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If p and q are positive integers, how many integers are lar

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If p and q are positive integers, how many integers are lar [#permalink]

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New post 02 Dec 2013, 20:35
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If p and q are positive integers, how many integers are larger than pq and smaller than p(q + 2)?

A. 3
B. P + 2
C. p – 2
D. 2p – 1
E. 2p + 1

Please solve using algebraic method because substitution will sometimes get the right answer but sometimes not.
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Re: If p and q are positive integers, how many integers are lar [#permalink]

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New post Updated on: 02 Dec 2013, 20:46
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If p and q are positive integers, how many integers are larger than pq and smaller than p(q + 2)?

First, we notice that:
p(q+2) = pq + 2p

So we are asking how many integers are larger than pq but smaller than pq+2p. Notice that the range of this is 2p, so our answer should be around 2p. When you count the number of integers in between two other integers, you take high number - low number - 1.

Answer: pq + 2p - pq - 1 = 2p -1

To see why this counting formula works, try it out.
How many numbers between 1 and 3? 3-1-1 = 1
How many numbers between 1 and 4? 4-1-1 = 2
-2 and 2? 2-(-2)-1=3

Originally posted by jamesphw on 02 Dec 2013, 20:45.
Last edited by jamesphw on 02 Dec 2013, 20:46, edited 1 time in total.
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Re: If p and q are positive integers, how many integers are lar [#permalink]

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New post 02 Dec 2013, 20:46
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smartyman wrote:
Please help....

If p and q are positive integers, how many integers are larger than pq and smaller than p(q + 2)?
A) 3
B) P + 2
C) p – 2
D) 2p – 1
E) 2p + 1

Please solve using algebraic method because substitution will sometimes get the right answer but sometimes not.


The number of integers larger than pq and smaller than p(q+2) will be the number of integers between the two values. So, we start with the difference between the two:

p(q+2) - pq
pq + 2p -pq
2p

Then, subtract 1 for the upper bound (p(q+2).

So, 2p-1 is the correct answer.
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Re: If p and q are positive integers, how many integers are lar [#permalink]

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New post 03 Dec 2013, 02:21
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smartyman wrote:
If p and q are positive integers, how many integers are larger than pq and smaller than p(q + 2)?

A. 3
B. P + 2
C. p – 2
D. 2p – 1
E. 2p + 1

Please solve using algebraic method because substitution will sometimes get the right answer but sometimes not.


The number of integers between x and y, where x>y is (x-y)-1. For example, the number of integers between 1 and 5 is (5-1)-1=3: 2, 3, and 4.

Thus, the number of integers between pq and p(q+2)=pq+2p is (pq+2p-pq)-1=2p-1.

Answer: D.
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If p and q are positive integers, how many integers are lar [#permalink]

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New post Updated on: 06 Jun 2018, 18:43
Hi smartyman,

Did you try TESTing VALUES ("plugging in", "substitution" or whatever else you might choose to call it)? What happened when you did?

When you mentioned that it "will sometimes get the right answer but sometimes not", I think that you mean to say that "in most cases it will get you the correct answer on the first try, but in some cases there might be more than one answer that comes back as a match, so you would have to do a bit more work and substitute a second time." TESTing VALUES is a remarkably fast and useful approach on many Quant questions on Test Day, so you shouldn't avoid that tactic outright.

Here, if we use...
P=3
Q=4
We have to figure out the number of integers that are greater than 12 and less than 18. THAT is not a very difficult task: 13, 14, 15, 16, 17 --> a total of 5 terms. Plugging P into the answer choices gives us two matches: B and D. One of those IS the correct answer, so now we just have to do a little more work to prove which one.

IF....
P=2
Q=3
Then we're looking for the number of integers that are greater than 6 and less than 10. That's 7, 8, 9 --> a total of 3 terms. Between Answers B and D, only Answer D matches.

Final Answer:

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Originally posted by EMPOWERgmatRichC on 26 Dec 2014, 13:55.
Last edited by EMPOWERgmatRichC on 06 Jun 2018, 18:43, edited 1 time in total.
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Re: If p and q are positive integers, how many integers are lar [#permalink]

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New post 05 Jan 2015, 05:33
This is how I did it:

We need numbers between the 2 given values.

So, I started like this: pq<x<p(q+2)

Then, I plugged in numbers:
p=1 | q=2
pq= 1*2=2 and p(q+2)=1(2+2)=4, so there is one number in between: 3.

p=1 | q=1
pq=1*1=1 and p(q+2)= 1(1+2)=3, so there is one number in between: 2.

Finally, I looked at the answer options:
A. I skipped it because according to my solution there is only 1 number in between and not 3
B. P + 2: using p=1, as the first plug in number we get 1+2=3, we don't like that, so move on.
C. p – 2: same plug in, we get 1-2=-1, not really possible to have minus number or numbers in between, so move on.
D. 2p – 1: same plug in, we get 2*1-2=2-1=1, we like that, but checking E just in case, since there is still time.
E. 2p + 1: same plug in, we get, 2*1+1-2+1=3, we don't like it.

Only D gave as a solution only 1 number in between. So, D must be correct.

Anything wrong wit this approac? Time wise it didn't take that long (less than 2 mins definitely).
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If p and q are positive integers, how many integers are lar [#permalink]

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New post 06 Jun 2018, 06:47
1
EMPOWERgmatRichC wrote:
Hi smartyman,

Did you try TESTing VALUES ("plugging in", "substitution" or whatever else you might choose to call it)? What happened when you did?

When you mentioned that it "will sometimes get the right answer but sometimes not", I think that you mean to say that "in most cases it will get you the correct answer on the first try, but in some cases there might be more than one answer that comes back as a match, so you would have to do a bit more work and substitute a second time." TESTing VALUES is a remarkably fast and useful approach on many Quant questions on Test Day, so you shouldn't avoid that tactic outright.

Here, if we use...
P=3
Q=4
We have to figure out the number of integers that are greater than 12 and less than 18. THAT is not a very difficult task: 13, 14, 15, 16, 17 --> a total of 5 terms. Plugging P into the answer choices gives us just one match:



GMAT assassins aren't born, they're made,
Rich

Hey Rich,

I usually plug in values too but in this case it doesn't work.
You said that plugging P into the answer choices gives us one match but there are two matches.

(B) P + 2 = 3 + 2 = 5

So B and D could work if we plug in numbers...
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Re: If p and q are positive integers, how many integers are lar [#permalink]

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New post 06 Jun 2018, 18:44
Hi Masterscorp,

Good catch! A bit more work was needed to prove the correct answer (and I've amended my explanation).

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Re: If p and q are positive integers, how many integers are lar   [#permalink] 06 Jun 2018, 18:44
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