At a certain hospital, 75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts. At the same time, 70% of the interns who receive 6 or more hours of sleep report no feelings of tiredness. If 80% of the interns receive fewer than 6 hours of sleep, what percent of the interns report no feelings of tiredness during their shifts?
6
14
19
20
81

I guess the answer is 19.

Consider 100 people in total.

"If 80% of the interns receive fewer than 6 hours of sleep, what percent of the interns report no feelings of tiredness during their shifts"

This means 80 people sleep less than 6 hrs and 20 more than 6.

Now subdivide these groups further into into interns who are tired and interns who are not tired.
Thus we have.

1st group---
People who have less than 6 hrs sleep(80) --this group consists of
a)people who are tired
b)people who are not tired

Also a+b =80

2nd group-----

People who have more than 6 hrs sleep(20) --this group consists of
c)people who are tired
d)people who are not tired

Also c+d =20


75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts. At the same time

Means a=75
Implies b=5

70% of the interns who receive 6 or more hours of sleep report no feelings of tiredness.

implies d=70/100 * 20=14
Therefore c=(20-14)=6

what percent of the interns report no feelings of tiredness during their shifts?

we have to find b+d=14+5=19 :)

If you have to take a guess analyse the choices.
Why?
Look at the choices.81 and 19 should strike you immediately because 81+ 19 is 100.So its like reading the mind of the person who set the question.81 out here is for the careless guy who will find the opposite of what has been asked.

Also 75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts.
This should make you strike out 81.So even if i had been clueless i had a good chance of guessing the right ans:)

Amit,

I cannot understand why you're subdiving the ist group into
a)people who are tired
b)people who are not tired

Does not the problem say that people who sleep for less than 6 hours report feeling tired during their shifts? My interpretation of "75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts" is that if there are 100 people, 75 of them receive fewer than 6 hours of sleep and report feeling tired during their shifts, and in the remaining 25 who sleep over 6 hours you've people who who are tired and people who are not tired. Based on my interpretation, the answer is 6.

Am I missing anything?

If 80% of the interns receive fewer than 6 hours of sleep......I guess this makes it clear that the 1st grp has to be sub-divided.
Capthan can we have the OA?

Capthan, would you kindly post the source of this question: MG Strategy Guide title, chapter, and problem number?

Rey, It is from Test 3.

Hi Capthan,
Thanks for citing our CAT exams as the source. I will move this thread to the MGMAT CAT Math folder, as that is the correct place for it.

Vinod asks:

You are right about the 75 who are tired with little sleep. What you are missing is that the remaining 25 people are not all people who slept more than 6 hours. Amit's point is a good one: 80 people got less than 6 hours sleep, which leaves 5 unaccounted for once we subtract the 75 of them who are tired. Those 5 enviable people are not tired despite their lack of sleep.

There is such a strong possibility of getting confused by the wording of these problems that we recommend you always use a Double Set Matrix. You would set up two columns: less than 6 hours sleep/more than 6 hours sleep. Set up two rows: tired/not tired. What results is 4 boxes: the categories of people Amit listed above. The rows should each subtotal across, the columns subtotal down, and the subtotal row and column both should grand total to 100 (i.e. 100%).

_________________
Emily Sledge
Instructor
ManhattanGMAT

I got 34 using double set matrix, can someone tell where I'm wrong?

<6 hrs >6hrs Total

Tired 60 6 66


Not Tired 20 14 34


Total 80 20 100

Hope I can Help..

<>Tired <> Not Tired <> Total


Less than 6 <> 75 <> 5 <> 80

More than 6 <> 6 <> 14 <> 20

Total <> 81 <> 19 <> 100


Assume total =100;
80% of the interns receive fewer than 6 hours of sleep=>Less than 6 && Total=80
Therefore, More than 6 && Total combination=100-80=20


70% of the interns who receive 6 or more hours of sleep report no feelings of tiredness. This implies 70% of 20=Morethan 6hours && not tired combination=14
Therefore, More than 6 && Tired combination =20-14=6

Finally we have , 75% of the interns receive fewer than 6 hours of sleep and report feeling tired during their shifts which implies Less than 6hours && tired combination=75. Therefore less than 6 && not tired combination= 80-75=5


Thus percent of the interns report no feelings of tiredness during their shifts =19%

beyice, it looks as though you simply misread the very first number, 75% of the interns as 60% of the interns.

I can't see any reason to misinterpret it that way, no obvious trap.

Your reasoning after that error is fine.

Assume Total is 100
Tired Not Tired Total
Sleep < 6 3/4 or 75% of 80 = 60 1/4 or 25% of 80 = 20 80
Sleep >= 6 3/10 or 30% of 20 = 6 7/10 or 70% of 20 = 14 20
Total 66 34 100


Guess I am perplexed that 75% of 100 is being used rather than 75% of 80 and 25% 80. Yeah, using the matrix is helpful; however, (75% + 25%) of 80, should also add up to 1 (3/4 + 1/4) horizontally. Simple plugging in 5 after 75 seems a bit disingenious without the 5 having the same lineage back to the original 80%, in proportion.

-Ad

the question says 75% of the interns, not 75% of some subset. so using your numbers, that is 75% of 100, not 75% of 80..

_________________
Tim Sanders
Manhattan GMAT Instructor