Problem Solving

Hello BTG

Would appreciate some help on the following question.
Thanks a lot.

Let p = the product of all the odd integers between 500 and 598, and let q = the product of all the odd integers between 500 and 602. In terms of q, what is the value of 1/p+1/q?

a) 1/600q

b) 1/359,99q

c) 1,200/q

d) 360,000/q

e) 359,999q

Let p = the product of all the odd integers between 500 and 598, and let q = the product of all the odd integers between 500 and 602. In terms of q, what is the value of 1/p+1/q?

A.1/600q
B.1/359999q
C.1200/q
D.360000/q
E.359999*q

p = (501)(503)...(595)(597).
q = (501)(503)...(595)(597)(599)(601).
Notice the OVERLAP between p and q.
Implication:
q = (599)(601)p
p = q/(599*601).

Since the answer choices are IN TERMS OF A VARIABLE, we can PLUG IN any values for p and q such that p = q/(599*601).

Let q=1.
Then p = 1/(599*601), implying that 1/p = 599*601 = (600-1)(600+1) = 360000 - 1.
Thus:
1/p + 1/q = (360000 - 1) + 1/1 = 360000. This is our target.

Now plug q=1 into the answers to see which yields our target of 360000.
A quick scan of the answers reveals that only D works:
360000/q = 360000/1 = 360000.

The correct answer is D.

lucas211 wrote:Let p = the product of all the odd integers between 500 and 598, and let q = the product of all the odd integers between 500 and 602. In terms of q, what is the value of 1/p+1/q?

a) 1/600q

b) 1/359,99q

c) 1,200/q

d) 360,000/q

e) 359,999q

p = 501 * 503 ... * 597

q = 501 * 503 ... * 597 * 599 * 601

The key thing to notice in doing this question is that while seemingly you are dealing with many factors and huge numbers, actually, because you are working in terms of q, you can simplify the situation to the following.

p = q/(599 * 601)

Therefore you can do the following.

1/p + 1/q = 1/(q/(599 * 601)) + 1/q =

(599 * 601)/q + 1/q =

{Might be a good time to do some trick math.}

((600 * 600 - 600) + 599)/q + 1/q =

(359,999 + 1)/q = 360,000/q

The correct answer is D.