rst=4 --> r=2^2/3
Not sufficient

2) Not sufficient

1+2) since three of p, q, r, s, t are integers Case 2 cannot take place. Answer C.

I guess you are referring to r as third root of 4, but it looks to be 4\3 so got a bit confused.

Can you please explain this example further, not very clear.
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Re: If pqrst = 4, then is p = 1/q ?  [#permalink]

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1
If we modify the original condition and the question, it is p=1/q?, 4/qrst=1/q?. If we divide each side with q, we get 4/rst=1?, rst=4?.
There are 5 variables (p,q,r,s,t) and 1 equation (pqrst=4). In order to match the number of variables to the number of equations, we need 4 more equations. Since the condition 1) and the condition 2) each has 1 equation, we lack 2 equations. Hence, there is high chance that E is the correct answer. Using both the condition 1) and the condition 2), it states r=s=t are 3 integers, we get r=s=t=1, -1, 2, -2,...... However, in any case we cannot obtain rst=4. Hence, the answer is no and the conditions are sufficient. Therefore, the answer is C.

- For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.
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Re: If pqrst = 4, then is p = 1/q ?  [#permalink]

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Bunuel wrote:
If pqrst = 4, then is p = 1/q ?

(1) r = s = t

(2) Three of p, q, r, s, t are integers

Bunuel chetan2u

I understand that individual statements are insufficient to answer the question. However, even using both statements it is quite possible that p could be or could not be equal to 1/q.

We know that r = s = t and all three of them are integers. So, r, s, and t could be positive/negative integers.

Let's say r = s = t = 1 => rst = 1 => pq = 4. To satisfy this equation, p and q can take any of the following values:
p = 4, q = 1/4 (or vice versa) => pq = 4
p = 8, q = 1/2 (or vice versa) => pq = 4
p = 16, q = 1/4 (or vice versa) => pq = 4
etc.

Let's say r = s = t = 2 => rst = 8 => pq = 1/2. Again, to satisfy this equation, it is not necessary that p must be equal to 1/q.

Let's say r = s = t = -2 => rst = -8 => pq = -1/2. Again, to satisfy this equation, it is not necessary that p must be equal to 1/q.

Let's say r = s = t = -1 => rst = -1 => pq = -4. Again, to satisfy this equation, it is not necessary that p must be equal to 1/q.

We are not given any information about p or q. So, the answer should be (E) not (C), right?

Please let me know what am I missing here. Thanks.
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Re: If pqrst = 4, then is p = 1/q ?  [#permalink]

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jayshah0621 wrote:
Bunuel wrote:
If pqrst = 4, then is p = 1/q ?

(1) r = s = t

(2) Three of p, q, r, s, t are integers

Bunuel chetan2u

I understand that individual statements are insufficient to answer the question. However, even using both statements it is quite possible that p could be or could not be equal to 1/q.

We know that r = s = t and all three of them are integers. So, r, s, and t could be positive/negative integers.

Let's say r = s = t = 1 => rst = 1 => pq = 4. To satisfy this equation, p and q can take any of the following values:
p = 4, q = 1/4 (or vice versa) => pq = 4
p = 8, q = 1/2 (or vice versa) => pq = 4
p = 16, q = 1/4 (or vice versa) => pq = 4
etc.

Let's say r = s = t = 2 => rst = 8 => pq = 1/2. Again, to satisfy this equation, it is not necessary that p must be equal to 1/q.

Let's say r = s = t = -2 => rst = -8 => pq = -1/2. Again, to satisfy this equation, it is not necessary that p must be equal to 1/q.

Let's say r = s = t = -1 => rst = -1 => pq = -4. Again, to satisfy this equation, it is not necessary that p must be equal to 1/q.

We are not given any information about p or q. So, the answer should be (E) not (C), right?

Please let me know what am I missing here. Thanks.

Notice that if p = 1/q, then pq = q*1/q = 1. In neither of your examples pq is 1. So, even you have a definite NO answer to the question.
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Re: If pqrst = 4, then is p = 1/q ?  [#permalink]

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Bunuel wrote:
jayshah0621 wrote:
Bunuel wrote:
If pqrst = 4, then is p = 1/q ?

(1) r = s = t

(2) Three of p, q, r, s, t are integers

Bunuel chetan2u

I understand that individual statements are insufficient to answer the question. However, even using both statements it is quite possible that p could be or could not be equal to 1/q.

We know that r = s = t and all three of them are integers. So, r, s, and t could be positive/negative integers.

Let's say r = s = t = 1 => rst = 1 => pq = 4. To satisfy this equation, p and q can take any of the following values:
p = 4, q = 1/4 (or vice versa) => pq = 4
p = 8, q = 1/2 (or vice versa) => pq = 4
p = 16, q = 1/4 (or vice versa) => pq = 4
etc.

Let's say r = s = t = 2 => rst = 8 => pq = 1/2. Again, to satisfy this equation, it is not necessary that p must be equal to 1/q.

Let's say r = s = t = -2 => rst = -8 => pq = -1/2. Again, to satisfy this equation, it is not necessary that p must be equal to 1/q.

Let's say r = s = t = -1 => rst = -1 => pq = -4. Again, to satisfy this equation, it is not necessary that p must be equal to 1/q.

We are not given any information about p or q. So, the answer should be (E) not (C), right?

Please let me know what am I missing here. Thanks.

Notice that if p = 1/q, then pq = q*1/q = 1. In neither of your examples pq is 1. So, even you have a definite NO answer to the question.

Uh oh. Silly me  Thanks for the explanation! Kudos!
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Re: If pqrst = 4, then is p = 1/q ?  [#permalink]

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gauz wrote:
gmat6nplus1 wrote:
1) Case 1: r = s = t = 1 then pq = 4
Case 2: pq = 1 and r = s = t --> rst=4 --> r=2^2/3
Not sufficient

2) Not sufficient

1+2) since three of p, q, r, s, t are integers Case 2 cannot take place. Answer C.

I guess you are referring to r as third root of 4, but it looks to be 4\3 so got a bit confused.

Can you please explain this example further, not very clear.

^Figured it out, i might be high when posting this doubt _________________
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Re: If pqrst = 4, then is p = 1/q ?  [#permalink]

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Is p = 1/q --> Is pq = 1 --> Is rst = 4 ?

(1) r = s = t . If r = s = t = 1 -> No, If r = s = t = $$\sqrt{4}$$ -> Yes. --> Not sufficient.
(2) Three of p, q, r, s, t are integers. If r, s, t are integers -> No. If r, s, t are not integers at the same time, then Yes (i.e. r=1, s=3, t=4/3) or No (i.e. r=1, s=3, r=2/3). --> Not sufficient.

(1) + (2): {p, q, r, s, t} = {p, q, r, r, r} --> p, q, r are integers --> All 5 variables p, q, r, s, t are integer. --> $$rst\neq{4}$$ --> No. --> Sufficient.

Bunuel wrote:
If pqrst = 4, then is p = 1/q ?

(1) r = s = t

(2) Three of p, q, r, s, t are integers

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