prernamalhotra wrote:A researcher has determined that she requires a minimum of n responses to a survey for the results to be valid. If p% of the surveyed individuals fail to respond to the survey, how many individuals, in terms of n and p, must the researcher survey to produce twice the minimum required number of responses?

1) 200n / 100-p
2) 2n/100-p
3) 200n/p
4) 2n(100+p)/ 100
5) 2n+2np/100

Strategy:
Plug in values, working from the END of the problem to the BEGINNING.

If n=10 responses are required, then twice the minimum = 20.
If p=50% of the individuals fail to respond, then 40 individuals must be surveyed to yield 20 responses.
The question stem asks for the number of individuals who must be surveyed: 40.
This is our target.

Now plug n=10 and p=50 into the answers to see which yields our target of 40.
Only A works:
200n/(100-p) = (200*10)/(100-50) = 2000/50 = 40.

The correct answer is A.

Algebraically:

Let the number interviewed = i.
Since p% don't respond, the number who don't respond = (p/100)i.
Thus, the number who DO respond = i - (p/100)i.
Since this result must be equal to 2n -- twice the minimum number of required responses -- we get:

i - (p/100)i = 2n

i(1 - p/100) = 2n

i*(100-p)/100 = 2n

i = 200n/(100-p).
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If x individuals must be surveyed.

p% of the surveyed individuals fail to respond

Which means, the number of individuals who did NOT fail to respond = x-[x*(p/100)]
The above must be equal to 2n, so:

x-[x*(p/100)]=2n

x(1-(p/100))=2n

x=(200n) / {100-p}.

Answer: A.

Of course one can also use plug-in method to solve this problem.

S = number of successes
F = number of fails
T = number of trials
P% = P/100
(100 - P)/100 x T = n
Therefore T = 100n/(100-P)
If we double n, we get: T = 200n/(100-P)