Positive integers P and Q

[dslewis]

Are positive integers P and Q both greater than N?

(1) P-Q is greater than N
(2) Q>P

Answer is C. I got the question right, but these questions generally confuse me and just like in the practice test I usually spend way to much time thinking about the possibilites of P and Q. Is there a rule for the concept of adding a subtracting 2 integers and comparing it to a third?

I started this problem by recognizing that from statement one you can only figure out that P is greater than N. I moved on to statement 2 and noticed it said nothing about N so it was insufficient. If Q >P and P-Q>N than both Pand Q are greater than N.

I guess my question is - is there a concept I can use inorder to figure this question out or do I have to use trail and error. Also was my assumption above about statement 1 correct when I said you could only figure out that P is greater than N from the statement P-Q>N. What about if the statement was changed and it was P+Q>N.

[dslewis]

Suppose these weren't positive integers would that change the outcome? Is it safe to say that if the question didnt have the limitation of P and Q as positive integers you wouldn't have enough info to answer the question assuming everything else stays the same?

[RonPurewal]

If Q >P and P-Q>N than both Pand Q are greater than N.

this is true, but it's less complicated to use the conclusion you already reached about statement (1) before, which is that p > n.

if you have statement (1), which tells you that p > n, and statement (2), which tells you that q > p, then you have q > p > n. therefore, both p and q are greater than n.

is there a concept I can use inorder to figure this question out or do I have to use trail and error.

well, it appears that you already figured out a certain 'method': try to rephrase the statements as much as possible. you did this with statement (1): you took the rather ugly statement that p - q > n, and reinterpreted it as p > n. (presumably, you rearranged the inequailty to p > n + q; but if p is greater than n plus some positive number, then it follows that p > n.)

in most problems like this, at least one of the following will be true:
(1) you'll be able to rephrase one or both of the statements (as you did with statement 1)
(2) the question can be rephrased in terms of some sort of number properties, which can usually be deduced from the material content of the problem. for instance, if the problem includes absolute values, then the number properties have to do with positives/negatives/zero; if the problem compares different powers of a variable, then the number properties have to do with the variable being greater/less than 1.

that's it as far as general strategies; there are too many different kinds of these problems to say anything more specific, other than on a problem-by-problem basis such as is done in this thread.

Also was my assumption above about statement 1 correct when I said you could only figure out that P is greater than N from the statement P-Q>N.

this question could mean one of two things, so i'll address both.
(1, unlikely) you could be asking whether this is the only one of the two statements that lets us figure out whether p > n. if so, then yes.
(2, likely) you could be making sure that you can only deduce this about p, not q. if so, that's also true: q could be any positive integer. for instance, 7 - 1 > 6 (with 1 not greater than 6), but 76 - 70 > 6 (with 70 greater than 6).

What about if the statement was changed and it was P+Q>N.

then you couldn't figure out anything. for instance, 4 + 5 > 6 (although neither 4 nor 5 is greater than 6), but also 400 + 500 > 6 (and both 400 and 500 are greater than 6).

[RonPurewal]

Suppose these weren't positive integers would that change the outcome? Is it safe to say that if the question didnt have the limitation of P and Q as positive integers you wouldn't have enough info to answer the question assuming everything else stays the same?

yeah, that would totally change the game, because n + q no longer has to be greater than n. therefore, the conclusion that p > n (from statement 1) is no longer valid.

you can prove the answer is e by examining the following p, q, n. (notice that n MUST be negative if both statements are true, because p - q is negative if q is greater than p.)
* p = 5, q = 6, n = -10: both statements are satisfied; answer = yes
* p = -6, q = -5, n = -3: both statements are satisfied; answer = no

--

interestingly, another way to prove that the answer to the original question is c is to realize that, as stated above, n must be negative if both statements are true. since p and q are positive integers, they must be greater than n, which is negative.

[mobenny]

Is it possible to add stament one and two together, which will result in n<0 bc
-q>n-p
+
q>p
_________
0>n

therefore, C.

[RonPurewal]

Is it possible to add stament one and two together, which will result in n<0 bc
-q>n-p
+
q>p
_________
0>n

therefore, C.

yep, well done.

takeaway:
you can ADD TWO INEQUALITIES TOGETHER if the inequalities are BOTH "<" OR BOTH ">".

(you can't add them if one is "<" and the other is ">". if that's the situation, then you should multiply one of the inequalities by -1, or just turn it around, so that both of them are either "<" or ">".)

[mario.jr.russo]

Please consider this:

p-q>n can be spit into both
(a) p>0,
-q>n,

and
(b) p>n
-q>0.

Since we know that both p and q are +ve, (b) cannot be true and only (a) is still valid.

So, if n<-q and q is +ve, n must always be -ve, so the statement 1 alone is sufficient and the right answer to the question is A.

What about?

[jnelson0612]

Please consider this:

p-q>n can be spit into both
(a) p>0,
-q>n,

and
(b) p>n
-q>0.

Since we know that both p and q are +ve, (b) cannot be true and only (a) is still valid.

So, if n<-q and q is +ve, n must always be -ve, so the statement 1 alone is sufficient and the right answer to the question is A.

What about?

Look at this one:

p-q>n can be spit into both
(a) p>0,
-q>n,

Not necessarily. For example, plug real numbers to test this out:
p=-2
q=-10

Thus, using these numbers, p-q is -2-(-10) or 8. Thus, 8 is greater than n.

Notice here that all of these variables fit the original equation p-q>n, but p IS NOT greater than 0, as you are assuming.

[mario.jr.russo]

Thank you very much Jamie.
However I cannot understand how can p and q be -ve, since the opposite is stated as a premise in the quiz.

[tim]

that's a correct observation. but you still cannot split p-q>n into p>0, -q>n. what if p=20, q=4, and n=10?